diff --git a/print.html b/print.html index c3b020d40b..687852384f 100644 --- a/print.html +++ b/print.html @@ -5187,7 +5187,7 @@

Poseidon-Fp

Poseidon-Fq

You can find the MDS matrix and round constants we use in /poseidon/src/pasta/fp_kimchi.rs.

Test vectors

-

We have test vectors contained in /poseidon/src/tests/test_vectors/kimchi.json.

+

We have test vectors contained in /poseidon/tests/test_vectors/kimchi.json.

Pointers to the OCaml code

Fields§

§cols: Box<[T; N_WIT]>

A witness row is represented by an array of N witness columns When T is a vector, then the witness describes the rows of the circuit.

-

Implementations§

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impl<const N_WIT: usize, T> Witness<N_WIT, T>

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pub fn len(&self) -> usize

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pub fn is_empty(&self) -> bool

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impl<const N_WIT: usize, T: Zero + Clone> Witness<N_WIT, Vec<T>>

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pub fn zero_vec(domain_size: usize) -> Self

source

pub fn to_pub_columns<const NPUB: usize>(&self) -> Witness<NPUB, Vec<T>>

Trait Implementations§

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impl<const N_WIT: usize, T: Clone> Clone for Witness<N_WIT, T>

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fn clone(&self) -> Witness<N_WIT, T>

Returns a copy of the value. Read more
1.0.0 · source§

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source. Read more
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impl<const N_WIT: usize, T: Debug> Debug for Witness<N_WIT, T>

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fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter. Read more
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impl<const N_WIT: usize, T: Zero + Clone> Default for Witness<N_WIT, T>

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fn default() -> Self

Returns the “default value” for a type. Read more
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impl<const N: usize, F: FftField> Foldable<F> for Witness<N, Evaluations<F, Radix2EvaluationDomain<F>>>

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fn combine(a: Self, b: Self, challenge: F) -> Self

Combine two objects ‘a’ and ‘b’ into a new object using the challenge.
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impl<const N_WIT: usize, G: Send + Debug> FromParallelIterator<G> for Witness<N_WIT, G>

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fn from_par_iter<I>(par_iter: I) -> Selfwhere +

Implementations§

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impl<const N_WIT: usize, T> Witness<N_WIT, T>

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pub fn len(&self) -> usize

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pub fn is_empty(&self) -> bool

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impl<const N_WIT: usize, T: Zero + Clone> Witness<N_WIT, Vec<T>>

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pub fn zero_vec(domain_size: usize) -> Self

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pub fn to_pub_columns<const NPUB: usize>(&self) -> Witness<NPUB, Vec<T>>

Trait Implementations§

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impl<const N_WIT: usize, T: Clone> Clone for Witness<N_WIT, T>

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fn clone(&self) -> Witness<N_WIT, T>

Returns a copy of the value. Read more
1.0.0 · source§

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source. Read more
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impl<const N_WIT: usize, T: Debug> Debug for Witness<N_WIT, T>

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fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter. Read more
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impl<const N_WIT: usize, T: Zero + Clone> Default for Witness<N_WIT, T>

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fn default() -> Self

Returns the “default value” for a type. Read more
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impl<const N: usize, F: FftField> Foldable<F> for Witness<N, Evaluations<F, Radix2EvaluationDomain<F>>>

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fn combine(a: Self, b: Self, challenge: F) -> Self

Combine two objects ‘a’ and ‘b’ into a new object using the challenge.
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impl<const N_WIT: usize, G: Send + Debug> FromParallelIterator<G> for Witness<N_WIT, G>

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fn from_par_iter<I>(par_iter: I) -> Selfwhere I: IntoParallelIterator<Item = G>,

Creates an instance of the collection from the parallel iterator par_iter. Read more
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impl<const N_WIT: usize, T: Hash> Hash for Witness<N_WIT, T>

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fn hash<__H: Hasher>(&self, state: &mut __H)

Feeds this value into the given Hasher. Read more
1.3.0 · source§

fn hash_slice<H>(data: &[Self], state: &mut H)where H: Hasher, Self: Sized,

Feeds a slice of this type into the given Hasher. Read more
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impl<const N_WIT: usize, T> Index<usize> for Witness<N_WIT, T>

§

type Output = T

The returned type after indexing.
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fn index(&self, index: usize) -> &Self::Output

Performs the indexing (container[index]) operation. Read more
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impl<'lt, const N_WIT: usize, G> IntoIterator for &'lt Witness<N_WIT, G>

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type Item = &'lt G

The type of the elements being iterated over.
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type IntoIter = IntoIter<&'lt G, Global>

Which kind of iterator are we turning this into?
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fn into_iter(self) -> Self::IntoIter

Creates an iterator from a value. Read more
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impl<const N_WIT: usize, F: Clone> IntoIterator for Witness<N_WIT, F>

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fn into_iter(self) -> Self::IntoIter

Iterate over the columns in the circuit.

@@ -20,7 +20,7 @@ Vec<G>: IntoParallelIterator,
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fn into_par_iter(self) -> Self::Iter

Iterate over the columns in the circuit, in parallel.

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type Iter = <Vec<G, Global> as IntoParallelIterator>::Iter

The parallel iterator type that will be created.
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type Item = <Vec<G, Global> as IntoParallelIterator>::Item

The type of item that the parallel iterator will produce.
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impl<const N_WIT: usize, T: PartialEq> PartialEq<Witness<N_WIT, T>> for Witness<N_WIT, T>

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fn eq(&self, other: &Witness<N_WIT, T>) -> bool

This method tests for self and other values to be equal, and is used by ==.
1.0.0 · source§

fn ne(&self, other: &Rhs) -> bool

This method tests for !=. The default implementation is almost always -sufficient, and should not be overridden without very good reason.
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impl<const N_WIT: usize, T> TryFrom<Vec<T, Global>> for Witness<N_WIT, T>

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type Error = String

The type returned in the event of a conversion error.
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fn try_from(value: Vec<T>) -> Result<Self, Self::Error>

Performs the conversion.
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impl<const N_WIT: usize, T: Eq> Eq for Witness<N_WIT, T>

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impl<const N_WIT: usize, T> StructuralEq for Witness<N_WIT, T>

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impl<const N_WIT: usize, T> StructuralPartialEq for Witness<N_WIT, T>

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impl<const N: usize, G: CommitmentCurve> Witness<G> for Witness<N, Evaluations<G::ScalarField, Radix2EvaluationDomain<G::ScalarField>>>

Auto Trait Implementations§

§

impl<const N_WIT: usize, T> RefUnwindSafe for Witness<N_WIT, T>where +sufficient, and should not be overridden without very good reason.

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impl<const N_WIT: usize, T> TryFrom<Vec<T, Global>> for Witness<N_WIT, T>

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type Error = String

The type returned in the event of a conversion error.
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fn try_from(value: Vec<T>) -> Result<Self, Self::Error>

Performs the conversion.
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impl<const N_WIT: usize, T: Eq> Eq for Witness<N_WIT, T>

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impl<const N_WIT: usize, T> StructuralEq for Witness<N_WIT, T>

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impl<const N_WIT: usize, T> StructuralPartialEq for Witness<N_WIT, T>

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impl<const N: usize, G: CommitmentCurve> Witness<G> for Witness<N, Evaluations<G::ScalarField, Radix2EvaluationDomain<G::ScalarField>>>

Auto Trait Implementations§

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impl<const N_WIT: usize, T> RefUnwindSafe for Witness<N_WIT, T>where T: RefUnwindSafe,

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impl<const N_WIT: usize, T> Send for Witness<N_WIT, T>where T: Send,

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impl<const N_WIT: usize, T> Sync for Witness<N_WIT, T>where T: Sync,

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impl<const N_WIT: usize, T> Unpin for Witness<N_WIT, T>

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impl<const N_WIT: usize, T> UnwindSafe for Witness<N_WIT, T>where @@ -42,12 +42,12 @@ &'data mut I: IntoParallelIterator,

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type Iter = <&'data mut I as IntoParallelIterator>::Iter

The type of iterator that will be created.
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type Item = <&'data mut I as IntoParallelIterator>::Item

The type of item that will be produced; this is typically an &'data mut T reference.
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fn par_iter_mut( &'data mut self -) -> <I as IntoParallelRefMutIterator<'data>>::Iter

Creates the parallel iterator from self. Read more
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impl<T> Pointable for T

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const ALIGN: usize = mem::align_of::<T>()

The alignment of pointer.
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type Init = T

The type for initializers.
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unsafe fn init(init: <T as Pointable>::Init) -> usize

Initializes a with the given initializer. Read more
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unsafe fn deref<'a>(ptr: usize) -> &'a T

Dereferences the given pointer. Read more
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unsafe fn deref_mut<'a>(ptr: usize) -> &'a mut T

Mutably dereferences the given pointer. Read more
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unsafe fn drop(ptr: usize)

Drops the object pointed to by the given pointer. Read more
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impl<G, W> RelaxableWitness<G, W> for Wwhere +) -> <I as IntoParallelRefMutIterator<'data>>::Iter

Creates the parallel iterator from self. Read more
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impl<T> Pointable for T

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const ALIGN: usize = mem::align_of::<T>()

The alignment of pointer.
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type Init = T

The type for initializers.
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unsafe fn init(init: <T as Pointable>::Init) -> usize

Initializes a with the given initializer. Read more
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unsafe fn deref<'a>(ptr: usize) -> &'a T

Dereferences the given pointer. Read more
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unsafe fn deref_mut<'a>(ptr: usize) -> &'a mut T

Mutably dereferences the given pointer. Read more
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unsafe fn drop(ptr: usize)

Drops the object pointed to by the given pointer. Read more
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impl<G, W> RelaxableWitness<G, W> for Wwhere G: CommitmentCurve, - W: Witness<G>,

§

fn relax( + W: Witness<G>,

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fn relax( self, zero_poly: &Evaluations<<G as AffineRepr>::ScalarField, Radix2EvaluationDomain<<G as AffineRepr>::ScalarField>> -) -> RelaxedWitness<G, W>

This method takes a witness and a vector of evaluations to the zero +) -> RelaxedWitness<G, W>

This method takes a witness and a vector of evaluations to the zero polynomial, returning a relaxed witness which is composed by the extended witness and the error vector that is set to the zero polynomial.

diff --git a/rustdoc/poly_commitment/commitment/fn.absorb_commitment.html b/rustdoc/poly_commitment/commitment/fn.absorb_commitment.html index 657f471ddd..99c0c2dcd3 100644 --- a/rustdoc/poly_commitment/commitment/fn.absorb_commitment.html +++ b/rustdoc/poly_commitment/commitment/fn.absorb_commitment.html @@ -1,4 +1,4 @@ -absorb_commitment in poly_commitment::commitment - Rust
pub fn absorb_commitment<Fq: Field, G: Clone, Fr: PrimeField, EFqSponge: FqSponge<Fq, G, Fr>>(
+absorb_commitment in poly_commitment::commitment - Rust
pub fn absorb_commitment<Fq: Field, G: Clone, Fr: PrimeField, EFqSponge: FqSponge<Fq, G, Fr>>(
     sponge: &mut EFqSponge,
     commitment: &PolyComm<G>
 )
\ No newline at end of file diff --git a/rustdoc/poly_commitment/commitment/fn.squeeze_challenge.html b/rustdoc/poly_commitment/commitment/fn.squeeze_challenge.html index d1d24fbe8c..234138913e 100644 --- a/rustdoc/poly_commitment/commitment/fn.squeeze_challenge.html +++ b/rustdoc/poly_commitment/commitment/fn.squeeze_challenge.html @@ -1,4 +1,4 @@ -squeeze_challenge in poly_commitment::commitment - Rust
pub fn squeeze_challenge<Fq: Field, G, Fr: PrimeField, EFqSponge: FqSponge<Fq, G, Fr>>(
+squeeze_challenge in poly_commitment::commitment - Rust
pub fn squeeze_challenge<Fq: Field, G, Fr: PrimeField, EFqSponge: FqSponge<Fq, G, Fr>>(
     endo_r: &Fr,
     sponge: &mut EFqSponge
 ) -> Fr
\ No newline at end of file diff --git a/rustdoc/poly_commitment/commitment/fn.squeeze_prechallenge.html b/rustdoc/poly_commitment/commitment/fn.squeeze_prechallenge.html index 8f3d834806..5aee728f7f 100644 --- a/rustdoc/poly_commitment/commitment/fn.squeeze_prechallenge.html +++ b/rustdoc/poly_commitment/commitment/fn.squeeze_prechallenge.html @@ -1,3 +1,3 @@ -squeeze_prechallenge in poly_commitment::commitment - Rust
pub fn squeeze_prechallenge<Fq: Field, G, Fr: Field, EFqSponge: FqSponge<Fq, G, Fr>>(
+squeeze_prechallenge in poly_commitment::commitment - Rust
pub fn squeeze_prechallenge<Fq: Field, G, Fr: Field, EFqSponge: FqSponge<Fq, G, Fr>>(
     sponge: &mut EFqSponge
-) -> ScalarChallenge<Fr>
\ No newline at end of file +) -> ScalarChallenge<Fr>
\ No newline at end of file diff --git a/rustdoc/poly_commitment/commitment/struct.BatchEvaluationProof.html b/rustdoc/poly_commitment/commitment/struct.BatchEvaluationProof.html index 27bc6c4ea8..7a2ec0ccf3 100644 --- a/rustdoc/poly_commitment/commitment/struct.BatchEvaluationProof.html +++ b/rustdoc/poly_commitment/commitment/struct.BatchEvaluationProof.html @@ -1,6 +1,6 @@ BatchEvaluationProof in poly_commitment::commitment - Rust
pub struct BatchEvaluationProof<'a, G, EFqSponge, OpeningProof>where
     G: AffineRepr,
-    EFqSponge: FqSponge<G::BaseField, G, G::ScalarField>,{
+    EFqSponge: FqSponge<G::BaseField, G, G::ScalarField>,{
     pub sponge: EFqSponge,
     pub evaluations: Vec<Evaluation<G>>,
     pub evaluation_points: Vec<G::ScalarField>,
diff --git a/rustdoc/poly_commitment/commitment/trait.CommitmentCurve.html b/rustdoc/poly_commitment/commitment/trait.CommitmentCurve.html
index a0732b1bfa..f955ba4282 100644
--- a/rustdoc/poly_commitment/commitment/trait.CommitmentCurve.html
+++ b/rustdoc/poly_commitment/commitment/trait.CommitmentCurve.html
@@ -1,6 +1,6 @@
 CommitmentCurve in poly_commitment::commitment - Rust
pub trait CommitmentCurve: AffineRepr + Sub<Output = Self::Group> {
     type Params: SWCurveConfig;
-    type Map: GroupMap<Self::BaseField>;
+    type Map: GroupMap<Self::BaseField>;
 
     // Required methods
     fn to_coordinates(&self) -> Option<(Self::BaseField, Self::BaseField)>;
@@ -8,4 +8,4 @@
 }
Expand description

A useful trait extending AffineRepr for commitments. Unfortunately, we can’t specify that AffineRepr<BaseField : PrimeField>, so usage of this traits must manually bind G::BaseField: PrimeField.

-

Required Associated Types§

source

type Params: SWCurveConfig

source

type Map: GroupMap<Self::BaseField>

Required Methods§

source

fn to_coordinates(&self) -> Option<(Self::BaseField, Self::BaseField)>

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fn of_coordinates(x: Self::BaseField, y: Self::BaseField) -> Self

Implementations on Foreign Types§

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impl<P: SWCurveConfig + Clone> CommitmentCurve for SWJAffine<P>

§

type Params = P

§

type Map = BWParameters<P>

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fn to_coordinates(&self) -> Option<(Self::BaseField, Self::BaseField)>

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fn of_coordinates(x: P::BaseField, y: P::BaseField) -> SWJAffine<P>

Implementors§

\ No newline at end of file +

Required Associated Types§

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type Params: SWCurveConfig

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type Map: GroupMap<Self::BaseField>

Required Methods§

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fn to_coordinates(&self) -> Option<(Self::BaseField, Self::BaseField)>

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fn of_coordinates(x: Self::BaseField, y: Self::BaseField) -> Self

Implementations on Foreign Types§

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impl<P: SWCurveConfig + Clone> CommitmentCurve for SWJAffine<P>

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type Params = P

§

type Map = BWParameters<P>

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fn to_coordinates(&self) -> Option<(Self::BaseField, Self::BaseField)>

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fn of_coordinates(x: P::BaseField, y: P::BaseField) -> SWJAffine<P>

Implementors§

\ No newline at end of file diff --git a/rustdoc/poly_commitment/commitment/trait.EndoCurve.html b/rustdoc/poly_commitment/commitment/trait.EndoCurve.html index 2a252d2dfd..bd3cf988b9 100644 --- a/rustdoc/poly_commitment/commitment/trait.EndoCurve.html +++ b/rustdoc/poly_commitment/commitment/trait.EndoCurve.html @@ -6,7 +6,7 @@ _endo_q: Self::BaseField, g1: &[Self], g2: &[Self], - x2: ScalarChallenge<Self::ScalarField> + x2: ScalarChallenge<Self::ScalarField> ) -> Vec<Self> { ... } fn combine( g1: &[Self], @@ -23,7 +23,7 @@ _endo_q: Self::BaseField, g1: &[Self], g2: &[Self], - x2: ScalarChallenge<Self::ScalarField> + x2: ScalarChallenge<Self::ScalarField> ) -> Vec<Self>

Combine where x1 = one

source

fn combine( g1: &[Self], @@ -35,7 +35,7 @@ endo_q: Self::BaseField, g1: &[Self], g2: &[Self], - x2: ScalarChallenge<Self::ScalarField> + x2: ScalarChallenge<Self::ScalarField> ) -> Vec<Self>

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fn combine( g1: &[Self], g2: &[Self], diff --git a/rustdoc/poly_commitment/ipa/struct.OpeningProof.html b/rustdoc/poly_commitment/ipa/struct.OpeningProof.html index e82a9d790f..abc3cbd5cf 100644 --- a/rustdoc/poly_commitment/ipa/struct.OpeningProof.html +++ b/rustdoc/poly_commitment/ipa/struct.OpeningProof.html @@ -6,12 +6,12 @@ pub sg: G, }

Fields§

§lr: Vec<(G, G)>

Vector of rounds of L & R commitments

§delta: G§z1: G::ScalarField§z2: G::ScalarField§sg: G

A final folded commitment base

-

Implementations§

source§

impl<G: AffineRepr> OpeningProof<G>

source

pub fn prechallenges<EFqSponge: FqSponge<G::BaseField, G, G::ScalarField>>( +

Implementations§

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impl<G: AffineRepr> OpeningProof<G>

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pub fn prechallenges<EFqSponge: FqSponge<G::BaseField, G, G::ScalarField>>( &self, sponge: &mut EFqSponge -) -> Vec<ScalarChallenge<G::ScalarField>>

Computes a log-sized vector of scalar challenges for +) -> Vec<ScalarChallenge<G::ScalarField>>

Computes a log-sized vector of scalar challenges for recombining elements inside the IPA.

-
source

pub fn challenges<EFqSponge: FqSponge<G::BaseField, G, G::ScalarField>>( +

source

pub fn challenges<EFqSponge: FqSponge<G::BaseField, G, G::ScalarField>>( &self, endo_r: &G::ScalarField, sponge: &mut EFqSponge @@ -38,7 +38,7 @@ sponge: EFqSponge, rng: &mut RNG ) -> Selfwhere - EFqSponge: Clone + FqSponge<<G as AffineRepr>::BaseField, G, <G as AffineRepr>::ScalarField>, + EFqSponge: Clone + FqSponge<<G as AffineRepr>::BaseField, G, <G as AffineRepr>::ScalarField>, RNG: RngCore + CryptoRng,

Create an opening proof for a batch of polynomials. The parameters are the following: Read more
source§

fn verify<EFqSponge, RNG>( srs: &Self::SRS, @@ -46,7 +46,7 @@ batch: &mut [BatchEvaluationProof<'_, G, EFqSponge, Self>], rng: &mut RNG ) -> boolwhere - EFqSponge: FqSponge<<G as AffineRepr>::BaseField, G, <G as AffineRepr>::ScalarField>, + EFqSponge: FqSponge<<G as AffineRepr>::BaseField, G, <G as AffineRepr>::ScalarField>, RNG: RngCore + CryptoRng,

Verify the opening proof
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impl<G> Serialize for OpeningProof<G>where G: CanonicalDeserialize + CanonicalSerialize + AffineRepr,

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fn serialize<__S>(&self, __serializer: __S) -> Result<__S::Ok, __S::Error>where __S: Serializer,

Serialize this value into the given Serde serializer. Read more

Auto Trait Implementations§

§

impl<G> RefUnwindSafe for OpeningProof<G>where diff --git a/rustdoc/poly_commitment/ipa/struct.SRS.html b/rustdoc/poly_commitment/ipa/struct.SRS.html index 14857cf35f..97af99b2fb 100644 --- a/rustdoc/poly_commitment/ipa/struct.SRS.html +++ b/rustdoc/poly_commitment/ipa/struct.SRS.html @@ -13,7 +13,7 @@ batch: &mut [BatchEvaluationProof<'_, G, EFqSponge, OpeningProof<G>>], rng: &mut RNG ) -> boolwhere - EFqSponge: FqSponge<G::BaseField, G, G::ScalarField>, + EFqSponge: FqSponge<G::BaseField, G, G::ScalarField>, RNG: RngCore + CryptoRng, G::BaseField: PrimeField,

This function verifies a batch of polynomial commitment opening proofs. Return true if the verification is successful, false otherwise.

@@ -36,7 +36,7 @@
Safety
sponge: EFqSponge, rng: &mut RNG ) -> OpeningProof<G>where - EFqSponge: Clone + FqSponge<G::BaseField, G, G::ScalarField>, + EFqSponge: Clone + FqSponge<G::BaseField, G, G::ScalarField>, RNG: RngCore + CryptoRng, G::BaseField: PrimeField, G: EndoCurve,

Trait Implementations§

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impl<G: Clone> Clone for SRS<G>

source§

fn clone(&self) -> SRS<G>

Returns a copy of the value. Read more
1.0.0 · source§

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source. Read more
source§

impl<G: Debug> Debug for SRS<G>

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fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter. Read more
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impl<G: Default> Default for SRS<G>

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fn default() -> SRS<G>

Returns the “default value” for a type. Read more
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impl<'de, G> Deserialize<'de> for SRS<G>where diff --git a/rustdoc/poly_commitment/kzg/struct.KZGProof.html b/rustdoc/poly_commitment/kzg/struct.KZGProof.html index 92e2affaae..d35374738f 100644 --- a/rustdoc/poly_commitment/kzg/struct.KZGProof.html +++ b/rustdoc/poly_commitment/kzg/struct.KZGProof.html @@ -41,7 +41,7 @@ _sponge: EFqSponge, _rng: &mut RNG ) -> Selfwhere - EFqSponge: Clone + FqSponge<<G as AffineRepr>::BaseField, G, F>, + EFqSponge: Clone + FqSponge<<G as AffineRepr>::BaseField, G, F>, RNG: RngCore + CryptoRng,

Parameters:

  • srs: the structured reference string
  • @@ -58,7 +58,7 @@ batch: &mut [BatchEvaluationProof<'_, G, EFqSponge, Self>], _rng: &mut RNG ) -> boolwhere - EFqSponge: FqSponge<G::BaseField, G, F>, + EFqSponge: FqSponge<G::BaseField, G, F>, RNG: RngCore + CryptoRng,
    Verify the opening proof
source§

impl<Pair: Pairing> Serialize for KZGProof<Pair>where Pair::G1Affine: CanonicalDeserialize + CanonicalSerialize,

source§

fn serialize<__S>(&self, __serializer: __S) -> Result<__S::Ok, __S::Error>where __S: Serializer,

Serialize this value into the given Serde serializer. Read more

Auto Trait Implementations§

§

impl<Pair> RefUnwindSafe for KZGProof<Pair>where diff --git a/rustdoc/poly_commitment/trait.OpenProof.html b/rustdoc/poly_commitment/trait.OpenProof.html index a8a0bd5fba..bf50e9f769 100644 --- a/rustdoc/poly_commitment/trait.OpenProof.html +++ b/rustdoc/poly_commitment/trait.OpenProof.html @@ -12,7 +12,7 @@ sponge: EFqSponge, rng: &mut RNG ) -> Self - where EFqSponge: Clone + FqSponge<<G as AffineRepr>::BaseField, G, <G as AffineRepr>::ScalarField>, + where EFqSponge: Clone + FqSponge<<G as AffineRepr>::BaseField, G, <G as AffineRepr>::ScalarField>, RNG: RngCore + CryptoRng; fn verify<EFqSponge, RNG>( srs: &Self::SRS, @@ -20,7 +20,7 @@ batch: &mut [BatchEvaluationProof<'_, G, EFqSponge, Self>], rng: &mut RNG ) -> bool - where EFqSponge: FqSponge<G::BaseField, G, G::ScalarField>, + where EFqSponge: FqSponge<G::BaseField, G, G::ScalarField>, RNG: RngCore + CryptoRng; }

Required Associated Types§

source

type SRS: SRS<G>

Required Methods§

source

fn open<EFqSponge, RNG, D: EvaluationDomain<<G as AffineRepr>::ScalarField>>( srs: &Self::SRS, @@ -32,7 +32,7 @@ sponge: EFqSponge, rng: &mut RNG ) -> Selfwhere - EFqSponge: Clone + FqSponge<<G as AffineRepr>::BaseField, G, <G as AffineRepr>::ScalarField>, + EFqSponge: Clone + FqSponge<<G as AffineRepr>::BaseField, G, <G as AffineRepr>::ScalarField>, RNG: RngCore + CryptoRng,

Create an opening proof for a batch of polynomials. The parameters are the following:

Verify the opening proof

Implementors§

source§

impl<BaseField: PrimeField, G: AffineRepr<BaseField = BaseField> + CommitmentCurve + EndoCurve> OpenProof<G> for OpeningProof<G>

§

type SRS = SRS<G>

source§

impl<F: PrimeField, G: CommitmentCurve<ScalarField = F>, G2: CommitmentCurve<ScalarField = F>, Pair: Pairing<G1Affine = G, G2Affine = G2>> OpenProof<G> for KZGProof<Pair>

§

type SRS = PairingSRS<Pair>

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For the Rust documentation, see here . Note that this book is a work in progress, not necessarily reflecting the current state of Mina. Authors: Izaak Meckler, Vanishree Rao, Matthew Ryan, Anaïs Querol, Joseph Spadavecchia, David Wong, Xiang Xie In memory of Vitaly Zelov.","breadcrumbs":"Introduction » Introduction","id":"0","title":"Introduction"},"1":{"body":"This document is intended primarily to communicate mathematical ideas to programmers with some experience of math. To that end, we will often be ambivalent about the difference between sets and types (in whatever programming language, but usually I am imagining some idealized form of Rust or OCaml). So for example we may write A is a type A is a set A is a type and these should all be interpreted the same way, assuming it makes sense in the context. Similarly, we may write either of the following, and potentially mean the same thing: a:A a∈A a : A We use → for function types ↦ for defining functions Also, we usually assume functions are computed by a program (in whatever sense of \"program\" makes sense in the given context). So if we say \"let f:A→B\", we typically mean that A and B are types in some programming language which makes sense in the context (or maybe that they are sets, again depending on context) f is actually a program computing a function of the type A→B (again, in whatever sense of program makes sense in context)","breadcrumbs":"Terminology » Terminology","id":"1","title":"Terminology"},"10":{"body":"The purpose of this document is to give the reader mathematical, cryptographic, and programming context sufficient to become an effective practitioner of zero-knowledge proofs and ZK-SNARKs specifically.","breadcrumbs":"Fields » Intro","id":"10","title":"Intro"},"100":{"body":"Unlike the latest version of vanilla PlonK that implements the final check using a polynomial opening (via Maller's optimization), we implement it manually. (That is to say, Izaak implemented Maller's optimization for 5-wires.) But the check is not exactly f(ζ)=ZH​(ζ)t(ζ). This note describes how and why the implementation deviates a little.","breadcrumbs":"Final Check » Understanding the implementation of the f(ζ)=ZH​(ζ)t(ζ) check","id":"100","title":"Understanding the implementation of the f(ζ)=ZH​(ζ)t(ζ) check"},"101":{"body":"If you look at how the evaluation of f(z) is computed on the prover side (or the commitment of f is computed on the verifier side), you can see that f is missing two things: the public input part some terms in the permutation What is it missing in the permutation part? Let's look more closely. This is what we have: −1⋅⋅⋅⋅⋅⋅⋅⋅​z(ζω)⋅zkpm(ζ)⋅αPERM0(w[0](ζ)+β⋅σ[0](ζ)+γ)(w[1](ζ)+β⋅σ[1](ζ)+γ)(w[2](ζ)+β⋅σ[2](ζ)+γ)(w[3](ζ)+β⋅σ[3](ζ)+γ)(w[4](ζ)+β⋅σ[4](ζ)+γ)(w[5](ζ)+β⋅σ[5](ζ)+γ)β⋅σ[6](x)​​ In comparison, here is the list of the stuff we needed to have: the two sides of the coin: z(ζ)⋅⋅⋅⋅⋅⋅⋅⋅​zkpm(ζ)⋅αPERM0(w[0](ζ)+β⋅shift[0](ζ)+γ)(w[1](ζ)+β⋅shift[1](ζ)+γ)(w[2](ζ)+β⋅shift[2](ζ)+γ)(w[3](ζ)+β⋅shift[3](ζ)+γ)(w[4](ζ)+β⋅shift[4](ζ)+γ)(w[5](ζ)+β⋅shift[5](ζ)+γ)(w[6](ζ)+β⋅shift[6](ζ)+γ)​​ with shift[0]=1 and −1⋅⋅⋅⋅⋅⋅⋅⋅​z(ζω)⋅zkpm(ζ)⋅αPERM0(w[0](ζ)+β⋅σ[0](ζ)+γ)(w[1](ζ)+β⋅σ[1](ζ)+γ)(w[2](ζ)+β⋅σ[2](ζ)+γ)(w[3](ζ)+β⋅σ[3](ζ)+γ)(w[4](ζ)+β⋅σ[4](ζ)+γ)(w[5](ζ)+β⋅σ[5](ζ)+γ)(w[6](ζ)+β⋅σ[6](ζ)+γ)​​ the initialization: (z(ζ)−1)⋅L1​(ζ)⋅αPERM1 and the end of the accumulator: (z(ζ)−1)⋅Ln−k​(ζ)⋅αPERM2 You can read more about why it looks like that in this post . We can see clearly that the permutation stuff we have in f is clearly lacking. It's just the subtraction part (equation 2), and even that is missing some terms. It is missing exactly this: −1 ⋅⋅⋅⋅⋅⋅⋅⋅​z(ζω)⋅zkpm(ζ)⋅αPERM0(w[0](ζ)+β⋅σ[0](ζ)+γ)(w[1](ζ)+β⋅σ[1](ζ)+γ)(w[2](ζ)+β⋅σ[2](ζ)+γ)(w[3](ζ)+β⋅σ[3](ζ)+γ)(w[4](ζ)+β⋅σ[4](ζ)+γ)(w[5](ζ)+β⋅σ[5](ζ)+γ)(w[6](ζ)+γ)​​ So at the end, when we have to check for the identity f(ζ)=ZH​(ζ)t(ζ) we'll actually have to check something like this (I colored the missing parts on the left hand side of the equation): f(ζ)​+pub(ζ)+z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+βζ+γ)⋅(w[1](ζ)+β⋅shift[0](ζ)+γ)⋅(w[2](ζ)+β⋅shift[1](ζ)+γ)⋅(w[3](ζ)+β⋅shift[2](ζ)+γ)⋅(w[4](ζ)+β⋅shift[3](ζ)+γ)⋅(w[5](ζ)+β⋅shift[4](ζ)+γ)⋅(w[6](ζ)+β⋅shift[5](ζ)+γ)−z(ζω)⋅zkpm(ζ)⋅αPERM0⋅⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅⋅(w[6](ζ)+γ)+(z(ζ)−1)⋅L1​(ζ)⋅αPERM1+(z(ζ)−1)⋅Ln−k​(ζ)⋅αPERM2=t(ζ)(ζn−1)​​ This is not exactly what happens in the code. But if we move things around a bit, we end up with what's implemented. I highlight what changes in each steps. First, we just move things around: f(ζ)​+pub(ζ)+z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅shift[0]ζ+γ)⋅(w[1](ζ)+β⋅shift[1]ζ+γ)⋅(w[2](ζ)+β⋅shift[2]ζ+γ)⋅(w[3](ζ)+β⋅shift[3]ζ+γ)⋅(w[4](ζ)+β⋅shift[4]ζ+γ)⋅(w[5](ζ)+β⋅shift[5]ζ+γ)⋅(w[6](ζ)+β⋅shift[6]ζ+γ)−z(ζω)⋅zkpm(ζ)⋅αPERM0⋅⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅(w[6](ζ)+γ)−t(ζ)(ζn−1)=(1−z(ζ))L1​(ζ)αPERM1+(1−z(ζ))Ln−k​(ζ)αPERM2​​ here are the actual lagrange basis calculated with the formula here , oh and we actually use L0​ in the code, not L1​, so let's change that as well: f(ζ)++−−=​pub(ζ)z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅shift[0]ζ+γ)⋅(w[1](ζ)+β⋅shift[1]ζ+γ)⋅(w[2](ζ)+β⋅shift[2]ζ+γ)⋅(w[3](ζ)+β⋅shift[3]ζ+γ)⋅(w[4](ζ)+β⋅shift[4]ζ+γ)⋅(w[5](ζ)+β⋅shift[5]ζ+γ)⋅(w[6](ζ)+β⋅shift[6]ζ+γ)+z(ζω)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅(w[6](ζ)+γ)+t(ζ)(ζn−1)(1−z(ζ))[n(ζ−1)(ζn−1)​αPERM1+n(ζ−ωn−k)ωn−k(ζn−1)​αPERM2]​​ finally we extract some terms from the lagrange basis: ​[f(ζ)+pub(ζ)+z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅shift[0]ζ+γ)⋅(w[1](ζ)+β⋅shift[1]ζ+γ)⋅(w[2](ζ)+β⋅shift[2]ζ+γ)⋅(w[3](ζ)+β⋅shift[3]ζ+γ)⋅(w[4](ζ)+β⋅shift[4]ζ+γ)⋅(w[5](ζ)+β⋅shift[5]ζ+γ)⋅(w[6](ζ)+β⋅shift[6]ζ+γ)+−z(ζω)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅(w[6](ζ)+γ)−t(ζ)(ζn−1)]⋅(ζ−1)(ζ−ωn−k)=(1−z(ζ))[n(ζn−1)(ζ−ωn−k)​αPERM1+nωn−k(ζn−1)(ζ−1)​αPERM2]​​​ with αPERM0=α17,αPERM1=α18,αPERM2=α19 Why do we do things this way? Most likely to reduce Also, about the code: the division by n in the αPERM1 and the αPERM2 terms is missing (why?) the multiplication by wn−k in the αPERM2 term is missing (why?) As these terms are constants, and do not prevent the division by ZH​ to form t, t also omits them. In other word, they cancel one another.","breadcrumbs":"Final Check » What's f and what's missing in the final equation?","id":"101","title":"What's f and what's missing in the final equation?"},"102":{"body":"In verifier.rs you can see the following code (cleaned to remove anything not important) // compute the witness polynomials $w_0, \\cdots, w_14$ and the permutation polynomial $z$ in evaluation forms on different domains\nlet lagrange = index.cs.evaluate(&w, &z); // compute parts of the permutation stuff that is included in t\nlet (perm, bnd) = index.cs.perm_quot(&lagrange, &oracles, &z, &alpha[range::PERM])?; // divide contributions with vanishing polynomial\nlet (mut t, res) = (perm + some_other_stuff).divide_by_vanishing_poly() // add the other permutation stuff\nt += &bnd; // t is evaluated at zeta and sent... Notice that bnd is not divided by the vanishing polynomial . Also what's perm? Let's answer the latter TESTREMOVEME: perm(x)= −​aPERM0⋅zkpl(x)⋅[z(x)⋅(w[0](x)+γ+x⋅β⋅shift[0])⋅(w[1](x)+γ+x⋅β⋅shift[1])⋅…z(x⋅w)⋅(w[0](x)+γ+σ[0]⋅β) ⋅(w[1](x)+γ+σ[1]⋅β)⋅…]​​ and bnd is: bnd(x)=aPERM1⋅x−1z(x)−1​+aPERM2⋅x−sid[n−k]z(x)−1​ you can see that some terms are missing in bnd, specifically the numerator xn−1. Well, it would have been cancelled by the division by the vanishing polynomial, and this is the reason why we didn't divide that term by ZH​(x)=xn−1. Also, note that the same constant terms that were missing in f are missing in t.","breadcrumbs":"Final Check » What about t?","id":"102","title":"What about t?"},"103":{"body":"This document proposes a protocol change for kimchi .","breadcrumbs":"Maller's Optimization for Kimchi » Maller's optimization for Kimchi","id":"103","title":"Maller's optimization for Kimchi"},"104":{"body":"See the section on Maller's optimization for background.","breadcrumbs":"Maller's Optimization for Kimchi » What is Maller's optimization?","id":"104","title":"What is Maller's optimization?"},"105":{"body":"We want the verifier to form commitment to the following polynomial: L(x)=f(x)−ZH​(ζ)⋅t(x) They could do this like so: com(L)=com(f)−ZH​(ζ)⋅com(t) Since they already know f, they can produce com(f), the only thing they need is com(t). So the protocol looks like that: maller 15 1 In the rest of this document, we review the details and considerations needed to implement this change in kimchi .","breadcrumbs":"Maller's Optimization for Kimchi » Overview","id":"105","title":"Overview"},"106":{"body":"There's one challenge that prevents us from directly using this approach: com(t) is typically sent and received in several commitments (called chunks or segments throughout the codebase). As t is the largest polynomial, usually exceeding the size of the SRS (which is by default set to be the size of the domain).","breadcrumbs":"Maller's Optimization for Kimchi » How to deal with a chunked t?","id":"106","title":"How to deal with a chunked t?"},"107":{"body":"Thus, the verifier will have to produce split commitments of L and combine them with powers of ζn to verify an evaluation proof. Let's define L as: L=L0​+xnL1​+x2nL1​+⋯ where every Li​ is of degree n−1 at most. Then we have that com(L)=com(L0​)+com(xn⋅L1​)+com(x2n⋅L2​)+⋯ Which the verifier can't produce because of the powers of xn, but we can linearize it as we already know which x we're going to evaluate that polynomial with: com(L~)=com(L0​)+ζn⋅com(L1​)+ζ2n⋅com(L2​)+⋯","breadcrumbs":"Maller's Optimization for Kimchi » The verifier side","id":"107","title":"The verifier side"},"108":{"body":"This means that the prover will produce evaluation proofs on the following linearized polynomial: L~(x)=1⋅L0​(x)+ζn⋅L1​(x)+ζ2n⋅L2​(x)+⋯ which is the same as L(x) only if evaluated at ζ. As the previous section pointed out, we will need L (ζω) and L (ζω)=L(ζω).","breadcrumbs":"Maller's Optimization for Kimchi » The prover side","id":"108","title":"The prover side"},"109":{"body":"To create an evaluation proof, the prover also needs to produce the blinding factor rL​ associated with the chunked commitment of: L =f ​−(ζn−1)t~ To compute it, there are two rules to follow: when two commitment are added together, their associated blinding factors get added as well: com(a)+com(b)⟹ra​+rb​ when scaling a commitment, its blinding factor gets scalled too: n⋅com(a)⟹n⋅ra​ As such, if we know rf​ and rt​, we can compute: rL ​=rf ​​+(ζn−1)⋅rt~​ The prover knows the blinding factor of the commitment to t, as they committed to it. But what about f? They never commit to it really, and the verifier recreates it from scratch using: The commitments we sent them . In the linearization process, the verifier actually gets rid of most prover commitments, except for the commitment to the last sigma commitment Sσ6​. (TODO: link to the relevant part in the spec) As this commitment is public, it is not blinded. The public commitments . Think commitment to selector polynomials or the public input polynomial. These commitments are not blinded, and thus do not impact the calculation of the blinding factor. The evaluations we sent them . Instead of using commitments to the wires when recreating f, the verifier uses the (verified) evaluations of these in ζ. If we scale our commitment com(z) with any of these scalars, we will have to do the same with rz​. Thus, the blinding factor of L is (ζn−1)⋅rt ​.","breadcrumbs":"Maller's Optimization for Kimchi » Evaluation proof and blinding factors","id":"109","title":"Evaluation proof and blinding factors"},"11":{"body":"In this section we'll discuss the fundamental objects used in the construction of most ZK-SNARKs in use today. These objects are used extensively and without ceremony, so it's important to understand them very well. If you find you don't understand something at first, don't worry: practice is often the best teacher and in using these objects you will find they will become like the bicycle you've had for years: like an extension of yourself that you use without a thought.","breadcrumbs":"Fields » Some fundamental mathematical objects","id":"11","title":"Some fundamental mathematical objects"},"110":{"body":"The prover actually does not send a commitment to the full f polynomial. As described in the last check section . The verifier will have to compute the evaluation of L~(ζ) because it won't be zero. It should be equal to the following: ​f ​(ζ)−t (ζ)(ζn−1)=(ζ−1)(ζ−ωn−k)1−z(ζ)​[n(ζn−1)(ζ−ωn−k)​αPERM1+nωn−k(ζn−1)(ζ−1)​αPERM2]−pub(ζ)−z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅shift[0]ζ+γ)⋅(w[1](ζ)+β⋅shift[1]ζ+γ)⋅(w[2](ζ)+β⋅shift[2]ζ+γ)⋅(w[3](ζ)+β⋅shift[3]ζ+γ)⋅(w[4](ζ)+β⋅shift[4]ζ+γ)⋅(w[5](ζ)+β⋅shift[5]ζ+γ)⋅(w[6](ζ)+β⋅shift[6]ζ+γ)+z(ζω)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅(w[6](ζ)+γ)+​​ Because we use the inner product polynomial commitment , we also need: L (ζω)=f ​(ζω)−ZH​(ζ)⋅t~(ζω) Notice the ZH​(ζ). That evaluation must be sent as part of the proof as well.","breadcrumbs":"Maller's Optimization for Kimchi » The evaluation of L~","id":"110","title":"The evaluation of L~"},"111":{"body":"Now here's how we need to modify the current protocol: The evaluations f(ζ),f(ζω),t(ζ),t(ζω) (and their corresponding evaluation proofs) don't have to be part of the protocol anymore. The prover must still send the chunked commitments to t. The prover must create a linearized polynomial L by creating a linearized polynomial f ​ and a linearized polynomial t and computing: L =f ​+(ζn−1)⋅t While the verifier can compute the evaluation of L (ζ) by themselves, they don't know the evaluation of L (ζω), so the prover needs to send that. The verifier must recreate com(L ), the commitment to L , themselves so that they can verify the evaluation proofs of both L (ζ) and L (ζω). The evaluation of L~(ζω) must be absorbed in both sponges (Fq and Fr). maller 15 2 The proposal is implemented in #150 with the following details: the L~ polynomial is called ft. the evaluation of L~(ζ) is called ft_eval0. the evaluation L~(ζω) is called ft_eval1.","breadcrumbs":"Maller's Optimization for Kimchi » The actual protocol changes","id":"111","title":"The actual protocol changes"},"112":{"body":"In 2020, Plookup showed how to create lookup proofs. Proofs that some witness values are part of a lookup table . Two years later, an independent team published plonkup showing how to integrate Plookup into PlonK. This document specifies how we integrate Plookup in kimchi. It assumes that the reader understands the basics behind Plookup.","breadcrumbs":"Lookup Tables » Plookup in Kimchi","id":"112","title":"Plookup in Kimchi"},"113":{"body":"We integrate Plookup in kimchi with the following differences: we snake-ify the sorted table instead of wrapping it around (see later) we allow fixed-ahead-of-time linear combinations of columns of the queries we make we implemented different tables, like RangeCheck and XOR. we allow several lookups (or queries) to be performed within the same row zero-knowledgeness is added in a specific way (see later) The following document explains the protocol in more detail","breadcrumbs":"Lookup Tables » Overview","id":"113","title":"Overview"},"114":{"body":"As per the Plookup paper, the prover will have to compute three vectors: f, the (secret) query vector , containing the witness values that the prover wants to prove are part of the lookup table. t, the (public) lookup table . s, the (secret) concatenation of f and t, sorted by t (where elements are listed in the order they are listed in t). Essentially, Plookup proves that all the elements in f are indeed in the lookup table t if and only if the following multisets are equal: {(1+β)f,diff(t)} diff(sorted(f,t)) where diff is a new set derived by applying a \"randomized difference\" between every successive pairs of a vector, and (f,t) is the set union of f et t. More precisely, for a set S={s0​,s1​,⋯,sn​}, diff(S) is defined as the set {s0​+βs1​,s1​+βs2​,⋯,sn−1​+βsn​}. For example, with: f={5,4,1,5} t={1,4,5} we have: sorted(f,t)={1,1,4,4,5,5,5} {(1+β)f,diff(t)}={(1+β)5,(1+β)4,(1+β)1,(1+β)5,1+β4,4+β5} diff(sorted(f,t))={1+β1,1+β4,4+β4,4+β5,5+β5,5+β5} Note: This assumes that the lookup table is a single column. You will see in the next section how to address lookup tables with more than one column. The equality between the multisets can be proved with the permutation argument of PlonK, which would look like enforcing constraints on the following accumulator: init: acc0​=1 final: accn​=1 for every 0 RAM loading, disk -> RAM loading, and RAM r/w. This gives the maximum flexibility for circuit designers, and each primitive has uses for projects that we are pursuing. This design has been refined over several iterations to reduce the number of components exposed in the proof and the amount of computation that the verifier requires. The largest remaining parts represent the data itself and the method for combining them, where simplifying assumptions have been made to minimize their impact while still satisfying the design constraints.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » Why is this design the best in the space of possible designs?","id":"138","title":"Why is this design the best in the space of possible designs?"},"139":{"body":"Other designs building upon the lookup primitive reduce on some subset of this design, or on relaxing a design constraint that would remove some useful / required functionality.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What other designs have been considered and what is the rationale for not choosing them?","id":"139","title":"What other designs have been considered and what is the rationale for not choosing them?"},"14":{"body":"For a finite field Fp​ where p fits in n bits (i.e., p<2n) we have Addition, subtraction: O(n) Multiplication O(n2) Division O(n2) I believe, in practice significantly slower than multiplication. Actually, on a practical level, it's more accurate to model the complexity in terms of the number of limbs rather than the number of bits where a limb is 64 bits. Asymptotically it makes no difference but concretely it's better to think about the number of limbs for the most part. As a result you can see it's the smaller n is the better, especially with respect to multiplication, which dominates performance considerations for implementations of zk-SNARKs, since they are dominated by elliptic curve operations that consist of field operations. While still in development, Mina used to use a field of 753 bits or 12 limbs and now uses a field of 255 bits or 4 limbs. As a result, field multiplication became automatically sped up by a factor of 122/42=9, so you can see it's very useful to try to shrink the field size.","breadcrumbs":"Fields » Algorithmics of prime order fields","id":"14","title":"Algorithmics of prime order fields"},"140":{"body":"We are currently unable to expose certain useful functionality to users of Snapps / SnarkyJS, and are missing the ability to use that same functionality for important parts of the zkOracles ecosystem (such as parsing / scraping). We also are currently forced to hard-code any constraints that we need for particular uses in the verifier circuit, and users are unable to remove the parts they do not need and to replace them with other useful constraints that their circuits might benefit from. The Mina transaction model is unable to support a bytecode interpreter without this or similar extensions.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What is the impact of not doing this?","id":"140","title":"What is the impact of not doing this?"},"141":{"body":"(Elided)","breadcrumbs":"Lookup Tables » Extended Lookup Tables » Prior art","id":"141","title":"Prior art"},"142":{"body":"","breadcrumbs":"Lookup Tables » Extended Lookup Tables » Unresolved questions","id":"142","title":"Unresolved questions"},"143":{"body":"The maximum number of tables / entries of each kind that we want to support in the 'standard' protocol. Validation (or rejection) of the simplifying assumptions. Confirmation of the design constraints, or simplifications thereto. Necessity / utility of the permutation argument.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What parts of the design do you expect to resolve through the RFC process before this gets merged?","id":"143","title":"What parts of the design do you expect to resolve through the RFC process before this gets merged?"},"144":{"body":"None, this RFC has been developed in part from an initial design and iteration on implementations of parts of that design.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What parts of the design do you expect to resolve through the implementation of this feature before merge?","id":"144","title":"What parts of the design do you expect to resolve through the implementation of this feature before merge?"},"145":{"body":"Bytecode design and implementation details. Mina transaction interface for interacting with these features. Snarky / SnarkyJS interface for interacting with these features.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What related issues do you consider out of scope for this RFC that could be addressed in the future independently of the solution that comes out of this RFC?","id":"145","title":"What related issues do you consider out of scope for this RFC that could be addressed in the future independently of the solution that comes out of this RFC?"},"146":{"body":"This section explains how to design and add a custom constraint to our proof-systems library. PLONK is an AIOP. That is, it is a protocol in which the prover sends polynomials as messages and the verifier sends random challenges, and then evaluates the prover's polynomials and performs some final checks on the outputs. PLONK is very flexible. It can be customized with constraints specific to computations of interest. For example, in Mina, we use a PLONK configuration called kimchi that has custom constraints for poseidon hashing, doing elliptic curve operations, and more. A \"PLONK configuration\" specifies The set of types of constraints that you would like to be able to enforce. We will describe below how these types of constraints are specified. A number of \"eq-able\" columns W A number of \"advice\" columns A Under such configuration, a circuit is specified by A number of rows n A vector cs of constraint-types of length n. I.e., a vector that specifies, for each row, which types of constraints should be enforced on that row. A vector eqs : Vec<(Position, Position)> of equalities to enforce, where struct Position { row: usize, column: usize }. E.g., if the pair (Position { row: 0, col: 8 }, Position { row: 10, col: 2 }) is in eqs, then the circuit is saying the entries in those two positions should be equal, or in other words that they refer to the same value. This is where the distinction between \"eq-able\" and \"advice\" columns comes in. The column field of a position in the eqs array can only refer to one of the first W columns. Equalities cannot be enforced on entries in the A columns after that. Then, given such a circuit, PLONK lets you produce proofs for the statement I know W + A \"column vectors\" of field elements vs: [Vec; W + A] such that for each row index i < n, the constraint of type cs[i] holds on the values [vs[0][i], ..., vs[W+A - 1][i], vs[0][i+1], ..., vs[W+A - 1][i+1] and all the equalities in eqs hold. I.e., for (p1, p2) in eqs we have vs[p1.col][p1.row] == vs[p2.col][p2.row]. So, a constraint can check the values in two adjacent rows. Mathematically speaking, a constraint is a multivariate polynomial over the variables cCurr,i​,…,vCurr,W+A−1​,vNext,0​,…,vNext,W+A−1​. In other words, there is one variable corresponding to the value of each column in the \"current row\" and one variable correspond to the value of each column in the \"next row\". In Rust, vr,i​ is written E::cell(Column::Witness(i), r). So, for example, the variable vNext,3​ is written E::cell(Column::Witness(3), CurrOrNext::Next). let w = |i| v(Column::Witness(i)); Let's","breadcrumbs":"Custom Constraints » Specifying a constraint","id":"146","title":"Specifying a constraint"},"147":{"body":"The art in proof systems comes from knowing how to design a PLONK configuration to ensure maximal efficiency for the sorts of computations you are trying to prove. That is, how to choose the numbers of columns W and A, and how to define the set of constraint types. Let's describe the trade-offs involved here. The majority of the proving time for the PLONK prover is in committing to the W + A column polynomials, which have length equal to the number of rows n committing to the \"permutation accumulator polynomial, which has length n. committing to the quotient polynomial, which reduces to computing max(k, W) MSMs of size n, where k is the max degree of a constraint. performing the commitment opening proof, which is mostly dependent on the number of rows n. So all in all, the proving time is approximately equal to the time to perform W + A + 1 + max(k - 1, W) MSMs of size n, plus the cost of an opening proof for polynomials of degree n - 1. and maybe computing the combined constraint polynomial, which has degree k * n where k is the maximum degree of a constraint Increasing W and A increase proof size, and they potentially impact the prover-time as the prover must compute polynomial commitments to each column, and computing a polynomial commitment corresponds to doing one MSM (multi-scalar multiplication, also called a multi-exponentiation.) However, often increasing the number of columns allows you to decrease the number of rows required for a given computation. For example, if you can perform one Poseidon hash in 36 rows with 5 total columns, then you can also perform it in 12 (= 36 / 3) rows with 15 (= 5 * 3) total columns. Decreasing the number of rows (even while keeping the total number of table entries the same) is desirable because it reduces the cost of the polynomial commitment opening proof, which is dominated by a factor linear in the number of rows, and barely depends on the number of columns. Increasing the number of columns also increases verifier time, as the verifier must perform one scalar-multiplication and one hash per column. Proof length is also affected by a larger number of columns, as more polynomials need to be committed and sent along to the verifier. There is typically some interplay between these","breadcrumbs":"Custom Constraints » Defining a PLONK configuration","id":"147","title":"Defining a PLONK configuration"},"148":{"body":"The goal of this section is to draw the connections between the practical aspect of designing custom gates and the theory behind it. First, suppose we had the following layout of gates. This means that our circuit has four gates: a (single) generic gate, a (non-optimized) complete add, and then two different Example gates that we just make up for the example; ExamplePairs and ExampleTriples. row GateType:: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 Generic l r o 1 CompleteAdd x1 y1 x2 y2 x3 y3 inf same_x s inf_z x21_inv 2 ExamplePairs a e i o u y 3 ExampleTriples a b c d e f g h i j k l m n o The most important aspect to take into account when you are designing a new custom gate is writing the actual description of the new functionality. For this, we are using the Expression framework that allows us to write equations generically by representing each column of the table (wire) as a variable. Then we can write operations between them and define the gate constraints . From a mathematical perspective, these will have the shape of multivariate polynomials of as many variables as columns in the table (currently 15 for Kimchi) that is constrained to equal 0. In our example, we would define equations that represent the operations carried out by each of the gates. In the following equations we use the variable names that we wrote in the cells above, and we don't imply that the same name in a different row correspond to the same value (we are not imposing equalities between gates yet): Generic: l ⋅cl​+ r ⋅cr​+ o ⋅co​+ l ⋅ r ⋅cm​+cc​=0 CompleteAdd: x21_inv ⋅( x2 − x1 )−(1− same_x)=0 same_x ⋅( x2 − x1 )=0 same_x ⋅(2⋅ s ⋅ y1 −3⋅ x12)+(1− same_x )⋅((x2 − x1 )⋅ s − y2 + y1 )=0 x1 + x2 + x3 − s2=0 s ⋅( x1 − x3 )− y1 − y3=0 ( y2 − y1 )⋅( same_x − inf )=0 ( y2 − y1 )⋅ inf_z − inf =0 ExamplePairs: a − e =0 i − o =0 u − y =0 ExampleTriples: a ⋅ b ⋅ c =0 d ⋅ e ⋅ f =0 g ⋅ h ⋅ i =0 j ⋅ k ⋅ l =0 m ⋅ n ⋅ o =0 Nonetheless, when we write these constraints in proof-systems, we use a shared framework that helps us use the columns in each row as placeholders. For example, the i-th column of the current row Curr is represented as: E::::cell(Column::Witness(i), CurrOrNext::Curr) where the E type is a shorthand for Expr. More succinctly, we refer to the above simply as: witness_curr(i) For readability, let's just refer to them here as w0..w14. Then, the above constraints would look like the following: Generic: w0 ⋅cl​+ w1 ⋅cr​+ w2 ⋅co​+ w0 ⋅ w1 ⋅cm​+cc​=0 CompleteAdd: w10 ⋅( w2 − w0 )−(1− w7)=0 w7 ⋅( w2 − w0 )=0 w7 ⋅(2⋅ w8 ⋅ w1 −3⋅ w02)+(1− w7 )⋅((w2 − w0 )⋅ w8 − w3 + w1 )=0 w0 + w2 + w4 − w82=0 w8 ⋅( w0 − w4 )− w1 − w4=0 ( w3 − w1 )⋅( w7 − w6 )=0 ( w3 − w1 )⋅ w9 − w6 =0 ExamplePairs: w0 − w1 =0 w2 − w3 =0 w4 − w5 =0 ExampleTriples: w0 ⋅ w1 ⋅ w2 =0 w3 ⋅ w4 ⋅ w5 =0 w6 ⋅ w7 ⋅ w8 =0 w9 ⋅ w10 ⋅ w11 =0 w12 ⋅ w13 ⋅ w14 =0 Looking at the list of equations, each of them look like polynomials in the variables w0..w14, constrained to zero. But so far, the type of equations we had studied were univariate (i.e. f(X) rather than f(X0​,...,X14​)). But this will not be a problem, since these multivariate equations will soon become univariate with a simple trick: interpolation . Right now, we are describing the gate constraints row-wise (i.e. horizontally ). But as we said before, our domain will be determined by the number of rows; we will transform these equations to a vertical and univariate behaviour. The strategy to follow will be obtaining witness polynomials that evaluate to the cells of the execution trace. First , note that so far we didn't mention any values for the w0..w14 terms, but instead they are placeholders for what is yet to come . And what is this you may ask: the witness of the relation . Put it another way, the witness will be the instantiation of the cells of the table that will make all of the above constraints hold. And where do we get these values from? Well, the prover will know or will compute them . Second , we count the number of rows that we ended up with: here it's four (conveniently, a power of two). What we want to do is to create as many witness polynomials as the number of columns in the execution trace. These polynomials, that we can call w0​(X),...,w14​(X), must be design to evaluate to the cells of the trace. Formally we define each wj​(X) as: G=:wj​(gi)=trace[i,j] Meaning that wi​(x) returns the i-th column of the x-th row of the execution trace. And how can we create such polynomials? We will use interpolation, which can be efficiently computed over power-of-two-sized groups, as it is the case in the example. Recalling Lagrangian terms , they act like linearly independent selectors over a set, in particular: Let G be a multiplicative group formed by the powers of a generator g. Given any element x in the group, the i-th Lagrangian term evaluates to 1 iff x=gi and 0 iff x=gi. ∀ x∈G:LgiG​(x)=1⟺x=θ and LgiG​(x)=0⟺x=θ Given that our group has size 4, the number of rows (ignoring ZK for now), G will be the set {1,g,g2,g3}. This means that we can build such 15 witness polynomials as: ∀j∈[0..15):wj​(X)=i=0∑3​trace[i,j]⋅LgiG​(X) Then, the above constraints will become the following equations using our 15 univariate witness polynomials: Generic: w0​(X)⋅cl​+w1​(X)⋅cr​+w2​(X)⋅co​+w0​(X)⋅w1​(X)⋅cm​+cc​=0 CompleteAdd: w10​(X)⋅(w2​(X)−w0​(X))−(1−w7​(X))=0 w7​(X)⋅(w2​(X)−w0​(X))=0 w7​(X)⋅(2⋅w8​(X)⋅w1​(X)−3⋅w0​(X)2)+(1−w7​(X))⋅((w2​(X)−w0​(X))⋅w8​(X)−w3​(X)+w1​(X))=0 w0​(X)+w2​(X)+w4​(X)−w8​(X)2=0 w8​(X)⋅(w0​(X)−w4​(X))−w1​(X)−w4​(X)=0 (w3​(X)−w1​(X))⋅(w7​(X)−w6​(X))=0 (w3​(X)−w1​(X))⋅w9​(X)−w6​(X)=0 ExamplePairs: w0​(X)−w1​(X)=0 w2​(X)−w3​(X)=0 w4​(X)−w5​(X)=0 ExampleTriples: w0​(X)⋅w1​(X)⋅w2​(X)=0 w3​(X)⋅w4​(X)⋅w5​(X)=0 w6​(X)⋅w7​(X)⋅w8​(X)=0 w9​(X)⋅w10​(X)⋅w11​(X)=0 w12​(X)⋅w13​(X)⋅w14​(X)=0 These are the 16 constraints that will need to be satisfied by a correct witness for the relation. We already know how to check many equations at once (combining them with powers of alpha), and how to check that the above holds for the full domain G (by providing the quotient polynomial of the huge equation after dividing by the vanishing polynomial on G). All there's left to do is making a correspondence between the constraints that need to be satisfied, and the particular row in the execution trace to which they apply. This implies, that before even aggregating the 16 constraints into a single one, we need to impose this restriction. In proof-systems, we represent this restriction with the GateType selector. In particular, we would multiply each of the constraints above by the Expr terms: Index(GateType::YourType), which will be transformed (again) into mutually exclusive selector polynomials. This means, that each row can only have one of these polynomials being nonzero (equal to 1), whereas the evaluation of all of the other selector polynomials for the given row should be 0. These polynomials are known by the verifier, who could check that this is indeed the case. But what will these polynomials look like? I am sure the above paragraph is familiar for you already. Once again, we will be able to build them using interpolation with our 4 Lagrangian terms for G. In our example: generic(X) will equal 1 when X=g0 and 0 otherwise; add(X) will equal 1 when X=g1 and 0 otherwise; pairs(X) will equal 1 when X=g2 and 0 otherwise; and triples(X) will equal 1 when X=g3 and 0 otherwise. These polynomials are, in fact, a description of the circuit. Note that we managed to make constraints for different gates already independent from each other (thanks to the mutually exclusivity). Nonetheless, granting independence within the gate is still needed. Here's where we will be using the powers of alpha: within a gate the powers cannot be repeated, but they can be reused among gates. Then, we will need 7 different powers of alpha for this stage of the protocol, as this is the largest number of different constraints within the same gate. Putting all of the above together, this means that our LONG and single constraint will look like: gates(X)= generic(X)⋅ α0⋅(w0​(X)⋅cl​+w1​(X)⋅cr​+w2​(X)⋅co​+w0​(X)⋅w1​(X)⋅cm​+cc​) +add(X)⋅( α0⋅(w10​(X)⋅(w2​(X)−w0​(X))−(1−w7​(X))) +α1⋅(w7​(X)⋅(w2​(X)−w0​(X))) +α2⋅(w7​(X)⋅(2⋅w8​(X)⋅w1​(X)−3⋅w0​(X)2)+(1−w7​(X))⋅((w2​(X)−w0​(X))⋅w8​(X)−w3​(X)+w1​(X))) +α3⋅(w0​(X)+w2​(X)+w4​(X)−w8​(X)2) +α4⋅(w8​(X)⋅(w0​(X)−w4​(X))−w1​(X)−w4​(X)) +α5⋅((w3​(X)−w1​(X))⋅(w7​(X)−w6​(X))) +α6⋅((w3​(X)−w1​(X))⋅w9​(X)−w6​(X))) +pairs(X)⋅( α0⋅(w0​(X)−w1​(X)) +α1⋅(w2​(X)−w3​(X)) +α2⋅(w4​(X)−w5​(X))) +triples(X)⋅( α0⋅(w0​(X)⋅w1​(X)⋅w2​(X)) +α1⋅(w3​(X)⋅w4​(X)⋅w5​(X)) +α2⋅(w6​(X)⋅w7​(X)⋅w8​(X)) +α3⋅(w9​(X)⋅w10​(X)⋅w11​(X)) +α4⋅(w12​(X)⋅w13​(X)⋅w14​(X))) Finally, providing q(X)=gates(X)/vG​(X) and performing the check on a random X∈F \\ G would give the verifier the assurance of the following: The prover knows a polynomial gates(X) that equals zero on any x∈{1,g,g2,g3}. Nonetheless, it would still remain to verify that gates(X) actually corresponds to the encoding of the actual constraints. Meaning, checking that this polynomial encodes the column witnesses, and the agreed circuit. So instead of providing just gates(X) (actually, a commitment to it), the prover can send commitments to each of th 15 witness polynomials, so that the verifier can reconstruct the huge constraint using the encodings of the circuit (which is known ahead).","breadcrumbs":"Custom Gates » Custom gates","id":"148","title":"Custom gates"},"149":{"body":"This document is meant to explain the foreign field addition gate in Kimchi.","breadcrumbs":"Custom Gates » Foreign Field Addition » Foreign Field Addition RFC","id":"149","title":"Foreign Field Addition RFC"},"15":{"body":"The next object to know about is polynomials. A polynomial is a syntactic object that also defines a function. Specifically, let R be a field (or more generally it could even be a ring, which is like a field but without necessarily having division, like the integers Z). And pick a variable name like x. Then R[x] (pronounced, \"R adjoin x\" or \"polynomials in x over R\" or \"polynomials in x with R coefficients\") is the set of syntactic expressions of the form a0​+a1​x+a2​x2+⋯+ad​xd where each ai​∈R and d is any natural number. If we wanted to be very formal about it, we could define R[x] to be the set of lists of elements of R (the lists of coefficients), and the above is just a suggestive way of writing that list. An important fact about polynomials over R is that they can be interpreted as functions R→R. In other words, there is a function eval:R[x]→(R→R) defined by eval(a0​+a1​x+⋯+ad​xd)=(y:R)↦a0​+a1​y+⋯+ad​yd where x is just a variable name without a value assigned to it and eval maps it to a field element y and evaluates the function as the inner product between the list of coefficients and the powers of y. It is important to remember that polynomials are different than the functions that they compute. Polynomials are really just lists of coefficients. Over some fields, there are polynomials p1 and p2 such that eval(p1)=eval(p2), but p1=p2. For example, consider in F2​[x] the polynomials x and x2. These map to the same function F2​→F2​ (meaning for x∈R=F2​={0,1} it holds that x=x2), but are distinct polynomials. Some definitions and notation If f is a polynomial in R[x] and a∈R, we write f(a) for eval(f)(a) If f is a polynomial, the degree of f is the largest d for which the coefficient of xd is non-zero in f. For example, the degree of x3+2x10 is 10. We will use the notation R[x]n. In order to deal with this, we will divide foreign field elements into limbs that fit in our native field. We want to be compatible with the foreign field multiplication gate, and thus the parameters we will be using are the following: 3 limbs of 88 bits each (for a total of 264 bits) So f<2264, concretely: The modulus f of the foreign field Ff​ is 256 bit The modulus of our the native field Fn​ is 255 bit In other words, using 3 limbs of 88 bits each allows us to represent any foreign field element in the range [0,2264) for foreign field addition, but only up to 2259 for foreign field multiplication. Thus, with the current configuration of our limbs, our foreign field must be smaller than 2259 (because 2264⋅2255>22592+2259, more on this in Foreign Field Multiplication or the original FFmul RFC .","breadcrumbs":"Custom Gates » Foreign Field Addition » Overview","id":"150","title":"Overview"},"151":{"body":"Let's take a closer look at what we have, if we split our variables in limbs (using little endian) bits 0..............87|88...........175|176...........263 a = (-------a0-------|-------a1-------|-------a2-------)\n+/-\nb = (-------b0-------|-------b1-------|-------b2-------)\n=\nr = (-------r0-------|-------r1-------|-------r2-------) mod(f) We will perform the addition in 3 steps, one per limb. Now, when we split long additions in this way, we must be careful with carry bits between limbs. Also, even if a and b are foreign field elements, it could be the case that a+b is already larger than the modulus (in this case a+b could be at most 2f−2). But it could also be the case that the subtraction produces an underflow because a > >\n= c_0 c_1 c_2\nq = -1 | 0 | 1\n·\nf = (-------f0-------|-------f1-------|-------f2-------)\n+\nr = (-------r0-------|-------r1-------|-------r2-------) Our witness computation is currently using the BigUint library, which takes care of all of the intermediate carry limbs itself so we do not have to address all casuistries ourselves (such as propagating the low carry flag to the high limb in case the middle limb is all zeros).","breadcrumbs":"Custom Gates » Foreign Field Addition » Splitting the addition","id":"151","title":"Splitting the addition"},"152":{"body":"Last but not least, we should perform some range checks to make sure that the result r is contained in Ff​. This is important because there could be other values of the result which still fit in <2264 but are larger than f, and we must make sure that the final result is the minimum one (that we will be referring to as rmin​ in the following). Ideally, we would like to reuse some gates that we already have. In particular, we can perform range checks for 0≤X<23ℓ=23⋅88. But we want to check that 0≤rmin​ >\n= k_0 k_1\nf\n+\nu = (-------u0-------|-------u1-------|-------u2-------) Following the steps above, and representing this equation in limb form, we have: u0​u1​u2​​=r0​+0−f0​−k0​⋅288=r1​+0−f1​−k1​⋅288+k0​=r2​+288−f2​+k1​​ But now we also have to check that 0≤r. But this is implicitly covered by r being a field element of at most 264 bits (range check). When we have a chain of additions ai​+bi​=ri​ with ai​=ri−1​, we could apply the field membership check naively to every intermediate ri​, however it is sufficient to apply it only once at the end of the computations for rn​, and keep intermediate ri​∈[0,2264), in a \"raw\" form. Generally speaking, treating intermediate values lazily helps to save a lot of computation in many different FF addition and multiplication scenarios.","breadcrumbs":"Custom Gates » Foreign Field Addition » Upper bound check for foreign field membership","id":"152","title":"Upper bound check for foreign field membership"},"153":{"body":"Mathematically speaking, a subtraction within a field is no more than an addition over that field. Negative elements are not different from \"positive\" elements in finite fields (or in any modular arithmetic). Our witness computation code computes negative sums by adding the modulus to the result. To give a general example, the element −e within a field Fm​ of order m and e > > c_0 c_1 0 Inspired by the halving approach in foreign field multiplication, an optimized version of the above gate results in a reduction by 2 in the number of constraints and by 1 in the number of witness cells needed. The main idea is to condense the claims about the low and middle limbs in one single larger limb of 176 bits, which fit in our native field. This way, we can get rid of the low carry flag, its corresponding carry check, and the three result checks become just two. bits 0..............87|88...........175|176...........263\nr = (-------r0------- -------r1-------|-------r2-------)\n=\na = (-------a0------- -------a1-------|-------a2-------)\n+\ns = 1 | -1\n·\nb = (-------b0------- -------b1-------|-------b2-------)\n-\nq = -1 | 0 | 1\n·\nf = (-------f0------- -------f1-------|-------f2-------) > > c 0 These are the new equations: rbot​rtop​​=(a0​+288⋅a1​)+s⋅(b0​+288⋅b1​)−q⋅(f0​+288⋅f1​)−c⋅2176=a2​+s⋅b2​−q⋅f2​+c​ with rtop​=r2​ and c=c1​.","breadcrumbs":"Custom Gates » Foreign Field Addition » Optimization","id":"154","title":"Optimization"},"155":{"body":"The full foreign field addition/subtraction gadget will be composed of: 1 public input row containing the value 1; n rows with ForeignFieldAdd gate type: for the actual n chained additions or subtractions; 1 ForeignFieldAdd row for the bound addition; 1 Zero row for the final bound addition. 1 RangeCheck gadget for the first left input of the chain a1​:=a; Then, n of the following set of rows: 1 RangeCheck gadget for the i-th right input bi​; 1 RangeCheck gadget for the i-th result which will correspond to the (i+1)-th left input of the chain ri​=ai+1​. 1 final RangeCheck gadget for the bound check u. A total of 20 rows with 15 columns in Kimchi for 1 addition. All ranges below are inclusive. Row(s) Gate type(s) Witness 0 PublicInput 1 1..n ForeignFieldAdd +/- n+1 ForeignFieldAdd bound n+2 Zero bound n+3..n+6 multi-range-check left n+7+8i..n+10+8i multi-range-check right n+11+8i..n+14+8i multi-range-check r 9n+7..9n+10 multi-range-check u This mechanism can chain foreign field additions together. Initially, there are n foreign field addition gates, followed by a foreign field addition gate for the bound addition (whose current row corresponds to the next row of the last foreign field addition gate), and an auxiliary Zero row that holds the upper bound. At the end, an initial left input range check is performed, which is followed by a n pairs of range check gates for the right input and intermediate result (which become the left input for the next iteration). After the chained inputs checks, a final range check on the bound takes place. For example, chaining the following set of 3 instructions would result in a full gadget with 37 rows: add(add(add(a,b),c),d) Row(s) Gate type(s) Witness 0 PublicInput 1 1 ForeignFieldAdd a+b 2 ForeignFieldAdd (a+b)+c 3 ForeignFieldAdd ((a+b)+c)+d 4 ForeignFieldAdd bound 5 Zero bound 6..9 multi-range-check a 10..13 multi-range-check b 14..17 multi-range-check a+b 18..21 multi-range-check c 22..25 multi-range-check a+b+c 26..29 multi-range-check d 30..33 multi-range-check a+b+c+d 34..37 multi-range-check bound Nonetheless, such an exhaustive set of checks are not necessary for completeness nor soundness. In particular, only the very final range check for the bound is required. Thus, a shorter gadget that is equally valid and takes (8⋅n+4) fewer rows could be possible if we can assume that the inputs of each addition are correct foreign field elements. It would follow the next layout (with inclusive ranges): Row(s) Gate type(s) Witness 0 public input row for soundness of bound overflow flag 1 1..n ForeignFieldAdd n+1 ForeignFieldAdd n+2 Zero n+3..n+6 multi-range-check for bound u Otherwise, we would need range checks for each new input of the chain, but none for intermediate results; implying 4⋅n fewer rows. Row(s) Gate type(s) Witness 0 public input row for soundness of bound overflow flag 1 1..n ForeignFieldAdd ai​+bi​ n+1 ForeignFieldAdd n+2 Zero n+3..n+6 multi-range-check for first left input a1​ n+7+4i..n+10+4i multi-range-check for i-th right input bi​ 5n+7..5n+10 multi-range-check for bound u For more details see the Bound Addition section in Foreign Field Multiplication or the original Foreign Field Multiplication RFC .","breadcrumbs":"Custom Gates » Foreign Field Addition » Gadget","id":"155","title":"Gadget"},"156":{"body":"For the full mode of tests of this gate, we need to perform 4 range checks to assert that the limbs of a,b,r,u have a correct size, meaning they fit in 288 (and thus, range-checking a,b,r,u for 2264). Because each of these elements is split into 3 limbs, we will have to use 3 copy constraints between the RangeCheck gates and the ForeignFieldAdd rows (per element). That amounts to 12 copy constraints. Recalling that Kimchi only allows for the first 7 columns of each row to host a copy constraint, we necessarily have to use 2 rows for the actual addition gate. The layout of these two rows is the following: Curr Next ... Final Column ForeignFieldAdd ForeignFieldAdd Zero 0 a0​ (copy) r0​ (copy) u0​ (copy) 1 a1​ (copy) r1​ (copy) u1​ (copy) 2 a2​ (copy) r2​ (copy) u2​ (copy) 3 b0​ (copy) 4 b1​ (copy) 5 b2​ (copy) 6 q 7 c 8 9 10 11 12 13 14","breadcrumbs":"Custom Gates » Foreign Field Addition » Layout","id":"156","title":"Layout"},"157":{"body":"So far, we have pointed out the following sets of constraints: Main addition 0=a0​+b0​−r0​−q⋅f0​−288⋅c0​ 0=a1​+b1​−r1​−q⋅f1​+c0​−288⋅c1​ 0=a2​+b2​−r2​−q⋅f2​+c1​ Carry checks 0=c0​⋅(c0​+1)⋅(c0​−1) 0=c1​⋅(c0​+1)⋅(c1​−1) Sign checks TODO: decide if we really want to keep this check or leave it implicit as it is a coefficient value 0=(s+1)⋅(s−1) Overflow checks 0=q⋅(q−s)","breadcrumbs":"Custom Gates » Foreign Field Addition » Constraints","id":"157","title":"Constraints"},"158":{"body":"When we use this gate as part of a larger chained gadget, we should optimize the number of range check rows to avoid redundant checks. In particular, if the result of an addition becomes one input of another addition, there is no need to check twice that the limbs of that term have the right length. The sign is now part of the coefficients of the gate. This will allow us to remove the sign check constraint and release one witness cell element. But more importantly, it brings soundness to the gate as it is now possible to wire the 1 public input to the overflow witness of the final bound check of every addition chain.","breadcrumbs":"Custom Gates » Foreign Field Addition » Optimizations","id":"158","title":"Optimizations"},"159":{"body":"This document is an original RFC explaining how we constrain foreign field multiplication (i.e. non-native field multiplication) in the Kimchi proof system. For a more recent RFC, see FFmul RFC . Changelog Author(s) Date Details Joseph Spadavecchia and Anais Querol October 2022 Design of foreign field multiplication in Kimchi","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Foreign Field Multiplication","id":"159","title":"Foreign Field Multiplication"},"16":{"body":"An important fact for zk-SNARKs about polynomials is that we can use them to encode arrays of field elements. In other words, there is a way of converting between polynomials of degree d and arrays of field elements of length d+1. This is important for a few reasons It will allow us to translate statements about arrays of field elements into statements about polynomials, which will let us construct zk-SNARKs. It will allow us to perform certain operations on polynomials more efficiently. So let's get to defining and proving this connection between arrays of field elements. The first thing to understand is that we won't be talking about arrays directly in this section. Instead, we'll be talking about functions A→F where A is a finite subset of F. The idea is, if A has size n, and we put an ordering on A, then we can think of a function A→F as the same thing as an immutable array of length n, since such a function can be thought of as returning the value of an array at the input position. With this understanding in hand, we can start describing the \"fundamental theorem of polynomials\". If A⊆F has size d+1, this theorem will define an isomorphism between functions A→F and F[x]≤d​, the set of polynomials of degree at most d. Now let's start defining this isomorphism. One direction of it is very easy to define. It is none other than the evaluation map, restricted to A: evalA​evalA​​:F[x]≤d​→(A→F)(c0​+⋯+cd​xd)=a↦i≤d∑​ci​ai​ We would now like to construct an inverse to this map. What would that mean? It would be a function that takes as input a function φ:A→F (remember, basically an array of length ∣A∣), and returns a polynomial f which agrees with φ on the set A. In other words, we want to construct a polynomial that interpolates between the points (a,φ(a)) for a∈A. Our strategy for constructing this polynomial will be straightforward. For each a∈A, we will construct a polynomial fa​ that is equal to φ(a) when evaluated at a, and equal to 0 when evaluated anywhere else in the set A. Then, our final polynomial will be f:=∑a∈A​fa​. Then, when f is evaluated at a0​∈A, only the fa0​​ term will be non-zero (and equal to φ(a0​) as desired), all the other terms will disappear. Constructing the interpolation map requires a lemma. Lemma. (construction of vanishing polynomials) Let S⊆F. Then there is a polynomial vS​∈F[x] of degree ∣S∣ such that vS​ evaluates to 0 on S, and is non-zero off of S. vS​ is called the vanishing polynomial of S. Proof. Let vS​=∏s∈S​(x−s). Clearly vS​ has degree ∣S∣. If t∈S, then vS​(t)=0 since x−t is one of the terms in the product defining vS​, so when we evaluate at t, t−t=0 is one of the terms. If t∈/S, then all the terms in the product are non-zero, and thus vS​(t) is non-zero as well. □ Now we can define the inverse map. Define interpA​interpA​​:(A→F)→F[x]≤d​(f)=a∈A∑​vA∖{a}​(a)f(a)​⋅vA∖{a}​​ Since each vA∖{a}​ has degree d, this polynomial has degree d. Now we have, for any b∈A, evalA​(interpA​(f))(b)​=a∈A∑​vA∖{a}​(a)f(a)​⋅vA∖{a}​(b)=vA∖{b}​(b)f(b)​⋅vA∖{b}​(b)+a=b∑​vA∖{a}​(a)f(a)​⋅vA∖{a}​(b)=f(b)+a=b∑​vA∖{a}​(a)f(a)​⋅0=f(b)​ Thus evalA​∘interpA​ is the identity. So we have successfully devised a way of interpolating a set of d+1 points with a polynomial of degree d. What remains to be seen is that these two maps are inverse in the other direction. That is, that for any polynomial f∈F[x], we have interpA​(evalA​(f))=f This says that if we interpolate the function that has the values of f on A, we get back f itself. This essentially says that there is only one way to interpolate a set of d+1 points with a degree d polynomial. So let's prove it by proving that statement. Lemma: polynomials that agree on enough points are equal. Let F be a field and suppose f,g∈F[x] have degree at most d. Let A⊆F have size d+1. Suppose for all a∈A, f(a)=g(a). Then f=g as polynomials. Proof. Define h:=f−g. Our strategy will be to show that h is the zero polynomial. This would then imply that f=g as polynomials. Note that h vanishes on all of A since f and g are equal on A. To that end, let a∈A. Then we can apply the polynomial division algorithm to divide h by x−a and obtain polynomials qa​,ra​ such that h=qa​(x−a)+ra​ and ra​ has degree less than 1. I.e., ra​ is a constant in F. Now, 0=h(a)=qa​(a)⋅0+ra​=ra​ so ra​=0 and thus h=qa​(x−a). Note that qa​ is 0 on all elements b∈A with b=a since h(b)=qa​(b)(b−a)=0, but b−a=0. Thus, if we iterate the above, enumerating the elements of A as a0​,…,ad​, we find that h​=(x−a0​)qa0​​=(x−a0​)(x−a1​)qa1​​=…=(x−a0​)(x−a1​)…(x−ad​)qad​​​ Now, if qad​​ is not the zero polynomial, then h will have degree at least d+1 since it would have as a factor the d+1 linear terms (x−ai​). But since f,g both have degree at most d and h=f−g, h has degree at most d as well. Thus, qad​​=0, which means h=0 as well. This gives us the desired conclusion, that f=g as polynomials. □ Now we can easily show that interpolating the evaluations of a polynomial yields that polynomial itself. Let f∈F[x]≤d​. Then interpA​(evalA​(f)) is a polynomial of degree at most d that agrees with f on A, a set of size d+1. Thus, by the lemma, they are equal as polynomials. So indeed interpA​(evalA​(f))=f for all f∈F[x]. So far we have proven that interpA​ and evalA​ give an isomorphism of sets (i.e., a bijection) between the sets A→F and F[x]≤d​. But we can take this theorem a bit further. The set of functions A→F can be given the structure of a ring, where addition and multiplication happen pointwise. I.e., for f,g:A→F we define f+g:=a↦f(a)+g(a) and f⋅g:=a↦f(a)⋅g(a). Then we can strengthen our theorem to say Fundamental theorem of polynomials (final version) Let d∈N and let A⊆F with ∣A∣=d+1. With evalA​:F[x]≤d​→(A→F)interpA​:(A→F)→F[x]≤d​ defined as above, these two functions define an isomorphism of rings. That is is, they are mutually inverse and each one respects addition, subtraction and multiplication. The fundamental theorem of polynomials is very important when it comes to computing operations on polynomials. As we will see in the next section , the theorem will help us to compute the product of degree n polynomials in time O(nlogn), whereas the naive algorithm takes time O(n2). To put this in perspective, if n=216, n2 is 4096 times larger than nlogn and the gap only gets bigger as n grows.","breadcrumbs":"Polynomials » Fundamental theorem of polynomials","id":"16","title":"Fundamental theorem of polynomials"},"160":{"body":"This gate constrains a⋅b=cmodf where a,b,c∈Ff​, a foreign field with modulus f, using the native field Fn​ with prime modulus n.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Overview","id":"160","title":"Overview"},"161":{"body":"In foreign field multiplication the foreign field modulus f could be bigger or smaller than the native field modulus n. When the foreign field modulus is bigger, then we need to emulate foreign field multiplication by splitting the foreign field elements up into limbs that fit into the native field element size. When the foreign modulus is smaller everything can be constrained either in the native field or split up into limbs. Since our projected use cases are when the foreign field modulus is bigger (more on this below) we optimize our design around this scenario. For this case, not only must we split foreign field elements up into limbs that fit within the native field, but we must also operate in a space bigger than the foreign field. This is because we check the multiplication of two foreign field elements by constraining its quotient and remainder. That is, renaming c to r, we must constrain a⋅b=q⋅f+r, where the maximum size of q and r is f−1 and so we have a⋅b​≤q(f−1)​​⋅f+r(f−1)​​≤f2−1.​ Thus, the maximum size of the multiplication is quadratic in the size of foreign field. Naïve approach Naïvely, this implies that we have elements of size f2−1 that must split them up into limbs of size at most n−1. For example, if the foreign field modulus is 256 and the native field modulus is 255 bits, then we'd need log2​((2256)2−1)≈512 bits and, thus, require 512/255≈3 native limbs. However, each limb cannot consume all 255 bits of the native field element because we need space to perform arithmetic on the limbs themselves while constraining the foreign field multiplication. Therefore, we need to choose a limb size that leaves space for performing these computations. Later in this document (see the section entitled \"Choosing the limb configuration\") we determine the optimal number of limbs that reduces the number of rows and gates required to constrain foreign field multiplication. This results in ℓ=88 bits as our optimal limb size. In the section about intermediate products we place some upperbounds on the number of bits required when constraining foreign field multiplication with limbs of size ℓ thereby proving that the computations can fit within the native field size. Observe that by combining the naïve approach above with a limb size of 88 bits, we would require 512/88≈6 limbs for representing foreign field elements. Each limb is stored in a witness cell (a native field element). However, since each limb is necessarily smaller than the native field element size, it must be copied to the range-check gate to constrain its value. Since Kimchi only supports 7 copyable witness cells per row, this means that only one foreign field element can be stored per row. This means a single foreign field multiplication would consume at least 4 rows (just for the operands, quotient and remainder). This is not ideal because we want to limit the number of rows for improved performance. Chinese remainder theorem Fortunately, we can do much better than this, however, by leveraging the chinese remainder theorem (CRT for short) as we will now show. The idea is to reduce the number of bits required by constraining our multiplication modulo two coprime moduli: 2t and n. By constraining the multiplication with both moduli the CRT guarantees that the constraints hold with the bigger modulus 2t⋅n. The advantage of this approach is that constraining with the native modulus n is very fast, allowing us to speed up our non-native computations. This practically reduces the costs to constraining with limbs over 2t, where t is something much smaller than 512. For this to work, we must select a value for t that is big enough. That is, we select t such that 2t⋅n>f2−1. As described later on in this document, by doing so we reduce the number of bits required for our use cases to 264. With 88 bit limbs this means that each foreign field element only consumes 3 witness elements, and that means the foreign field multiplication gate now only consumes 2 rows. The section entitled \"Choosing t\" describes this in more detail. Overall approach Bringing it all together, our approach is to verify that a⋅b=q⋅f+r​​ over Z+. In order to do this efficiently we use the CRT, which means that the equation holds mod M=2t⋅n. For the equation to hold over the integers we must also check that each side of the equation is less than 2t⋅n. The overall steps are therefore Apply CRT to equation (1) Check validity with binary modulus mod2t Check validity with native modulus modn Check each side of equation (1) is less than M a⋅b<2t⋅n q⋅f+r<2t⋅n This then implies that a⋅b=cmodf. where c=r. That is, a multiplied with b is equal to c where a,b,c∈Ff​. There is a lot more to each of these steps. That is the subject of the rest of this document. Other strategies Within our overall approach, aside from the CRT, we also use a number of other strategies to reduce the number and degree of constraints. Avoiding borrows and carries Intermediate product elimination Combining multiplications The rest of this document describes each part in detail.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Approach","id":"161","title":"Approach"},"162":{"body":"This section describes important parameters that we require and how they are computed. Native field modulus n Foreign field modulus f Binary modulus 2t CRT modulus 2t⋅n Limb size in bits ℓ Choosing t Under the hood, we constrain a⋅b=q⋅f+rmod2t⋅n. Since this is a multiplication in the foreign field f, the maximum size of q and r are f−1, so this means a⋅b​≤(f−1)⋅f+(f−1)≤f2−1.​ Therefore, we need the modulus 2t⋅n such that 2t⋅n>f2−1, which is the same as saying, given f and n, we must select t such that 2t⋅nt​≥f2≥2log2​(f)−log2​(n).​ Thus, we have an lower bound on t. Instead of dynamically selecting t for every n and f combination, we fix a t that will work for the different selections of n and f relevant to our use cases. To guide us, from above we know that tmin​=2log2​(f)−log2​(n) and we know the field moduli for our immediate use cases. vesta = 2^254 + 45560315531506369815346746415080538113 (255 bits)\npallas = 2^254 + 45560315531419706090280762371685220353 (255 bits)\nsecp256k1 = 2^256 - 2^32 - 2^9 - 2^8 - 2^7 - 2^6 - 2^4 - 1 (256 bits) So we can create a table n f tmin​ vesta secp256k1 258 pallas secp256k1 258 vesta pallas 255 pallas vesta 255 and know that to cover our use cases we need t≥258. Next, given our native modulus n and t, we can compute the maximum foreign field modulus supported . Actually, we compute the maximum supported bit length of the foreign field modulus f=2m. 2t⋅nt+log2​(n)​≥f2≥(2m)2=22m>log2​(22m)=2m​ So we get mmax​=2t+log2​(n)​. With t=258,n=255 we have mmax​​=2258+255​=256.5,​ which is not enough space to handle anything larger than 256 bit moduli. Instead, we will use t=264, giving mmax​=259 bits. The above formula is useful for checking the maximum number of bits supported of the foreign field modulus, but it is not useful for computing the maximum foreign field modulus itself (because 2mmax​ is too coarse). For these checks, we can compute our maximum foreign field modulus more precisely with maxmod​=⌊2t⋅n​⌋. The max prime foreign field modulus satisfying the above inequality for both Pallas and Vesta is 926336713898529563388567880069503262826888842373627227613104999999999999999607. Choosing the limb configuration Choosing the right limb size and the right number of limbs is a careful balance between the number of constraints (i.e. the polynomial degree) and the witness length (i.e. the number of rows). Because one limiting factor that we have in Kimchi is the 12-bit maximum for range check lookups, we could be tempted to introduce 12-bit limbs. However, this would mean having many more limbs, which would consume more witness elements and require significantly more rows. It would also increase the polynomial degree by increasing the number of constraints required for the intermediate products (more on this later). We need to find a balance between the number of limbs and the size of the limbs. The limb configuration is dependent on the value of t and our maximum foreign modulus (as described in the previous section). The larger the maximum foreign modulus, the more witness rows we will require to constrain the computation. In particular, each limb needs to be constrained by the range check gate and, thus, must be in a copyable (i.e. permuteable) witness cell. We have at most 7 copyable cells per row and gates can operate on at most 2 rows, meaning that we have an upperbound of at most 14 limbs per gate (or 7 limbs per row). As stated above, we want the foreign field modulus to fit in as few rows as possible and we need to constrain operands a,b, the quotient q and remainder r. Each of these will require cells for each limb. Thus, the number of cells required for these is cells=4⋅limbs It is highly advantageous for performance to constrain foreign field multiplication with the minimal number of gates. This not only helps limit the number of rows, but also to keep the gate selector polynomial small. Because of all of this, we aim to constrain foreign field multiplication with a single gate (spanning at most 2 rows). As mentioned above, we have a maximum of 14 permuteable cells per gate, so we can compute the maximum number of limbs that fit within a single gate like this. limbsmax​​=⌊cells/4⌋=⌊14/4⌋=3​ Thus, the maximum number of limbs possible in a single gate configuration is 3. Using limbsmax​=3 and t=264 that covers our use cases (see the previous section), we are finally able to derive our limbs size ℓ​=limbsmax​t​=264/3=88​ Therefore, our limb configuration is: Limb size ℓ=88 bits Number of limbs is 3","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Parameter selection","id":"162","title":"Parameter selection"},"163":{"body":"When we constrain a⋅b−q⋅f=rmod2t we want to be as efficient as possible. Observe that the expansion of a⋅b−q⋅f into limbs would also have subtractions between limbs, requiring our constraints to account for borrowing. Dealing with this would create undesired complexity and increase the degree of our constraints. In order to avoid the possibility of subtractions we instead use a⋅b+q⋅f′ where f′​=−fmod2t=2t−f​ The negated modulus f′ becomes part of our gate coefficients and is not constrained because it is publicly auditable. Using the substitution of the negated modulus, we now must constrain a⋅b+q⋅f′=rmod2t. Observe that f′<2t since f<2t and that f′>f when f<2t−1.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Avoiding borrows","id":"163","title":"Avoiding borrows"},"164":{"body":"This section explains how we expand our constraints into limbs and then eliminate a number of extra terms. We must constrain a⋅b+q⋅f′=rmod2t on the limbs, rather than as a whole. As described above, each foreign field element x is split into three 88-bit limbs: x0​,x1​,x2​, where x0​ contains the least significant bits and x2​ contains the most significant bits and so on. Expanding the right-hand side into limbs we have ​(a0​+a1​⋅2ℓ+a2​⋅22ℓ)⋅(b0​+b1​⋅2ℓ+b2​⋅22ℓ)+(q0​+q1​⋅2ℓ+q2​⋅22ℓ)⋅(f0′​+f1′​⋅2ℓ+f2′​⋅22ℓ)= a0​⋅b0​+a0​⋅b1​⋅2ℓ+a0​⋅b2​⋅22ℓ +a1​⋅b0​⋅2ℓ+a1​⋅b1​⋅22ℓ+a1​⋅b2​⋅23ℓ +a2​⋅b0​⋅22ℓ+a2​⋅b1​⋅23ℓ+a2​⋅b2​⋅24ℓ+ q0​⋅f0′​+q0​⋅f1′​⋅2ℓ+q0​⋅f2′​⋅22ℓ +q1​⋅f0′​⋅2ℓ+q1​⋅f1′​⋅22ℓ+q1​⋅f2′​⋅23ℓ +q2​⋅f0′​⋅22ℓ+q2​⋅f1′​⋅23ℓ+q2​⋅f2′​⋅24ℓ=a0​⋅b0​+q0​⋅f0′​+2ℓ⋅(a0​⋅b1​+a1​⋅b0​+q0​⋅f1′​+q1​⋅f0′​)+22ℓ⋅(a0​⋅b2​+a2​⋅b0​+q0​⋅f2′​+q2​⋅f0′​+a1​⋅b1​+q1​⋅f1′​)+23ℓ⋅(a1​⋅b2​+a2​⋅b1​+q1​⋅f2′​+q2​⋅f1′​)+24ℓ⋅(a2​⋅b2​+q2​⋅f2′​)​ Since t=3ℓ, the terms scaled by 23ℓ and 24ℓ are a multiple of the binary modulus and, thus, congruent to zero mod2t. They can be eliminated and we don't need to compute them. So we are left with 3 intermediate products that we call p0​,p1​,p2​: Term Scale Product p0​ 1 a0​b0​+q0​f0′​ p1​ 2ℓ a0​b1​+a1​b0​+q0​f1′​+q1​f0′​ p2​ 22ℓ a0​b2​+a2​b0​+q0​f2′​+q2​f0′​+a1​b1​+q1​f1′​ So far, we have introduced these checked computations to our constraints Computation of p0​,p1​,p2​","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Intermediate products","id":"164","title":"Intermediate products"},"165":{"body":"Let's call p:=ab+qf′mod2t. Remember that our goal is to constrain that p−r=0mod2t (recall that any more significant bits than the 264th are ignored in mod2t). Decomposing that claim into limbs, that means 22ℓ(p2​−r2​)+2ℓ(p1​−r1​)+p0​−r0​=0mod2t.​​ We face two challenges Since p0​,p1​,p2​ are at least 2ℓ bits each, the right side of the equation above does not fit in Fn​ The subtraction of the remainder's limbs r0​ and r1​ could require borrowing For the moment, let's not worry about the possibility of borrows and instead focus on the first problem.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Constraining mod2t","id":"165","title":"Constraining mod2t"},"166":{"body":"The first problem is that our native field is too small to constrain 22ℓ(p2​−r2​)+2ℓ(p1​−r1​)+p0​−r0​=0mod2t. We could split this up by multiplying a⋅b and q⋅f′ separately and create constraints that carefully track borrows and carries between limbs. However, a more efficient approach is combined the whole computation together and accumulate all the carries and borrows in order to reduce their number. The trick is to assume a space large enough to hold the computation, view the outcome in binary and then split it up into chunks that fit in the native modulus. To this end, it helps to know how many bits these intermediate products require. On the left side of the equation, p0​ is at most 2ℓ+1 bits. We can compute this by substituting the maximum possible binary values (all bits set to 1) into p0​=a0​b0​+q0​f0′​ like this maxbits(p0​)​=log2​(a0​(2ℓ−1)​​b0​(2ℓ−1)​​+q0​(2ℓ−1)​​f0′​(2ℓ−1)​​)=log2​(2(22ℓ−2ℓ+1+1))=log2​(22ℓ+1−2ℓ+2+2).​ So p0​ fits in 2ℓ+1 bits. Similarly, p1​ needs at most 2ℓ+2 bits and p2​ is at most 2ℓ+3 bits. Term p0​ p1​ p2​ Bits 2ℓ+1 2ℓ+2 2ℓ+3 The diagram below shows the right hand side of the zero-sum equality from equation (2). That is, the value p−r. Let's look at how the different bits of p0​,p1​,p2​,r0​,r1​ and r2​ impact it. 0 L 2L 3L 4L\n|-------------|-------------|-------------|-------------|-------------| :\n|--------------p0-----------:-| 2L + 1 : |-------------:-p1------------| 2L + 2 p10➚ : p11➚ |----------------p2-------------| 2L + 3 :\n|-----r0------| : : |-----r1------| : |-----r2------|\n\\__________________________/ \\______________________________/ ≈ h0 ≈ h1 Within our native field modulus we can fit up to 2ℓ+δfk quickly overflows our CRT modulus 2t⋅n. For example, assuming our maximum foreign modulus of f=2259 and either of Kimchi's native moduli (i.e. Pallas or Vesta), fk>2t⋅n for k>2. That is, an overflow is possible for a chain of greater than 1 foreign field multiplication. Thus, we must check a,b,q and r are less than f for each multiplication. Fortunately, in many situations the input operands may already be checked either as inputs or as outputs of previous operations, so they may not be required for each multiplication operation. Thus, the q′ and r′ checks (or equivalently q and r) are our main focus because they must be done for every multiplication. Compute bound q′=q+f′ with addition gate (2 rows) Compute bound r′=r+f′ with addition gate (2 rows) Range check q′<2t (4 rows) Range check r′<2t (4 rows) This costs 12 rows per multiplication. In a subsequent section, we will reduce it to 8 rows.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Cost of bound checks","id":"174","title":"Cost of bound checks"},"175":{"body":"Due to the limited number of permutable cells per gate, we do not have enough cells for copy constraining q′ and r′ (or q and r) to their respective range check gadgets. To address this issue, we must decompose q′ into 2 limbs instead of 3, like so q′=q01′​+22ℓ⋅q2′​ and q01′​=q0′​+2ℓ⋅q1′​ Thus, q′ is decomposed into two limbs q01′​ (at most 2ℓ bits) and q2′​ (at most ℓ bits). Note that q′ must be range checked by a multi-range-check gadget. To do this the multi-range-check gadget must Store a copy of the limbs q0′​,q1′​ and q2′​ in its witness Range check that they are ℓ bit each Constrain that q01′​=q0′​+2ℓ⋅q1′​ (this is done with a special mode of the multi-range-check gadget) Have copy constraints for q01′​ and q2′​ as outputs of the ForeignFieldMul gate and inputs to the multi-range-check gadget Note that since the foreign field multiplication gate computes q′ from q which is already in the witness and q01′​ and q2′​ have copy constraints to a multi-range-check gadget that fully constrains their decomposition from q′, then the ForeignFieldMul gate does not need to store an additional copy of q0′​ and q1′​.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 2-limb decomposition","id":"175","title":"2-limb decomposition"},"176":{"body":"Since the q2ℓ where 2ℓ−1 is the maximum possible size of x1​ (before it overflows) and ϕ0​ is the overflow bit from the addition of the least significant limbs x0​ and f0′​. This means >2ℓ−ϕ0​−f1′​ We cannot allow x to overflow the foreign field, so we also have >x1​<(f−x0​)/22ℓ> Thus, >2ℓ−ϕ0​−f1′​<(f−x0​)/22ℓ=f/22ℓ−x0​/22ℓ> Since x0​/22ℓ=ϕ0​ we have >2ℓ−ϕ0​−f1′​ so >2ℓ−f1′​ Notice that f/22ℓ=f1​. Now we have >2ℓ−f1′​⟺>f1′​>2ℓ−f1​> However, this is a contradiction with the definition of our negated foreign field modulus limb f1′​=2ℓ−f1​. ■ We have proven that qcarry2′​ is always zero, so that allows use to simplify our constraints. We now have q0​∈[0,2ℓ) multi-range-check q1​∈[0,2ℓ) multi-range-check q01′​=q0′​+2ℓ⋅q1′​ multi-range-check qcarry01′​∈[0,2) ForeignFieldMul 22ℓ⋅qcarry01′​=s01​−q01′​ ForeignFieldMul q2​∈[0,2ℓ) multi-range-check q2′​=s2​+w01​ ForeignFieldMul In other words, we have eliminated constraint (7) and removed qcarry2′​ from the witness. Since we already needed to range-check q or q′, the total number of new constraints added is 4: 3 added to ForeignFieldMul and 1 added to multi-range-check gadget for constraining the decomposition of q01′​. This saves 2 rows per multiplication.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » An optimization","id":"176","title":"An optimization"},"177":{"body":"Due to the limited number of cells accessible to gates, we are not able to chain multiplications into multiplications. We can chain foreign field additions into foreign field multiplications, but currently do not support chaining multiplications into additions (though there is a way to do it).","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Chaining multiplications","id":"177","title":"Chaining multiplications"},"178":{"body":"Now we collect all of the checks that are required to constrain foreign field multiplication","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Constraining the computation","id":"178","title":"Constraining the computation"},"179":{"body":"If the a operand has not been constrained to [0,f) by any previous foreign field operations, then we constrain it like this Compute bound a′=a+f′ with addition gate (2 rows) ForeignFieldAdd Range check a′∈[0,2t) (4 rows) multi-range-check","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 1. Range constrain a","id":"179","title":"1. Range constrain a"},"18":{"body":"There are three common representations for polynomials used in computer implementations. Dense coefficient form. A degree d polynomial is represented as a vector of length d+1 of all the coefficients. Entry i of the vector corresponds to the coefficient ai​. This corresponds to the DensePolynomial type in arkworks. Sometimes this is also described as writing the polynomial \"in the monomial basis\", because it amounts to writing the polynomial as a linear combination of the monomials xi. Sparse coefficient form. If a polynomial does not have very many non-zero coefficients, the above representation is wasteful. In sparse coefficient form, you represent a polynomial as a vector (or potentially a hash-map) of pairs (usize, F) where F is the type of coefficients. The polynomial corresponding to the list [(i_0, b_0), ..., (i_n, b_n)] is b0​xi0​+⋯+bn​xin​ Evaluation form. We fix an index set A⊆F, with A={a0​,…,ad​}, and represent a polynomial f∈F[x]≤d​ as the vector [f(a_0), ..., f(a_d)]. By the fundamental theorem of polynomials, this is a valid way of representing the polynomials, since the coefficients form can be obtained by using the interpA​ function. The evaluation form is very important. The reason is that multiplying two polynomials in the evaluation form takes time O(n). You just multiply the two vectors entry-wise. By contrast, the coefficient forms naively require time O(n2) to multiply. Now, there is a trick. For certain sets A⊆F, we can efficiently translate between the dense coefficients form and the evaluation form. That is, for certain A, the functions interpA​ and evalA​ can be computed more efficiently than O(n2).","breadcrumbs":"Polynomials » Computer representation","id":"18","title":"Computer representation"},"180":{"body":"If the b operand has not been constrained to [0,f) by any previous foreign field operations, then we constrain it like this Compute bound b′=b+f′ with addition gate (2 rows) ForeignFieldAdd Range check b′∈[0,2t) (4 rows) multi-range-check","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 2. Range constrain b","id":"180","title":"2. Range constrain b"},"181":{"body":"The quotient q is constrained to [0,f) for each multiplication as part of the multiplication gate Compute bound q′=q+f′ with ForeignFieldMul constraints Range check q′∈[0,2t) (4 rows) multi-range-check Range check q≥0 ForeignFieldMul (implicit by storing q in witness)","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 3. Range constrain q","id":"181","title":"3. Range constrain q"},"182":{"body":"The remainder r is constrained to [0,f) for each multiplication using an external addition gate. Compute bound r′=r+f′ with addition gate (2 rows) ForeignFieldAdd Range check r′∈[0,2t) (4 rows) multi-range-check","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 4. Range constrain r","id":"182","title":"4. Range constrain r"},"183":{"body":"Compute and constrain the intermediate products p0​,p1​ and p2​ as: p0​=a0​⋅b0​+q0​⋅f0′​ ForeignFieldMul p1​=a0​⋅b1​+a1​⋅b0​+q0​⋅f1′​+q1​⋅f0′​ ForeignFieldMul p2​=a0​⋅b2​+a2​⋅b0​+a1​⋅b1​+q0​⋅f2′​+q2​⋅f0′​+q1​⋅f1′​ ForeignFieldMul where each of them is about 2ℓ-length elements.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 5. Compute intermediate products","id":"183","title":"5. Compute intermediate products"},"184":{"body":"Compute and constrain the native modulus values, which are used to check the constraints modulo n in order to apply the CRT an​=22ℓ⋅a2​+2ℓ⋅a1​+a0​modn bn​=22ℓ⋅b2​+2ℓ⋅b1​+b0​modn qn​=22ℓ⋅q2​+2ℓ⋅q1​+q0​modn rn​=22ℓ⋅r2​+2ℓ⋅r1​+r0​modn fn​=22ℓ⋅f2​+2ℓ⋅f1​+f0​modn","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 6. Native modulus checked computations","id":"184","title":"6. Native modulus checked computations"},"185":{"body":"Check that p1​=2ℓ⋅p11​+p10​: p1​=2ℓ⋅p11​+p10​ ForeignFieldMul Range check p10​∈[0,2ℓ) multi-range-check Range check p11​∈[0,2ℓ+2) p11​=p111​⋅2ℓ+p110​ ForeignFieldMul Range check p110​∈[0,2ℓ) multi-range-check Range check p111​∈[0,22) with a degree-4 constraint ForeignFieldMul","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 7. Decompose middle intermediate product","id":"185","title":"7. Decompose middle intermediate product"},"186":{"body":"Now we have to constrain the zero sum (p0​−r0​)+288(p1​−r1​)+2176(p2​−r2​)=0mod2t We constrain the first and the second halves as v0​⋅22ℓ=p0​+2ℓ⋅p10​−r0​−2ℓ⋅r1​ ForeignFieldMul v1​⋅2ℓ=(p111​⋅2ℓ+p110​)+p2​−r2​+v0​ ForeignFieldMul And some more range checks Check that v0​∈[0,22) with a degree-4 constraint ForeignFieldMul Check that v1​∈[0,2ℓ+3) Check v1​=v11​⋅288+v10​ ForeignFieldMul Check v11​∈[0,23] ForeignFieldMul Check v10​<2ℓ with range constraint multi-range-check To check that v11​∈[0,23) (i.e. that v11​ is at most 3 bits long) we first range-check v11​∈[0,212) with a 12-bit plookup. This means there can be no higher bits set beyond the 12-bits of v11​. Next, we scale v11​ by 29 in order to move the highest 12−3=9 bits beyond the 12th bit. Finally, we perform a 12-bit plookup on the resulting value. That is, we have Check v11​∈[0,212) with a 12-bit plookup (to prevent any overflow) Check scaledv11​​=29⋅v11​ Check scaledv11​​ is a 12-bit value with a 12-bit plookup Kimchi's plookup implementation is extremely flexible and supports optional scaling of the lookup target value as part of the lookup operation. Thus, we do not require two witness elements, two lookup columns, nor the scaledv11​​=29⋅v11​ custom constraint. Instead we can just store v11​ in the witness and define this column as a \"joint lookup\" comprised of one 12-bit plookup on the original cell value and another 12-bit plookup on the cell value scaled by 29, thus, yielding a 3-bit check. This eliminates one plookup column and reduces the total number of constraints.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 8. Zero sum for multiplication","id":"186","title":"8. Zero sum for multiplication"},"187":{"body":"Using the checked native modulus computations we constrain that an​⋅bn​−qn​⋅fn​−rn​=0modn.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 9. Native modulus constraint","id":"187","title":"9. Native modulus constraint"},"188":{"body":"Compute and constrain the intermediate sums s01​ and s2​ as: s01​=q01​+f01′​ s2​=q2​+f2′​ q01​=q0​+2ℓ⋅q1​ f01′​=f0′​+2ℓ⋅f1′​","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 10. Compute intermediate sums","id":"188","title":"10. Compute intermediate sums"},"189":{"body":"Check that q01′​=q0′​+2ℓ⋅q1′​. Done by (3) above with the multi-range-check on q′ q01′​=q0′​+2ℓ⋅q1′​ Range check q0′​∈[0,2ℓ) Range check q1′​∈[0,2ℓ)","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 11. Decompose the lower quotient bound","id":"189","title":"11. Decompose the lower quotient bound"},"19":{"body":"The algorithm that allows us to multiply polynomials in O(nlogn) is called the Cooley-Tukey fast Fourier transform , or FFT for short. It was discovered by Gauss 160 years earlier, but then separately rediscovered and publicized by Cooley-Tukey. The heart of Cooley-Tukey FFT is actually about converting between coefficient and evaluation representations, rather than the multiplication itself. Given polynomials p and q in dense coefficient representation, it works like this. Convert p and q from coefficient to evaluation form in O(nlogn) using Cooley-Tukey FFT Compute r=p∗q in evaluation form by multiplying their points pairwise in O(n) Convert r back to coefficient form in O(nlogn) using the inverse Cooley-Tukey FFT The key observation is that we can choose any n distinct evaluation points to represent any degree n−1 polynomial. The Cooley-Tukey FFT works by selecting points that yield an efficient FFT algorithm. These points are fixed and work for any polynomials of a given degree. The next section describes the Cooley-Tukey FFT in detail.","breadcrumbs":"Polynomials » Multiplying Polynomials » Multiplying polynomials","id":"19","title":"Multiplying polynomials"},"190":{"body":"We constrain that s01​−q01′​+22ℓ⋅(s2​−q2′​)=0mod2t by constraining the two halves 22ℓ⋅qcarry01′​=s01​−q01′​ 2ℓ⋅qcarry2′​=s2​+qcarry01′​−q2′​ We also need a couple of small range checks Check that qcarry01′​ is boolean ForeignFieldMul Check that q2′​ is boolean ForeignFieldMul","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 12. Zero sum for quotient bound addition","id":"190","title":"12. Zero sum for quotient bound addition"},"191":{"body":"Based on the constraints above, we need the following 12 values copied from the range check gates. a0, a1, a2, b0, b1, b2, q0, q1, q2, r0, r1, r2 Since we need 12 copied values for the constraints, they must span 2 rows. The q>s01​>>>​=q01​+f01′​=q0​+2ℓ⋅q1​+f0′​+2ℓ⋅f1′​=q0​+f0′​+2ℓ⋅(q1′​+f1′​)=q0′​+2ℓ⋅q1′​>​> where q0′​=q0​+f0′​ and q1′​=q1​+f1′​ can be done with checked computations. Next, for applying the CRT we compute an​=22ℓ⋅a2​+2ℓ⋅a1​+a0​modn bn​=22ℓ⋅b2​+2ℓ⋅b1​+b0​modn qn​=22ℓ⋅q2​+2ℓ⋅q1​+q0​modn rn​=22ℓ⋅r2​+2ℓ⋅r1​+r0​modn fn​=22ℓ⋅f2​+2ℓ⋅f1​+f0​modn","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Checked computations","id":"192","title":"Checked computations"},"193":{"body":"In total we require the following checks p111​∈[0,22) v10​∈[0,2ℓ) multi-range-check p110​∈[0,2ℓ) multi-range-check p10​∈[0,2ℓ) multi-range-check p11​=2ℓ⋅p111​+p110​ p1​=2ℓ⋅p11​+p10​ v0​∈[0,22) 22ℓ⋅v0​=p0​+2ℓ⋅p10​−r0​−2ℓ⋅r1​ v11​∈[0,23) v1​=2ℓ⋅v11​+v10​ 2ℓ⋅v1​=v0​+p11​+p2​−r2​ an​⋅bn​−qn​⋅fn​=rn​ q0′​∈[0,2ℓ) multi-range-check q1′​∈[0,2ℓ) multi-range-check q2′​∈[0,2ℓ) multi-range-check q01′​=q0′​+2ℓ⋅q1′​ multi-range-check qcarry01′​∈[0,2) 22ℓ⋅qcarry01′​=s01​−q01′​ q2′​=s2​+qcarry01′​","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Checks","id":"193","title":"Checks"},"194":{"body":"These checks can be condensed into the minimal number of constraints as follows. First, we have the range-check corresponding to (1) as a degree-4 constraint C1: p111​⋅(p111​−1)⋅(p111​−2)⋅(p111​−3) Checks (2) - (4) are all handled by a single multi-range-check gadget. C2: multi-range-check v10​,p10​,p110​ Next (5) and (6) can be combined into C3: 2ℓ⋅(2ℓ⋅p111​+p110​)+p10​=p1​ Now we have the range-check corresponding to (7) as another degree-4 constraint C4: v0​⋅(v0​−1)⋅(v0​−2)⋅(v0​−3) Next we have check (8) C5: 22ℓ⋅v0​=p0​+2ℓ⋅p10​−r0​−2ℓ⋅r1​ Up next, check (9) is a 3-bit range check C6: Plookup v11​ (12-bit plookup scaled by 29) Now checks (10) and (11) can be combined into C7: 2ℓ⋅(v11​⋅2ℓ+v10​)=p2​+p11​+v0​−r2​ Next, for our use of the CRT, we must constrain that a⋅b=q⋅f+rmodn. Thus, check (12) is C8: an​⋅bn​−qn​⋅fn​=rn​ Next we must constrain the quotient bound addition. Checks (13) - (16) are all combined into multi-range-check gadget C9: multi-range-check q0′​,q1′​,q2′​ and q01′​=q0′​+2ℓ⋅q1′​. Check (17) is a carry bit boolean check C10: qcarry01′​⋅(qcarry01′​−1) Next, check (18) is C11: 22ℓ⋅qcarry10′​=s01​−q01′​ Finally, check (19) is C12: q2′​=s2​+qcarry01′​ The Zero gate has no constraints and is just used to hold values required by the ForeignFieldMul gate.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Constraints","id":"194","title":"Constraints"},"195":{"body":"The following checks must be done with other gates to assure the soundness of the foreign field multiplication Range check input multi-range-check a multi-range-check b Range check witness data multi-range-check q′ and check q01′​=q0′​+2ℓq1′​ multi-range-check ℓ-bit limbs: p10​,p110​,p111​ Range check output multi-range-check either r or r′ Compute and constrain r′ the bound on r ForeignFieldAdd r+2t=1⋅f+r′ Copy constraints must connect the above witness cells to their respective input cells within the corresponding external check gates witnesses.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » External checks","id":"195","title":"External checks"},"196":{"body":"The Keccak gadget is comprised of 3 circuit gates (Xor16, Rot64, and Zero) Keccak works with 64-bit words. The state is represented using 5×5 matrix of 64 bit words. Each compression step of Keccak consists of 24 rounds. Let us denote the state matrix with A (indexing elements as A[x,y]), from which we derive further states as follows in each round. Each round then consists of the following 5 steps: C[x]D[x]E[x,y]B[y,2x+3y]F[x,y]Fp[0,0]​=A[x,0]⊕A[x,1]⊕A[x,2]⊕A[x,3]⊕A[x,4]=C[x−1]⊕ROT(C[x+1],1)=A[x,y]⊕D[x]=ROT(E[x,y],ρ[x,y])=B[x,y]⊕((NOT B[x+1,y])AND B[x+2,y])=F[0,0]⊕RC​​ for 0≤x,y≤4 and ρ[x,y] is the rotation offset defined for Keccak. The values are in the table below extracted from the Keccak reference https://keccak.team/files/Keccak-reference-3.0.pdf x = 3 x = 4 x = 0 x = 1 x = 2 y = 2 155 231 3 10 171 y = 1 55 276 36 300 6 y = 0 28 91 0 1 190 y = 4 120 78 210 66 253 y = 3 21 136 105 45 15","breadcrumbs":"Custom Gates » Keccak » Keccak Gate","id":"196","title":"Keccak Gate"},"197":{"body":"The atomic operations are XOR, ROT, NOT, AND. In the sections below, we will describe the gates for these operations. Below are some common approaches followed in their design. To fit within 15 wires, we first decompose each word into its lower and upper 32-bit components. A gate for an atomic operation works with those 32-bit components at a time. Before we describe the specific gate design approaches, below are some constraints in the Kimchi framework that dictated those approaches. only 4 lookups per row only first 7 columns are available to the permutation polynomial","breadcrumbs":"Custom Gates » Keccak » Design Approach:","id":"197","title":"Design Approach:"},"198":{"body":"It is clear from the definition of the rotation gate that its constraints are complete (meaning that honest instances always satisfy the constraints). It is left to be proven that the proposal is sound. In this section, we will give a proof that as soon as we perform the range checks on the excess and shifted parts of the input, only one possible assignment satisfies the constraints. This means that there is no dishonest instance that can make the constraints pass. We will also give an example where one could find wrong rotation witnesses that would satisfy the constraints if we did not check the range.","breadcrumbs":"Custom Gates » Keccak » Rot64","id":"198","title":"Rot64"},"199":{"body":"First of all, we will illustrate the necessity of range-checks with a simple example. For the sake of readability, we will use some toy field lengths. In particular, let us assume that our words have 4 bits, meaning all of the elements between 0x0 and 0xF. Next, we will be using the native field F32​. As we will later see, this choice of field lengths is not enough to perform any 4-bit rotation, since the operations in the constraints would overflow the native field. Nonetheless, it will be sufficient for our example where we will only rotate by 1 bit. Assume we want to rotate the word 0b1101 (meaning 13) by 1 bit to the left. This gives us the rotated word 0b1011 (meaning 11). The excess part of the word is 0b1, whereas the shifted part corresponds to 0b1010. We recall the constraints for the rotation gate: word⋅2rotrotated​=excess⋅2len+shifted=excess+shifted​ Applied to our example, this results in the following equations: 13⋅211​=excess⋅16+shifted=excess+shifted​ We can easily check that the proposed values of the shifted 0b1010=10 and the excess 0b1=1 values satisfy the above constraint because 26=1⋅16+10 and 11=1+10. Now, the question is: can we find another value for excess and shifted, such that their addition results in an incorrect rotated word? The answer to this question is yes, due to diophantine equations . We basically want to find x,y such that 26=x⋅16+y( mod 32). The solution to this equation is: ∀k∈[0…31]:xy​=k ∧=26−16⋅k​ We chose these word and field lengths to better understand the behaviour of the solution. Here, we can see two \"classes\" of evaluations. If we choose an even k, then y will have the following shape: 26−16⋅(2⋅n)⟺26−32n≡32​26 Meaning, if x=2n then y=26. If on the other hand, we chose an odd k, then y will have the following shape instead: 26−16⋅(2⋅n+1)⟺26−32n−16≡32​26−16=10 Meaning, if x=2n+1 then y=10. Thus, possible solutions to the diophantine equation are: xy​=0,1,2,3,4,5…=26,10,26,10,26,10…​ Note that our valid witness is part of that set of solutions, meaning x=1 and y=10. Of course, we can also find another dishonest instantiation such as x=0 and y=26. Perhaps one could think that we do not need to worry about this case, because the resulting rotation word would be 0+26=26, and if we later use that result as an input to a subsequent gate such as XOR, the value 26 would not fit and at some point the constraint system would complain. Nonetheless, we still have other solutions to worry about, such as (x=3,y=10) or (x=5,y=10), since they would result in a rotated word that would fit in the word length of 4 bits, yet would be incorrect (not equal to 11). All of the above incorrect solutions differ in one thing: they have different bit lengths. This means that we need to range check the values for the excess and shifted witnesses to make sure they have the correct length.","breadcrumbs":"Custom Gates » Keccak » Necessity of range checks","id":"199","title":"Necessity of range checks"},"2":{"body":"Groups are an algebraic structure which are used in most widely deployed signature schemes are used in most widely deployed zk-SNARKs generalize the notion of a collection of symmetries of an object generalize certain aspects of numbers I will not touch on the last two too much. First let's do a quick definition and then some examples. Definition. A group is a set G with a binary operation ∗:G×G→G, an identity e∈G for that operation, and an inverse map ()−1:G→G such that ∗ is associative: x∗(y∗z)=(x∗y)∗z for all x,y,z. e is an identity for ∗. I.e., e∗x=x∗e=x for all x. ()−1 is an inverse map for ∗ with identity e. I.e., x∗x−1=x−1∗x=e for all x. So basically, an invertible binary operation. Definition in hand, we can see some examples: The integers Z with +, 0 as identity, and negation as inverse. For any natural number n, the integers mod n. This can be thought of as the set of numbers {0,…,n−1} with the operation being + followed by taking remainder mod n. It can also be thought of as the group of rotations of an n-gon. For any set X, the set of invertible functions X→X (a.k.a permutations) with function composition as the operation, the identity function as the identity, and function inverse as the inverse operation. The set of translations of a plane R×R with composition of translations (since the composition of translations is a translation). The identity is translating by (0,0) and inverse is reversing the translation. This is equivalent to the group R×R with coordinate-wise addition as the operation and coordinate-wise negation as the inverse. The set of rotations of a sphere with composition as the operation, doing nothing as the identity, and performing the rotation in reverse as the inverse map. For any field F, the set F×:=F∖{0}, with field multiplication as the operation, 1 as the identity, and x↦1/x as the inverse map. This is called the group of units of F. Sometimes, instead of x∗y we simply write xy when ∗ is clear from context.","breadcrumbs":"Groups » Groups","id":"2","title":"Groups"},"20":{"body":"This section describes how the Cooley-Tukey fast Fourier transform works. As we learned in the previous section, the key is to select evaluation points that yield an efficient FFT algorithm. Specifically, say we have ω∈F such that ωn=1, and ωr=1 for any 00, then we will split f into two polynomials, recursively call FFT on them, and reconstruct the result from the recursive calls. To that end, define f0​ to be the polynomial whose coefficients are all the even-index coefficients of f and f1​ the polynomial whose coefficients are all the odd-index coefficients of f. In terms of the array representation, this just means splitting out every other entry into two arrays. So that can be done in time O(n). Write f=∑i<2k​ci​xi, so that f0​=∑i<2k−1​c2i​xi and f1​=∑i<2k−1​c2i+1​xi. Then f(x)​=i<2k∑​ci​xi=i<2k−1∑​c2i​x2i+i<2k−1∑​c2i+1​x2i+1=i<2k−1∑​c2i​(x2)i+i<2k−1∑​c2i+1​x⋅(x2)i=i<2k−1∑​c2i​(x2)i+xi<2k−1∑​c2i+1​(x2)i=f0​(x2)+xf1​(x2)​ Now, notice that if ω is a 2kth root of unity, then ω2 is a 2k−1th root of unity. Thus we can recurse with FFT(k−1,ω2,f0​) and similarly for f1​. Let [e0,0​,…,e0,2k−1−1​][e1,0​,…,e1,2k−1−1​]​=FFT(k−1,ω2,f0​)=FFT(k−1,ω2,f1​)​ By assumption ei,j​=fi​((ω2)j). So, for any j we have f(ωj)​=f0​((ω2)j)+ωjf1​((ω2)j)​ Now, since j may be larger than 2k−1−1, we need to reduce it mod 2k−1, relying on the fact that if τ is an nth root of unity then τj=τjmodn since τn=1. Thus, (ω2)j=(ω2)jmod2k−1 and so we have f(ωj)​=f0​((ω2)jmod2k−1)+ωjf1​((ω2)jmod2k−1)=e0,jmod2k−1​+ωje1,jmod2k−1​​ We can compute the array W=[1,ω,…,ω2k−1] in time O(n) (since each entry is the previous entry times ω). Then we can compute each entry of the output in O(1) as f(ωj)​=e0,jmod2k−1​+W[j]⋅e1,jmod2k−1​​ There are n such entries, so this takes time O(n). This concludes the recursive definition of the algorithm FFT(k,ω,f). Algorithm: computing evalAk​​ Input f=[c0​,…,c2k−1​] the coefficients of polynomial f(x)=∑i<2k​ci​xi Compute W←[1,ω,ω2,...,ω2k−1] FFT(k,ω,f)→[f(1),f(ω),f(ω2)…,f(ω2k−1)] if k==0 return f else Compute f0​=[c0​,c2​,...,c2k−2​] the even coefficients of f, corresponding to f0​(x)=∑i<2k−1​c2i​xi Compute f1​=[c1​,c3​,...,c2k−1​] the odd coefficients of f, corresponding to f1​(x)=∑i<2k−1​c2i+1​xi e0​←FFT(k−1,ω2,f0​) e1​←FFT(k−1,ω2,f1​) for j∈[0,2k−1] Fj​←e0,jmod2k−1​+W[j]⋅e1,jmod2k−1​ return F Now let's analyze the time complexity. Let T(n) be the complexity on an instance of size n (that is, for n=2k). Looking back at what we have done, we have done O(n) for computing f0​ and f1​ two recursive calls, each of size n/2 O(n) for computing the powers of ω O(n) for combining the results of the recursive calls In total, this is O(n)+2T(n/2). Solving this recurrence yields T(n)=O(n)⋅logn=O(nlogn). Basically, there are logn recursions before we hit the base case, and each step takes time O(n). □ Now, in practice there are ways to describe this algorithm non-recursively that have better concrete performance, but that's out of scope for this document. Read the code if you are interested. Using the FFT algorithm to compute interpAk​​ So far we have a fast way to compute evalAk​​(f) all at once, where Ak​ is the set of powers of a 2kth root of unity ω. For convenience let n=2k. Now we want to go the other way and compute a polynomial given an array of evaluations. Specifically, n evaluations [f(x0​),f(x1​),...,f(xn−1​)] uniquely define a degree n−1 polynomial. This can be written as a system of n equations f(x0​)f(x1​)⋮f(xn−1​)​=c0​+c1​x0​+…+cn−1​x0n−1​=c0​+c1​x1​+…+cn−1​x1n−1​=c0​+c1​xn−1​+…+cn−1​xn−1n−1​,​ which can be rewritten as a matrix vector product. ​f(x0​)f(x1​)⋮f(xn−1​)​​=​11⋮1​x0​x1​⋮xn−1​​⋯⋯⋱⋯​x0n−1​x1n−1​⋮xn−1n−1​​​×​c0​c1​⋮cn−1​​​ This n×n matrix is a Vandermonde matrix and it just so happens that square Vandermonde matrices are invertible, iff the xi​ are unique. Since we purposely selected our xi​ to be the powers of ω, a primitive n-th root of unity, by definition xi​=ωi are unique. Therefore, to compute the polynomial given the corresponding array of evaluations (i.e. interpolation) we can solve for the polynomial's coefficients using the inverse of the matrix. ​c0​c1​⋮cn−1​​​=​11⋮1​1ω⋮ωn−1​⋯⋯⋱⋯​1n−1ωn−1⋮ω(n−1)(n−1)​​−1×​f(1)f(ω)⋮f(ωn−1)​​ All we need now is the inverse of this matrix, which is slightly complicated to compute. I'm going to skip it for now, but if you have the details please make a pull request. Substituting in the inverse matrix we obtain the equation for interpolation. ​c0​c1​⋮cn−1​​​=n1​​11⋮1​1ω−1⋮ω−(n−1)​⋯⋯⋱⋯​1n−1ω−(n−1)⋮ω−(n−1)(n−1)​​×​f(1)f(ω)⋮f(ωn−1)​​ Observe that this equation is nearly identical to the original equation for evaluation, except with the following substitution. ωi⇒n1​ω−1i Consequently and perhaps surprisingly, we can reuse the FFT algorithm evalAk​​ in order to compute the inverse-- interpAk​​. So, suppose we have an array [a0​,…,an−1​] of field elements (which you can think of as a function Ak​→F) and we want to compute the coefficients of a polynomial f with f(ωi)=ai​. To this end, define a polynomial g by g=∑j264, then b′ cannot be smaller than 264 as it was assumed. CONTRADICTION. b=b′ and a=a′: then 264(a−a′)≡n​0 and this can only happen if a′=a+kn. But since n>2r, then a′ cannot be smaller than 2r as it was assumed. CONTRADICTION. a=a′ and b=b′: then we have something like 264α+β≡n​0. This means β≡n​−264α=k⋅n−264α for any k. According to the assumption, both 0≤a<2r and 0≤a′<2r. This means, the difference α:=(a−a′) lies anywhere in between the following interval: 1−2r≤α≤2r−1 We plug in this interval to the above equation to obtain the following interval for β: k⋅n−264(1−2r)≤β≤k⋅n−264(2r−1) We look at this interval from both sides of the inequality: β≥kn−264+264+r and β≤kn+264−264+r and we wonder if kn−264+264+r≤kn+264−264+r is at all possible. We rewrite as follows: 264+r−2642⋅264+r264+r​≤264−264+r≤2⋅264≤264​ But this can only happen if r≤0, which is impossible since we assume 0p. We have a proof system (Kimchi) over Fp​ and Fq​, where commitments to public inputs are: Pp​=⟨e,H⟩∈Ep​(Fq​)Pq​=⟨e,G⟩∈Eq​(Fp​) Respectively. See Pasta Curves for more details. When referring to the Fq​-side we mean the proof system for circuit over the field Fq​.","breadcrumbs":"Deferred Computation » Deferred Computation","id":"217","title":"Deferred Computation"},"218":{"body":"In pickles-rs we have the notion of \"passing\" a variable (including the transcript) from one side to the other. e.g. when a field element α∈Fp​ needs to be used as a scalar on Fq​. This document explains what goes on \"under the hood\". Let us start by understanding why: Let v∈Fq​ be a scalar which we want to use to do both: Field arithmetic in Fq​ Scalar operations on Eq​(Fp​) In order to do so efficiently, we need to split these operations across two circuits (and therefore proofs) because: Emulating Fq​ arithmetic in Fp​ is very expensive, e.g. computing v⋅w requires O(log(q)2) multiplications over Fp​: 100's of gates for a single multiplication. Since Eq​(Fp​)⊆Fp​×Fp​ we cannot compute [v]⋅H∈Eq​(Fp​) over Fq​ efficiently, because, like before, emulating Fp​ arithmetic in Fq​ is very expensive...","breadcrumbs":"Deferred Computation » Public Inputs / Why Passing","id":"218","title":"Public Inputs / Why Passing"},"219":{"body":"The solution is to \"pass\" a value v between the two proofs, in other words to have two values v ∈Fp​ and v∈Fq​ which are equal as integers i.e. lift(v)=lift(v )∈Z: they represent \"the same number\". A naive first attempt would be to simply add v~∈Fp​ to the witness on the Fp​-side, however this has two problems: Insufficient Field Size: pq; always the case for any cycle of curves, p is only ≈q​ smaller than q, by Hasse. Now \"v \" is \"represented\" by the two values h ,l~∈Fp​. Note that no decomposition is nessary if the \"original value\" v was in Fp​, since Fq​ is big enough to hold the lift of any element in Fp​. Problem 2: Compute Commitment to the Public Input of other side To solve the binding issue we will add l,h to the public inputs on the Fp​-side, for simplicity we will describe the case where l,h are the only public inputs in the Fp​-side, which means that the commitment Pp​∈Ep​(Fq​) to the public inputs on the Fp​ side is: Pp​=[h]⋅Gh​+[l]⋅Gl​∈Ep​(Fq​) At this point it is important to note that Ep​ is defined over Fq​! Which means that we can compute Pp​∈Ep​(Fq​) efficiently on the Fq​-side! Therefore to enforce the binding, we: Add a sub-circuit which checks: Pp​=[h]⋅Gh​+[l]⋅Gl​∈Ep​(Fq​),∣h∣=⌊logp⌋∣l∣=1 Add Pp​=(x,y)∈Fq​×Fq​ to the public input on the Fq​-side.","breadcrumbs":"Deferred Computation » Solution","id":"219","title":"Solution"},"22":{"body":"Implement types DensePolynomial and Evaluations that wrap a Vec and implement the FFT algorithms described above for converting between them Familiarize yourself with the types and functions provided by ark_poly","breadcrumbs":"Polynomials » Fast Fourier Transform » Exercises","id":"22","title":"Exercises"},"220":{"body":"At this point the statement of the proof in Fq​-side is: the Fq​-proof is sound, condition on providing an opening of Pp​ that satisifies the Fp​-relation. At this point you can stop and verify the proof (in the case of a \"step proof\" you would), by recomputing Pp​ outside the circuit while checking the Fp​-relation manually \"in the clear\". However, when recursing (e.g. wrapping in a \"wrap proof\") we need to \"ingest\" this public input Pp​; after all, to avoid blowup in the proof size everything (proofs/accumulators/public inputs etc.) must eventually become part of the witness and every computation covered by a circuit... To this end, the wrap proof is a proof for the Fp​-relation with the public input Pp​ which additionally verifies the Fq​-proof. The next \"step\" proof then verifies this wrap proof which means that Pp​ then becomes part of the witness!","breadcrumbs":"Deferred Computation » We recurse onwards...","id":"220","title":"We recurse onwards..."},"221":{"body":"We can arbitrarily choose which side should compute the public input of the other, in pickles we let \"wrap\" compute the commitment to the public input.","breadcrumbs":"Deferred Computation » In Pickles","id":"221","title":"In Pickles"},"222":{"body":"Enforces that the public input of the proof verified on the Fr side is equal to the Fp input computed on Fr side.","breadcrumbs":"Deferred Computation » Enforcing Equality","id":"222","title":"Enforcing Equality"},"223":{"body":"This section contains a series of diagrams giving an overview of implementation of pickles, closely following the structures and abstractions in the code. However, they are very technical and are primarily intended for developer use. The primary source of all the images in this section is pickles_structure.drawio file, and if one edits it one must re-generate the .svg renders by doing \"export -> selection\". The legend of the diagrams is quite straightforward: The black boxes are data structures that have names and labels following the implementation. MFNStep/MFNWrap is an abbreviation from MessagesForNextStep and MessagesForNextWrap that is used for brevity. Most other datatypes are exactly the same as in the codebase. The blue boxes are computations. Sometimes, when the computation is trivial or only vaguely indicated, it is denoted as a text sign directly on an arrow. Arrows are blue by default and denote moving a piece of data from one place to another with no (or very little) change. Light blue arrows are denoting witness query that is implemented through the handler mechanism. The \"chicken foot\" connector means that this arrow accesses just one field in an array: such an arrow could connect e.g. a input field of type old_a: A in a structure Vec<(A,B)> to an output new_a: A, which just means that we are inside a for loop and this computation is done for all the elemnts in the vector/array. Colour of the field is sometimes introduced and denotes how many steps ago was this piece of data created. The absense of the colour means either that (1) the data structure contains different subfields of different origin, or that (2) it was not coloured but it could be. The colours are assigned according to the following convention:","breadcrumbs":"Technical Diagrams » Pickles Technical Diagrams","id":"223","title":"Pickles Technical Diagrams"},"224":{"body":"The following is the diagram that explains the Wrap computation. The left half of it corresponds to the wrap.ml file and the general logic of proof creation, while the right part of it is wrap_main.ml/wrap_verifier.ml and explains in-circuit computation. This diagram explains the deferred_values computation which is in the heart of wrap.ml represented as a blue box in the middle of the left pane of the main Wrap diagram. Deferred values for Wrap are computed as follows:","breadcrumbs":"Technical Diagrams » Wrap Computatiton and Deferred Values","id":"224","title":"Wrap Computatiton and Deferred Values"},"225":{"body":"The following is the diagram that explains the Step computation, similarly to Wrap. The left half of it corresponds to the general logic in step.ml, and the right part of it is step_main.ml and explains in-circuit computation. We provide no deferred_values computation diagram for Step, but it is very conceptually similar to the one already presented for Wrap.","breadcrumbs":"Technical Diagrams » Step Computation","id":"225","title":"Step Computation"},"226":{"body":"Poseidon is a hash function that can efficiently run in a zk circuit. (See poseidon-hash.info ) It is based on the sponge function , with a state composed of field elements and a permutation based on field element operation (addition and exponentiation). The permutation contains an S-box (exponentiation of a group element), adding constants to the state, and matrix multiplication of the state (multiplications and additions) with an MDS matrix . Since a field element is around 255-bit, a single field element is enough as the capacity of the sponge to provide around 116-bit security. Note We might want to piggy back on the zcash poseidon spec at some point (perhaps by making this an extension of the zcash poseidon spec).","breadcrumbs":"Poseidon hash » Poseidon hash function","id":"226","title":"Poseidon hash function"},"227":{"body":"We define a base sponge, and a scalar sponge. Both must be instantiated when verifying a proof (this is due to recursion-support baked in Kimchi). External users of kimchi (or pickles) are most likely to interact with a wrap proof (see the pickles specification ). As such, the sponges they need to instantiate are most likely to be instantiated with: Poseidon-Fp for base sponge Poseidon-Fq for the scalar sponge","breadcrumbs":"Poseidon hash » APIs","id":"227","title":"APIs"},"228":{"body":"new(params) -> BaseSponge. Creates a new base sponge. BaseSponge.absorb(field_element). Absorbs a field element by calling the underlying sponge absorb function. BaseSponge.absorb_point(point). Absorbs an elliptic curve point. If the point is the point at infinity, absorb two zeros. Otherwise, absorb the x and y coordinates with two calls to absorb. BaseSponge.absorb_scalar(field_element_of_scalar_field). Absorbs a scalar. If the scalar field is smaller than the base field (e.g. Fp is smaller than Fq), then the scalar is casted to a field element of the base field and absorbed via absorb. Otherwise, the value is split between its least significant bit and the rest. Then both values are casted to field elements of the base field and absorbed via absorb (the high bits first, then the low bit). BaseSponge.digest() -> field_element. The squeeze function of the underlying sponge function is called and the first field element is returned. BaseSponge.digest_scalar() -> field_element_of_scalar_field. BaseSponge.challenge // TODO: specify. BaseSponge.challenge_fq // TODO: specify.","breadcrumbs":"Poseidon hash » Base Sponge","id":"228","title":"Base Sponge"},"229":{"body":"new(params) -> ScalarSponge ScalarSponge.absorb(scalar_field_element) ScalarSponge.digest() -> scalar_field_element ScalarSponge.challenge // TODO: specify","breadcrumbs":"Poseidon hash » Scalar Sponge","id":"229","title":"Scalar Sponge"},"23":{"body":"A \"commitment scheme\" is a cryptographic scheme that lets us provide a \"commitment\" to a given piece of data, in such a way that we can later \"open\" that commitment. I recommend checking out section 2.4.1 of David's book Real-World Cryptography. There are two properties we typically expect of our commitment schemes: Hiding. A commitment to a piece of data should not reveal anything about that data. Binding. There should be no ambiguity about what data is committed to. That is, it should not be possible for the committer to open the commitment to a piece of data other than the one they originally committed to. There are various kinds of commitments that allow opening in different ways, revealing only part of the data, or some function of the data. Sometimes it is even useful to elide the hiding property entirely--- so-called non-hiding commitments.","breadcrumbs":"Commitments » Commitments","id":"23","title":"Commitments"},"230":{"body":"Note that every operation is done in the field of the sponge. In this section we define the high-level algorithm behind the permutation and the sponge. The permutation is never used directly by users, it is used only by the sponge function.","breadcrumbs":"Poseidon hash » Algorithms","id":"230","title":"Algorithms"},"231":{"body":"In pseudo-code: def sbox(field_element): # modular exponentiation return field_element^7 # apply MDS matrix\ndef apply_mds(state): n = [0, 0, 0] n[0] = state[0] * mds[0][0] + state[1] * mds[0][1] + state[2] * mds[0][2] n[1] = state[0] * mds[1][0] + state[1] * mds[1][1] + state[2] * mds[1][2] n[2] = state[0] * mds[2][0] + state[1] * mds[2][1] + state[2] * mds[2][2] return n # a round in the permutation\ndef apply_round(round, state): # sbox state[0] = sbox(state[0]) state[1] = sbox(state[1]) state[2] = sbox(state[2]) # apply MDS matrix state = apply_mds(state) # add round constant state[0] += round_constants[round][0] state[1] += round_constants[round][1] state[2] += round_constants[round][2] # the permutation\ndef permutation(state): round_offset = 0 if ARK_INITIAL: constant = round_constants[0] state[0] += constant[0] state[1] += constant[1] state[2] += constant[2] round_offset = 1 for round in range(round_offset, ROUNDS + round_offset): apply_round(round, state)","breadcrumbs":"Poseidon hash » Permutation","id":"231","title":"Permutation"},"232":{"body":"In pseudo-code: def new(): return { \"state\": [0] * RATE, # `RATE` field elements set to 0 \"mode\": \"absorbing\", \"offset\": 0, } def absorb(sponge, field_element): # if we're changing mode, reset the offset if sponge.mode == \"squeezing\": sponge.mode = \"absorbing\" sponge.offset = 0 # we reached the end of the rate, permute elif sponge.offset == RATE: sponge.state = permutation(sponge.state) sponge.offset = 0 # absorb by adding to the state sponge.state[sponge.offset] += field_element sponge.offset += 1 def squeeze(sponge): # permute when changing mode or when we already squeezed everything if sponge.mode == \"absorbing\" or sponge.offset == RATE: sponge.mode = \"squeezing\" sponge.state = permutation(sponge.state) sponge.offset = 0 result = sponge.state[sponge.offset] sponge.offset += 1 return result","breadcrumbs":"Poseidon hash » Sponge","id":"232","title":"Sponge"},"233":{"body":"We instantiate two versions of Poseidon, one for the field Fp, and one for the field Fq (see the pasta specification ). Both share the following sponge configuration: capacity 1 rate: 2 and the following permutation configuration: number of full rounds: 55 sbox: 7 ARK_INITIAL: false","breadcrumbs":"Poseidon hash » Instantiations","id":"233","title":"Instantiations"},"234":{"body":"You can find the MDS matrix and round constants we use in /poseidon/src/pasta/fp_kimchi.rs .","breadcrumbs":"Poseidon hash » Poseidon-Fp","id":"234","title":"Poseidon-Fp"},"235":{"body":"You can find the MDS matrix and round constants we use in /poseidon/src/pasta/fp_kimchi.rs .","breadcrumbs":"Poseidon hash » Poseidon-Fq","id":"235","title":"Poseidon-Fq"},"236":{"body":"We have test vectors contained in /poseidon/src/tests/test_vectors/kimchi.json .","breadcrumbs":"Poseidon hash » Test vectors","id":"236","title":"Test vectors"},"237":{"body":"our ocaml implementation: https://github.com/minaprotocol/mina/blob/develop/src/lib/random_oracle/random_oracle.mli relies on random_oracle_input: https://github.com/minaprotocol/mina/blob/develop/src/lib/random_oracle_input/random_oracle_input.ml is instantiated with two types of fields: https://github.com/minaprotocol/mina/blob/develop/src/nonconsensus/snark_params/snark_params_nonconsensus.ml","breadcrumbs":"Poseidon hash » Pointers to the OCaml code","id":"237","title":"Pointers to the OCaml code"},"238":{"body":"Our polynomial commitment scheme is a bootleproof-type of commitment scheme, which is a mix of: The polynomial commitment scheme described in appendix A.1 of the PCD paper . The zero-knowledge opening described in section 3.1 of the HALO paper .","breadcrumbs":"Polynomial Commitment » Kimchi Polynomial Commitment","id":"238","title":"Kimchi Polynomial Commitment"},"239":{"body":"The two curves pallas and vesta (pa(llas ve)sta) created by the Zcash team. Each curve's scalar field is the other curve's base field, which is practical for recursion (see Pickles). supporting evidence mina parameters: https://github.com/o1-labs/proof-systems/tree/master/curves/src/pasta arkworks ark-pallas , pallas and vesta Note that in general Fq refers to the base field (in which the curve is defined over), while Fr refers to the scalar field (defined by the order of the curve). But in our code, because of the cycles: Fp refers to the base field of Pallas, and the scalar field of Vesta Fq refers to the base field of Vesta, and the scalar field of Pallas Note that ∣Fq​∣>∣Fp​∣. In pickles: Vesta is also referred to as the step curve, or the tick curve. Pallas is also referred to as the wrap curve, or the tock curve.","breadcrumbs":"Pasta Curves » Pasta Curves","id":"239","title":"Pasta Curves"},"24":{"body":"The simplest kind of a commitment is one in which opening reveals all of the underlying data. Let's give a simple construction of such a scheme for committing to field elements. Suppose we have Fp​ a prime order field, with p being rather large (say on the order of 2256). A hash function h:List(Fp​)→Fp​. Then we define commit(x,r)=h([x,r])open(c)=(x,r)verify(c,x,r)=true/false The argument r of the commit function is data that must only be known to the committer (until the commitment is opened). When a committer wants to commit to a field element x, they sample a random \"blinder\" r∈Fp​ and hash it together with x to form the commitment. To open the commitment c, they simply provide the committed value together with the blinder. Alternatively, if the verifier already knows x, they can just provide r, i.e. open(x,c)=r. Finally, given the commitment and the opening, we can verify whether the input was the value originally committed to using the verify function. If the hash function is collision-resistant, then this function is binding (because there's no way the committer could find another preimage of h([x,r])). If the hash function is one-way, then this commitment is also hiding (assuming x is revealed only as part of the opening). Commitments are often used in protocols between provers and verifiers. The following illustration provides an example with a prover named Peggy and a verifier named Victor. Peggyinputblinder​ }→commit​ commitment→open​ ​ ⟶⋮ ⟶⋮ ​Victorcommitment input,blinder commitmentinputblinder​ }→verify​→true/false​ ​ Here Peggy commits to an input using a blinder, obtains the commitment and sends it to Victor. The interlocutors continue their protocol, but eventually to convince Victor of her claims, Peggy must send the opening proof to her earlier commitment. Victor verifies the opening (i.e. the input and blinder) against the commitment. If the verification fails, then Victor knows that Peggy was trying to trick him, otherwise Victor has sufficient assurances that Peggy was telling the truth.","breadcrumbs":"Commitments » Simple commitments","id":"24","title":"Simple commitments"},"240":{"body":"curve equation: y2=x3+5 base field: 28948022309329048855892746252171976963363056481941560715954676764349967630337 scalar field: 28948022309329048855892746252171976963363056481941647379679742748393362948097 mina generator: (1,12418654782883325593414442427049395787963493412651469444558597405572177144507) arkworks generator: (−1,2) endo: endo_q = 2D33357CB532458ED3552A23A8554E5005270D29D19FC7D27B7FD22F0201B547\nendo_r = 397E65A7D7C1AD71AEE24B27E308F0A61259527EC1D4752E619D1840AF55F1B1 You can use sage to test this: Fp = GF(28948022309329048855892746252171976963363056481941560715954676764349967630337)\nPallas = EllipticCurve(Fp, [0, 5])\nPallas.count_points()","breadcrumbs":"Pasta Curves » Pallas","id":"240","title":"Pallas"},"241":{"body":"curve equation: y2=x3+5 base field: 28948022309329048855892746252171976963363056481941647379679742748393362948097 scalar field: 28948022309329048855892746252171976963363056481941560715954676764349967630337 mina generator: (1,11426906929455361843568202299992114520848200991084027513389447476559454104162) arkworks generator: (−1,2) endo: endo_q = 06819A58283E528E511DB4D81CF70F5A0FED467D47C033AF2AA9D2E050AA0E4F\nendo_r = 12CCCA834ACDBA712CAAD5DC57AAB1B01D1F8BD237AD31491DAD5EBDFDFE4AB9 You can use sage to test this: Fq = GF(28948022309329048855892746252171976963363056481941647379679742748393362948097)\nVesta = EllipticCurve(Fq, [0, 5])\nVesta.count_points()","breadcrumbs":"Pasta Curves » Vesta","id":"241","title":"Vesta"},"242":{"body":"This document specifies kimchi , a zero-knowledge proof system that's a variant of PLONK. This document does not specify how circuits are created or executed, but only how to convert a circuit and its execution into a proof. Table of content: Overview Tables used to describe a circuit Tables produced during proof creation Dependencies Polynomial Commitments Poseidon hash function Pasta Constraints Linearization Permutation Lookup The Lookup Tables The Lookup Selectors Producing the sorted table as the prover Gates Double Generic Gate Poseidon Elliptic Curve Addition Endo Scalar Endo Scalar Multiplication Scalar Multiplication Range Check Foreign Field Addition Foreign Field Multiplication Rotation Xor Gadgets Not And Setup Common Index Lookup Index Prover Index Verifier Index Proof Construction & Verification Proof Structure Proof Creation Proof Verification Fiat-Shamir argument Partial verification Batch verification of proofs Optimizations Security Considerations","breadcrumbs":"Kimchi » Kimchi","id":"242","title":"Kimchi"},"243":{"body":"There are three main algorithms to kimchi: Setup : takes a circuit and produces a prover index, and a verifier index. Proof creation : takes the prover index, and the execution trace of the circuit to produce a proof. Proof verification : takes the verifier index and a proof to verify. As part of these algorithms, a number of tables are created (and then converted into polynomials) to create a proof.","breadcrumbs":"Kimchi » Overview","id":"243","title":"Overview"},"244":{"body":"The following tables are created to describe the circuit: Gates . A circuit is described by a series of gates, that we list in a table. The columns of the tables list the gates, while the rows are the length of the circuit. For each row, only a single gate can take a value 1 while all other gates take the value 0. row Generic Poseidon CompleteAdd VarBaseMul EndoMul EndoMulScalar 0 1 0 0 0 0 0 1 0 1 0 0 0 0 Coefficients . The coefficient table has 15 columns, and is used to tweak the gates. Currently, only the Generic and the Poseidon gates use it (refer to their own sections to see how). All other gates set their values to 0. row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 / / / / / / / / / / / / / / / Wiring (or Permutation, or sigmas) . For gates to take the outputs of other gates as inputs, we use a wiring table to wire registers together. To learn about registers, see the next section. It is defined at every row, but only for the first 7 registers. Each cell specifies a (row, column) tuple that it should be wired to. Cells that are not connected to another cell are wired to themselves. Note that if three or more registers are wired together, they must form a cycle. For example, if register (0, 4) is wired to both registers (80, 6) and (90, 0) then you would have the following table: row 0 1 2 3 4 5 6 0 0,0 0,1 0,2 0,3 80,6 0,5 0,6 ... 80 80,0 80,1 80,2 80,3 80,4 80,5 90,0 ... 90 0,4 90,1 90,2 90,3 90,4 90,5 90,6 The lookup feature is currently optional, as it can add some overhead to the protocol. In the case where you would want to use lookups, the following tables would be needed: Lookup Tables . The different lookup tables that are used in the circuit. For example, the XOR lookup table: l r o 1 0 1 0 1 1 1 1 0 0 0 0 Lookup selectors . A lookup selector is used to perform a number of queries in different lookup tables. Any gate can advertise its use of a lookup selector (so a lookup selector can be associated to several gates), and on which rows they want to use them (current and/or next). In cases where a gate need to use lookups in its current row only, and is the only one performing a specific combination of queries, then its gate selector can be used in place of a lookup selector. As with gates, lookup selectors (including gates used as lookup selectors) are mutually exclusives (only one can be used on a given row). For example, suppose we have two lookup selectors: row ChaChaQuery ChaChaFinalQuery 0 0 0 1 1 0 Where each applies 4 queries. A query is a table describing which lookup table it queries, and the linear combination of the witness to use in the query. For example, the following table describes a query into the XOR table made out of linear combinations of registers (checking that r0​⊕r2​=2⋅r1​): table_id l r o XOR 1, r0 1, r2 2, r1","breadcrumbs":"Kimchi » Tables used to describe a circuit","id":"244","title":"Tables used to describe a circuit"},"245":{"body":"The following tables are created by the prover at runtime: Registers (or Witness) . Registers are also defined at every row, and are split into two types: the IO registers from 0 to 6 usually contain input or output of the gates (note that a gate can output a value on the next row as well). I/O registers can be wired to each other (they'll be forced to have the same value), no matter what row they're on (for example, the register at row:0, col:4 can be wired to the register at row:80, col:6). The rest of the registers, 7 through 14, are called advice registers as they can store values that useful only for the row's active gate. Think of them as intermediary or temporary values needed in the computation when the prover executes a circuit. row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 / / / / / / / / / / / / / / / Wiring (Permutation) trace . You can think of the permutation trace as an extra register that is used to enforce the wiring specified in the wiring table. It is a single column that applies on all the rows as well, which the prover computes as part of a proof. row pt 0 / Queries trace . These are the actual values made by queries, calculated by the prover at runtime, and used to construct the proof. Table trace . Represents the concatenation of all the lookup tables, combined into a single column at runtime by both the prover and the verifier. Sorted trace . Represents the processed (see the lookup section) concatenation of the queries trace and the table trace. It is produced at runtime by the prover. The sorted trace is long enough that it is split in several columns. Lookup (aggregation, or permutation) trace . This is a one column table that is similar to the wiring (permutation) trace we talked above. It is produced at runtime by the prover.","breadcrumbs":"Kimchi » Tables produced during proof creation","id":"245","title":"Tables produced during proof creation"},"246":{"body":"To specify kimchi, we rely on a number of primitives that are specified outside of this specification. In this section we list these specifications, as well as the interfaces we make use of in this specification.","breadcrumbs":"Kimchi » Dependencies","id":"246","title":"Dependencies"},"247":{"body":"Refer to the specification on polynomial commitments . We make use of the following functions from that specification: PolyCom.non_hiding_commit(poly) -> PolyCom::NonHidingCommitment PolyCom.commit(poly) -> PolyCom::HidingCommitment PolyCom.evaluation_proof(poly, commitment, point) -> EvaluationProof PolyCom.verify(commitment, point, evaluation, evaluation_proof) -> bool","breadcrumbs":"Kimchi » Polynomial Commitments","id":"247","title":"Polynomial Commitments"},"248":{"body":"Refer to the specification on Poseidon . We make use of the following functions from that specification: Poseidon.init(params) -> FqSponge Poseidon.update(field_elem) Poseidon.finalize() -> FieldElem specify the following functions on top: Poseidon.produce_challenge() (TODO: uses the endomorphism) Poseidon.to_fr_sponge() -> state_of_fq_sponge_before_eval, FrSponge With the current parameters: S-Box alpha: 7 Width: 3 Rate: 2 Full rounds: 55 Round constants: fp_kimchi , fq_kimchi MDS matrix: fp_kimchi , fq_kimchi","breadcrumbs":"Kimchi » Poseidon hash function","id":"248","title":"Poseidon hash function"},"249":{"body":"Kimchi is made to work on cycles of curves, so the protocol switch between two fields Fq and Fr, where Fq represents the base field and Fr represents the scalar field. See the Pasta curves specification .","breadcrumbs":"Kimchi » Pasta","id":"249","title":"Pasta"},"25":{"body":"Instead of a cryptographic hash function, we can use elliptic curve scalar multiplication to construct a commitment scheme. Here scalar multiplication is used like a one-way function based on the hardness assumption of the elliptic curve discrete logarithm problem (ECDLP). Suppose we have Fp​ a prime order field, with p being large (e.g. something like 2256). Publicly agreed generator point G over an elliptic curve E(Fp​) Another publicly agreed curve point H for which no one knows the discrete logarithm commit(x,r)=xG+rHopen(c)=(x,r) where x∈Fp​ is the value being committed to, r∈Fp​ is a random blinding factor and the commitment c=commit(x,r) is a curve point. These commitments are algebraic (i.e. they do not use a boolean-based cryptographic hash function) and have homomorphic properties: you can add commitments together to form another commitment of the added committed values. For example, if you have commitments A and B, you can perform: A+B​=commit(xa​,ra​)+commit(xb​,rb​)=xa​G+ra​H+xb​G+rb​H=(xa​+xb​)G+(ra​+rb​)H=commit(xa​+xb​,ra​+rb​)​ In other words, the sum of commitments A and B is equal to the commitment of the sum of the two committed values xa​ and xb​ and blinders ra​ and rb​. This is possible because in such a scheme scaling is commutative with adding scalars. As a cryptographic primitive, the ability to find a public curve point H for which no one knows the discrete logarithm may, at first, seem rather mind-blowing and powerful. Actually, it's as easy as it is awesome to find such a point--- simply perform rejection sampling by cryptographically hashing G (or, respectively, the hash output), using the output as the x-coordinate of a candidate point on E and checking whether it's valid. The first valid curve point obtained is H and by the hardness assumption of the ECDLP, no one knows it. Since approximately half of the hash outputs will be valid curve points on E, sampling will terminate very quickly. Indeed, as we will see later, this process can be used to sample many public curve points G1​,…,Gn​ for which the discrete logarithms are unknown; the so-called hash to curve algorithm.","breadcrumbs":"Commitments » Algebraic and homomorphic commitments","id":"25","title":"Algebraic and homomorphic commitments"},"250":{"body":"Kimchi enforces the correct execution of a circuit by creating a number of constraints and combining them together. In this section, we describe all the constraints that make up the main polynomial f once combined. We define the following functions: combine_constraints(range_alpha, constraints), which takes a range of contiguous powers of alpha and a number of constraints. It returns the sum of all the constraints, where each constraint has been multiplied by a power of alpha. In other words it returns: i∑​αi⋅constrainti​ The different ranges of alpha are described as follows: gates . Offset starts at 0 and 21 powers of α are used Permutation . Offset starts at 21 and 3 powers of α are used Note As gates are mutually exclusive (a single gate is used on each row), we can reuse the same range of powers of alpha across all the gates.","breadcrumbs":"Kimchi » Constraints","id":"250","title":"Constraints"},"251":{"body":"TODO","breadcrumbs":"Kimchi » Linearization","id":"251","title":"Linearization"},"252":{"body":"The permutation constraints are the following 4 constraints: The two sides of the coin (with shift0​=1): ​z(x)⋅zkpm(x)⋅αPERM0⋅(w0​(x)+β⋅shift0​x+γ)⋅(w1​(x)+β⋅shift1​x+γ)⋅(w2​(x)+β⋅shift2​x+γ)⋅(w3​(x)+β⋅shift3​x+γ)⋅(w4​(x)+β⋅shift4​x+γ)⋅(w5​(x)+β⋅shift5​x+γ)⋅(w6​(x)+β⋅shift6​x+γ)​​ and ​−1⋅z(xω)⋅zkpm(x)⋅αPERM0⋅(w0​(x)+β⋅σ0​(x)+γ)⋅(w1​(x)+β⋅σ1​(x)+γ)⋅(w2​(x)+β⋅σ2​(x)+γ)⋅(w3​(x)+β⋅σ3​(x)+γ)⋅(w4​(x)+β⋅σ4​(x)+γ)⋅(w5​(x)+β⋅σ5​(x)+γ)⋅(w6​(x)+β⋅σ6​(x)+γ)⋅​​ the initialization of the accumulator: (z(x)−1)L1​(x)αPERM1 and the accumulator's final value: (z(x)−1)Ln−k​(x)αPERM2 You can read more about why it looks like that in this post . The quotient contribution of the permutation is split into two parts perm and bnd. They will be used by the prover. perm(x)=​aPERM0⋅zkpl(x)⋅[z(x)⋅(w0​(x)+γ+x⋅β⋅shift0​)⋅(w1​(x)+γ+x⋅β⋅shift1​)⋅(w2​(x)+γ+x⋅β⋅shift2​)⋅(w3​(x)+γ+x⋅β⋅shift3​)⋅(w4​(x)+γ+x⋅β⋅shift4​)⋅(w5​(x)+γ+x⋅β⋅shift5​)⋅(w6​(x)+γ+x⋅β⋅shift6​)⋅−z(x⋅w)⋅(w0​(x)+γ+σ0​⋅β)⋅(w1​(x)+γ+σ1​⋅β)⋅(w2​(x)+γ+σ2​⋅β)⋅(w3​(x)+γ+σ3​⋅β)⋅(w4​(x)+γ+σ4​⋅β)⋅(w5​(x)+γ+σ5​⋅β)⋅(w6​(x)+γ+σ6​⋅β)⋅]​​ and bnd: bnd(x)=aPERM1⋅x−1z(x)−1​+aPERM2⋅x−sid[n−k]z(x)−1​ The linearization: scalar⋅σ6​(x) where scalar is computed as: z(ζω)βαPERM0zkpl(ζ)⋅(γ+βσ0​(ζ)+w0​(ζ))⋅(γ+βσ1​(ζ)+w1​(ζ))⋅(γ+βσ2​(ζ)+w2​(ζ))⋅(γ+βσ3​(ζ)+w3​(ζ))⋅(γ+βσ4​(ζ)+w4​(ζ))⋅(γ+βσ5​(ζ)+w5​(ζ))⋅​​ To compute the permutation aggregation polynomial, the prover interpolates the polynomial that has the following evaluations. The first evaluation represents the initial value of the accumulator: z(g0)=1 For i=0,⋅,n−4, where n is the size of the domain, evaluations are computed as: z(gi+1)=z1​/z2​ with z1​=​ (w0​(gi+sid(gi)⋅beta⋅shift0​+γ)⋅ (w1​(gi)+sid(gi)⋅beta⋅shift1​+γ)⋅ (w2​(gi)+sid(gi)⋅beta⋅shift2​+γ)⋅ (w3​(gi)+sid(gi)⋅beta⋅shift3​+γ)⋅ (w4​(gi)+sid(gi)⋅beta⋅shift4​+γ)⋅ (w5​(gi)+sid(gi)⋅beta⋅shift5​+γ)⋅ (w6​(gi)+sid(gi)⋅beta⋅shift6​+γ)​​ and z2​=​ (w0​(gi)+σ0​⋅beta+γ)⋅ (w1​(gi)+σ1​⋅beta+γ)⋅ (w2​(gi)+σ2​⋅beta+γ)⋅ (w3​(gi)+σ3​⋅beta+γ)⋅ (w4​(gi)+σ4​⋅beta+γ)⋅ (w5​(gi)+σ5​⋅beta+γ)⋅ (w6​(gi)+σ6​⋅beta+γ)​​ We randomize the evaluations at n - zk_rows + 1 and n - zk_rows + 2 in order to add zero-knowledge to the protocol. For a valid witness, we then have have z(gn−zkr​ows)=1.","breadcrumbs":"Kimchi » Permutation","id":"252","title":"Permutation"},"253":{"body":"Lookups in kimchi allows you to check if a single value, or a series of values, are part of a table. The first case is useful to check for checking if a value belongs to a range (from 0 to 1,000, for example), whereas the second case is useful to check truth tables (for example, checking that three values can be found in the rows of an XOR table) or write and read from a memory vector (where one column is an index, and the other is the value stored at that index). Note Similarly to the generic gate, each values taking part in a lookup can be scaled with a fixed field element. The lookup functionality is an opt-in feature of kimchi that can be used by custom gates. From the user's perspective, not using any gates that make use of lookups means that the feature will be disabled and there will be no overhead to the protocol. Please refer to the lookup RFC for an overview of the lookup feature. In this section, we describe the tables kimchi supports, as well as the different lookup selectors (and their associated queries) The Lookup Tables Kimchi currently supports two lookup tables: /// The table ID associated with the XOR lookup table.\npub const XOR_TABLE_ID: i32 = 0; /// The range check table ID.\npub const RANGE_CHECK_TABLE_ID: i32 = 1; XOR The lookup table for 4-bit xor. Note that it is constructed so that (0, 0, 0) is the last position in the table. This is because tables are extended to the full size of a column (essentially) by padding them with their final value. And, having the value (0, 0, 0) here means that when we commit to this table and use the dummy value in the lookup_sorted columns, those entries that have the dummy value of 0=0+j∗0+j2∗0 will translate into a scalar multiplication by 0, which is free. 12-bit Check The range check table is a single-column table containing the numbers from 0 to 2^12 (excluded). This is used to check that the value fits in 12 bits. Runtime tables Another type of lookup tables has been suggested in the Extended Lookup Tables . The Lookup Selectors XorSelector . Performs 4 queries to the XOR lookup table. l r o - l r o - l r o - l r o 1, r3 1, r7 1, r11 - 1, r4 1, r8 1, r12 - 1, r5 1, r9 1, r13 - 1, r6 1, r10 1, r14 Producing the sorted table as the prover Because of our ZK-rows, we can't do the trick in the plookup paper of wrapping around to enforce consistency between the sorted lookup columns. Instead, we arrange the LookupSorted table into columns in a snake-shape. Like so, _ _\n| | | | |\n| | | | |\n|_| |_| | or, imagining the full sorted array is [ s0, ..., s8 ], like s0 s4 s4 s8\ns1 s3 s5 s7\ns2 s2 s6 s6 So the direction (\"increasing\" or \"decreasing\" (relative to LookupTable) is if i % 2 = 0 { Increasing } else { Decreasing } Then, for each i < max_lookups_per_row, if i % 2 = 0, we enforce that the last element of LookupSorted(i) = last element of LookupSorted(i + 1), and if i % 2 = 1, we enforce that the first element of LookupSorted(i) = first element of LookupSorted(i + 1).","breadcrumbs":"Kimchi » Lookup","id":"253","title":"Lookup"},"254":{"body":"A circuit is described as a series of gates. In this section we describe the different gates currently supported by kimchi, the constraints associated to them, and the way the register table, coefficient table, and permutation can be used in conjunction. TODO: for each gate describe how to create it? Double Generic Gate The double generic gate contains two generic gates. A generic gate is simply the 2-fan in gate specified in the vanilla PLONK protocol that allows us to do operations like: addition of two registers (into an output register) or multiplication of two registers equality of a register with a constant More generally, the generic gate controls the coefficients ci​ in the equation: c0​⋅l+c1​⋅r+c2​⋅o+c3​⋅(l×r)+c4​ The layout of the gate is the following: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 l1 r1 o1 l2 r2 o2 where l1, r1, and o1 (resp. l2, r2, o2) are the left, right, and output registers of the first (resp. second) generic gate. The selectors are stored in the coefficient table as: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 l1 r1 o1 m1 c1 l2 r2 o2 m2 c2 with m1 (resp. m2) the mul selector for the first (resp. second) gate, and c1 (resp. c2) the constant selector for the first (resp. second) gate. The constraints: w0​⋅c0​+w1​⋅c1​+w2​⋅c2​+w0​⋅w1​⋅c3​+c4​ w3​⋅c5​+w4​⋅c6​+w5​⋅c7​+w3​w4​c8​+c9​ where the ci​ are the coefficients. Poseidon The poseidon gate encodes 5 rounds of the poseidon permutation. A state is represents by 3 field elements. For example, the first state is represented by (s0, s0, s0), and the next state, after permutation, is represented by (s1, s1, s1). Below is how we store each state in the register table: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 s0 s0 s0 s4 s4 s4 s1 s1 s1 s2 s2 s2 s3 s3 s3 s5 s5 s5 The last state is stored on the next row. This last state is either used: with another Poseidon gate on that next row, representing the next 5 rounds. or with a Zero gate, and a permutation to use the output elsewhere in the circuit. or with another gate expecting an input of 3 field elements in its first registers. Note As some of the poseidon hash variants might not use 5k rounds (for some k), the result of the 4-th round is stored directly after the initial state. This makes that state accessible to the permutation. We define Mr,c​ as the MDS matrix at row r and column c. We define the S-box operation as wS for S the SPONGE_BOX constant. We store the 15 round constants ri​ required for the 5 rounds (3 per round) in the coefficient table: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 The initial state, stored in the first three registers, are not constrained. The following 4 states (of 3 field elements), including 1 in the next row, are constrained to represent the 5 rounds of permutation. Each of the associated 15 registers is associated to a constraint, calculated as: first round: w6​−(r0​+(M0,0​w0S​+M0,1​w1S​+M0,2​w2S​)) w7​−(r1​+(M1,0​w0S​+M1,1​w1S​+M1,2​w2S​)) w8​−(r2​+(M2,0​w0S​+M2,1​w1S​+M2,2​w2S​)) second round: w9​−(r3​+(M0,0​w6S​+M0,1​w7S​+M0,2​w8S​)) w10​−(r4​+(M1,0​w6S​+M1,1​w7S​+M1,2​w8S​)) w11​−(r5​+(M2,0​w6S​+M2,1​w7S​+M2,2​w8S​)) third round: w12​−(r6​+(M0,0​w9S​+M0,1​w10S​+M0,2​w11S​)) w13​−(r7​+(M1,0​w9S​+M1,1​w10S​+M1,2​w11S​)) w14​−(r8​+(M2,0​w9S​+M2,1​w10S​+M2,2​w11S​)) fourth round: w3​−(r9​+(M0,0​w12S​+M0,1​w13S​+M0,2​w14S​)) w4​−(r10​+(M1,0​w12S​+M1,1​w13S​+M1,2​w14S​)) w5​−(r11​+(M2,0​w12S​+M2,1​w13S​+M2,2​w14S​)) fifth round: w0,next​−(r12​+(M0,0​w3S​+M0,1​w4S​+M0,2​w5S​)) w1,next​−(r13​+(M1,0​w3S​+M1,1​w4S​+M1,2​w5S​)) w2,next​−(r14​+(M2,0​w3S​+M2,1​w4S​+M2,2​w5S​)) where wi,next​ is the polynomial wi​(ωx) which points to the next row. Elliptic Curve Addition The layout is 0 1 2 3 4 5 6 7 8 9 10 x1 y1 x2 y2 x3 y3 inf same_x s inf_z x21_inv where (x1, y1), (x2, y2) are the inputs and (x3, y3) the output. inf is a boolean that is true iff the result (x3, y3) is the point at infinity. The rest of the values are inaccessible from the permutation argument, but same_x is a boolean that is true iff x1 == x2. The following constraints are generated: constraint 1: x0​=w2​−w0​ (w10​⋅x0​−F(1)−w7​) constraint 2: x0​=w2​−w0​ w7​⋅x0​ constraint 3: x0​=w2​−w0​ x1​=w3​−w1​ x2​=w0​⋅w0​ w7​⋅(2⋅w8​⋅w1​−2⋅x2​−x2​)+(F(1)−w7​)⋅(x0​⋅w8​−x1​) constraint 4: w0​+w2​+w4​−w8​⋅w8​ constraint 5: w8​⋅(w0​−w4​)−w1​−w5​ constraint 6: x1​=w3​−w1​ x1​⋅(w7​−w6​) constraint 7: x1​=w3​−w1​ x1​⋅w9​−w6​ Endo Scalar We give constraints for the endomul scalar computation. Each row corresponds to 8 iterations of the inner loop in \"Algorithm 2\" on page 29 of the Halo paper . The state of the algorithm that's updated across iterations of the loop is (a, b). It's clear from that description of the algorithm that an iteration of the loop can be written as (a, b, i) -> ( 2 * a + c_func(r_{2 * i}, r_{2 * i + 1}), 2 * b + d_func(r_{2 * i}, r_{2 * i + 1}) ) for some functions c_func and d_func. If one works out what these functions are on every input (thinking of a two-bit input as a number in {0,1,2,3}), one finds they are given by c_func(x), defined by c_func(0) = 0 c_func(1) = 0 c_func(2) = -1 c_func(3) = 1 d_func(x), defined by d_func(0) = -1 d_func(1) = 1 d_func(2) = 0 d_func(3) = 0 One can then interpolate to find polynomials that implement these functions on {0,1,2,3}. You can use sage , as R = PolynomialRing(QQ, 'x')\nc_func = R.lagrange_polynomial([(0, 0), (1, 0), (2, -1), (3, 1)])\nd_func = R.lagrange_polynomial([(0, -1), (1, 1), (2, 0), (3, 0)]) Then, c_func is given by 2/3 * x^3 - 5/2 * x^2 + 11/6 * x and d_func is given by 2/3 * x^3 - 7/2 * x^2 + 29/6 * x - 1 <=> c_func + (-x^2 + 3x - 1) We lay it out the witness as 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Type n0 n8 a0 b0 a8 b8 x0 x1 x2 x3 x4 x5 x6 x7 ENDO where each xi is a two-bit \"crumb\". We also use a polynomial to check that each xi is indeed in {0,1,2,3}, which can be done by checking that each xi​ is a root of the polyunomial below: crumb(x)\n= x (x - 1) (x - 2) (x - 3)\n= x^4 - 6*x^3 + 11*x^2 - 6*x\n= x *(x^3 - 6*x^2 + 11*x - 6) Each iteration performs the following computations Update n: ni+1​=2⋅ni​+xi​ Update a: ai+1​=2⋅ai​+ci​ Update b: bi+1​=2⋅bi​+di​ Then, after the 8 iterations, we compute expected values of the above operations as: expected_n8 := 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * (2 * n0 + x0) + x1 ) + x2 ) + x3 ) + x4 ) + x5 ) + x6 ) + x7 expected_a8 := 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * (2 * a0 + c0) + c1 ) + c2 ) + c3 ) + c4 ) + c5 ) + c6 ) + c7 expected_b8 := 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * (2 * b0 + d0) + d1 ) + d2 ) + d3 ) + d4 ) + d5 ) + d6 ) + d7 Putting together all of the above, these are the 11 constraints for this gate Checking values after the 8 iterations: Constrain n: 0 = expected_n8 - n8 Constrain a: 0 = expected_a8 - a8 Constrain b: 0 = expected_b8 - b8 Checking the crumbs, meaning each x is indeed in the range {0,1,2,3}: Constrain x0​: 0 = x0 * ( x0^3 - 6 * x0^2 + 11 * x0 - 6 ) Constrain x1​: 0 = x1 * ( x1^3 - 6 * x1^2 + 11 * x1 - 6 ) Constrain x2​: 0 = x2 * ( x2^3 - 6 * x2^2 + 11 * x2 - 6 ) Constrain x3​: 0 = x3 * ( x3^3 - 6 * x3^2 + 11 * x3 - 6 ) Constrain x4​: 0 = x4 * ( x4^3 - 6 * x4^2 + 11 * x4 - 6 ) Constrain x5​: 0 = x5 * ( x5^3 - 6 * x5^2 + 11 * x5 - 6 ) Constrain x6​: 0 = x6 * ( x6^3 - 6 * x6^2 + 11 * x6 - 6 ) Constrain x7​: 0 = x7 * ( x7^3 - 6 * x7^2 + 11 * x7 - 6 ) Endo Scalar Multiplication We implement custom gate constraints for short Weierstrass curve endomorphism optimised variable base scalar multiplication. Given a finite field Fq​ of order q, if the order is not a multiple of 2 nor 3, then an elliptic curve over Fq​ in short Weierstrass form is represented by the set of points (x,y) that satisfy the following equation with a,b∈Fq​ and 4a3+27b2=Fq​​0: E(Fq​):y2=x3+ax+b If P=(xp​,yp​) and T=(xt​,yt​) are two points in the curve E(Fq​), the goal of this operation is to perform the operation 2P±T efficiently as (P±T)+P. S = (P + (b ? T : −T)) + P The same algorithm can be used to perform other scalar multiplications, meaning it is not restricted to the case 2⋅P, but it can be used for any arbitrary k⋅P. This is done by decomposing the scalar k into its binary representation. Moreover, for every step, there will be a one-bit constraint meant to differentiate between addition and subtraction for the operation (P±T)+P: In particular, the constraints of this gate take care of 4 bits of the scalar within a single EVBSM row. When the scalar is longer (which will usually be the case), multiple EVBSM rows will be concatenated. Row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Type i xT yT Ø Ø xP yP n xR yR s1 s3 b1 b2 b3 b4 EVBSM i+1 = = xS yS n' xR' yR' s1' s3' b1' b2' b3' b4' EVBSM The layout of this gate (and the next row) allows for this chained behavior where the output point of the current row S gets accumulated as one of the inputs of the following row, becoming P in the next constraints. Similarly, the scalar is decomposed into binary form and n (n′ respectively) will store the current accumulated value and the next one for the check. For readability, we define the following variables for the constraints: endo := EndoCoefficient xq1 :=(1+(endo−1)⋅b1​)⋅xt​ xq2 :=(1+(endo−1)⋅b3​)⋅xt​ yq1 :=(2⋅b2​−1)⋅yt​ yq2 :=(2⋅b4​−1)⋅yt​ These are the 11 constraints that correspond to each EVBSM gate, which take care of 4 bits of the scalar within a single EVBSM row: First block: (xq1 - xp) * s1 = yq1 - yp (2 * xp – s1^2 + xq1) * ((xp – xr) * s1 + yr + yp) = (xp – xr) * 2 * yp (yr + yp)^2 = (xp – xr)^2 * (s1^2 – xq1 + xr) Second block: (xq2 - xr) * s3 = yq2 - yr (2*xr – s3^2 + xq2) * ((xr – xs) * s3 + ys + yr) = (xr – xs) * 2 * yr (ys + yr)^2 = (xr – xs)^2 * (s3^2 – xq2 + xs) Booleanity checks: Bit flag b1​: 0 = b1 * (b1 - 1) Bit flag b2​: 0 = b2 * (b2 - 1) Bit flag b3​: 0 = b3 * (b3 - 1) Bit flag b4​: 0 = b4 * (b4 - 1) Binary decomposition: Accumulated scalar: n_next = 16 * n + 8 * b1 + 4 * b2 + 2 * b3 + b4 The constraints above are derived from the following EC Affine arithmetic equations: (1) => (xq1​​−xp​)⋅s1​=yq1​​−yp​ (2&3) => (xp​–xr​)⋅s2​=yr​+yp​ (2) => (2⋅xp​+xq1​​–s12​)⋅(s1​+s2​)=2⋅yp​ <=> (2⋅xp​–s12​+xq1​​)⋅((xp​–xr​)⋅s1​+yr​+yp​)=(xp​–xr​)⋅2⋅yp​ (3) => s12​−s22​=xq1​​−xr​ <=> (yr​+yp​)2=(xp​–xr​)2⋅(s12​–xq1​​+xr​) (4) => (xq2​​−xr​)⋅s3​=yq2​​−yr​ (5&6) => (xr​–xs​)⋅s4​=ys​+yr​ (5) => (2⋅xr​+xq2​​–s32​)⋅(s3​+s4​)=2⋅yr​ <=> (2⋅xr​–s32​+xq2​​)⋅((xr​–xs​)⋅s3​+ys​+yr​)=(xr​–xs​)⋅2⋅yr​ (6) => s32​–s42​=xq2​​−xs​ <=> (ys​+yr​)2=(xr​–xs​)2⋅(s32​–xq2​​+xs​) Defining s2​ and s4​ as s2​:=2∗xP​+xT​−s12​2⋅yP​​−s1​ s4​:=2∗xR​+xT​−s32​2⋅yR​​−s3​ Gives the following equations when substituting the values of s2​ and s4​: (xq1 - xp) * s1 = (2 * b1 - 1) * yt - yp (2 * xp – s1^2 + xq1) * ((xp – xr) * s1 + yr + yp) = (xp – xr) * 2 * yp (yr + yp)^2 = (xp – xr)^2 * (s1^2 – xq1 + xr) (xq2 - xr) * s3 = (2 * b2 - 1) * yt - yr (2 * xr – s3^2 + xq2) * ((xr – xs) * s3 + ys + yr) = (xr – xs) * 2 * yr (ys + yr)^2 = (xr – xs)^2 * (s3^2 – xq2 + xs) Scalar Multiplication We implement custom Plonk constraints for short Weierstrass curve variable base scalar multiplication. Given a finite field Fq​ of order q, if the order is not a multiple of 2 nor 3, then an elliptic curve over Fq​ in short Weierstrass form is represented by the set of points (x,y) that satisfy the following equation with a,b∈Fq​ and 4a3+27b2=Fq​​0: E(Fq​):y2=x3+ax+b If P=(xp​,yp​) and Q=(xq​,yq​) are two points in the curve E(Fq​), the algorithm we represent here computes the operation 2P+Q (point doubling and point addition) as (P+Q)+Q. Info Point Q=(xq​,yq​) has nothing to do with the order q of the field Fq​. The original algorithm that is being used can be found in the Section 3.1 of https://arxiv.org/pdf/math/0208038.pdf , which can perform the above operation using 1 multiplication, 2 squarings and 2 divisions (one more squaring) if P=Q), thanks to the fact that computing the Y-coordinate of the intermediate addition is not required. This is more efficient to the standard algorithm that requires 1 more multiplication, 3 squarings in total and 2 divisions. Moreover, this algorithm can be applied not only to the operation 2P+Q, but any other scalar multiplication kP. This can be done by expressing the scalar k in biwise form and performing a double-and-add approach. Nonetheless, this requires conditionals to differentiate 2P from 2P+Q. For that reason, we will implement the following pseudocode from https://github.com/zcash/zcash/issues/3924 (where instead, they give a variant of the above efficient algorithm for Montgomery curves b⋅y2=x3+a⋅x2+x). Acc := [2]T\nfor i = n-1 ... 0: Q := (k_{i + 1} == 1) ? T : -T Acc := Acc + (Q + Acc)\nreturn (k_0 == 0) ? Acc - P : Acc The layout of the witness requires 2 rows. The i-th row will be a VBSM gate whereas the next row will be a ZERO gate. Row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Type i xT yT x0 y0 n n' x1 y1 x2 y2 x3 y3 x4 y4 VBSM i+1 x5 y5 b0 b1 b2 b3 b4 s0 s1 s2 s3 s4 ZERO The gate constraints take care of 5 bits of the scalar multiplication. Each single bit consists of 4 constraints. There is one additional constraint imposed on the final number. Thus, the VarBaseMul gate argument requires 21 constraints. For every bit, there will be one constraint meant to differentiate between addition and subtraction for the operation (P±T)+P: S = (P + (b ? T : −T)) + P We follow these criteria: If the bit is positive, the sign should be a subtraction If the bit is negative, the sign should be an addition Then, paraphrasing the above, we will represent this behavior as: S = (P - (2 * b - 1) * T ) + P Let us call Input the point with coordinates (xI, yI) and Target is the point being added with coordinates (xT, yT). Then Output will be the point with coordinates (xO, yO) resulting from O = ( I ± T ) + I Info Do not confuse our Output point (xO, yO) with the point at infinity that is normally represented as O. In each step of the algorithm, we consider the following elliptic curves affine arithmetic equations: s1​:=xi​−xt​yi​−(2⋅b−1)⋅yt​​ s2​:=2∗xi​+xt​−s12​2⋅yi​​−s1​ xo​:=xt​+s22​−s12​ yo​:=s2​⋅(xi​−xo​)−yi​ For readability, we define the following 3 variables in such a way that s2​ can be expressed as u / t: rx :=s12​−xi​−xt​ t :=xi​− rx ⟺2⋅xi​−s12​+xt​ u :=2⋅yi​− t ⋅s1​⟺2⋅yi​−s1​⋅(2⋅xi​−s12​+xt​) Next, for each bit in the algorithm, we create the following 4 constraints that derive from the above: Booleanity check on the bit b: 0 = b * b - b Constrain s1​: (xI - xT) * s1 = yI – (2b - 1) * yT Constrain Output X-coordinate xo​ and s2​: 0 = u^2 - t^2 * (xO - xT + s1^2) Constrain Output Y-coordinate yo​ and s2​: 0 = (yO + yI) * t - (xI - xO) * u When applied to the 5 bits, the value of the Target point (xT, yT) is maintained, whereas the values for the Input and Output points form the chain: [(x0, y0) -> (x1, y1) -> (x2, y2) -> (x3, y3) -> (x4, y4) -> (x5, y5)] Similarly, 5 different s0..s4 are required, just like the 5 bits b0..b4. Finally, the additional constraint makes sure that the scalar is being correctly expressed into its binary form (using the double-and-add decomposition) as: n′=25⋅n+24⋅b0​+23⋅b1​+22⋅b2​+21⋅b3​+b4​ This equation is translated as the constraint: Binary decomposition: 0 = n' - (b4 + 2 * (b3 + 2 * (b2 + 2 * (b1 + 2 * (b0 + 2*n))))) Range Check The multi range check gadget is comprised of three circuit gates (RangeCheck0, RangeCheck1 and Zero) and can perform range checks on three values (v0​,v1​ and v2​) of up to 88 bits each. Values can be copied as inputs to the multi range check gadget in two ways: (Standard mode) With 3 copies, by copying v0​,v1​ and v2​ to the first cells of the first 3 rows of the gadget. In this mode the first gate coefficient is set to 0. (Compact mode) With 2 copies, by copying v2​ to the first cell of the first row and copying v10​=v0​+2ℓ⋅v1​ to the 2nd cell of row 2. In this mode the first gate coefficient is set to 1. The RangeCheck0 gate can also be used on its own to perform 64-bit range checks by constraining witness cells 1-2 to zero. Byte-order: Each cell value is in little-endian byte order Limbs are mapped to columns in big-endian order (i.e. the lowest columns contain the highest bits) We also have the highest bits covered by copy constraints and plookups, so that we can copy the highest two constraints to zero and get a 64-bit lookup, which are envisioned to be a common case The values are decomposed into limbs as follows: L is a 12-bit lookup (or copy) limb, C is a 2-bit \"crumb\" limb (we call half a nybble a crumb). <----6----> <------8------> v0 = L L L L L L C C C C C C C C v1 = L L L L L L C C C C C C C C <2> <--4--> <---------------18----------------> v2 = C C L L L L C C C C C C C C C C C C C C C C C C Witness structure: Row Contents 0 v0​ 1 v1​ 2 v2​ 3 v0​,v1​,v2​ The first 2 rows contain v0​ and v1​ and their respective decompositions into 12-bit and 2-bit limbs The 3rd row contains v2​ and part of its decomposition: four 12-bit limbs and the 1st 10 crumbs The final row contains v0​'s and v1​'s 5th and 6th 12-bit limbs as well as the remaining 10 crumbs of v2​ Note Because we are constrained to 4 lookups per row, we are forced to postpone some lookups of v0 and v1 to the final row. Constraints: For efficiency, the limbs are constrained differently according to their type: 12-bit limbs are constrained with plookups 2-bit crumbs are constrained with degree-4 constraints x(x−1)(x−2)(x−3) Layout: This is how the three 88-bit inputs v0​,v1​ and v2​ are laid out and constrained. vipj is the jth 12-bit limb of value vi​ vicj is the jth 2-bit crumb limb of value vi​ Gates RangeCheck0 RangeCheck0 RangeCheck1 Zero Rows 0 1 2 3 Cols 0 v0 v1 v2 crumb v2c9 MS:1 copy v0p0 copy v1p0 optional v12 crumb v2c10 2 copy v0p1 copy v1p1 crumb v2c0 crumb v2c11 3 plookup v0p2 plookup v1p2 plookup v2p0 plookup v0p0 4 plookup v0p3 plookup v1p3 plookup v2p1 plookup v0p1 5 plookup v0p4 plookup v1p4 plookup v2p2 plookup v1p0 6 plookup v0p5 plookup v1p5 plookup v2p3 plookup v1p1 7 crumb v0c0 crumb v1c0 crumb v2c1 crumb v2c12 8 crumb v0c1 crumb v1c1 crumb v2c2 crumb v2c13 9 crumb v0c2 crumb v1c2 crumb v2c3 crumb v2c14 10 crumb v0c3 crumb v1c3 crumb v2c4 crumb v2c15 11 crumb v0c4 crumb v1c4 crumb v2c5 crumb v2c16 12 crumb v0c5 crumb v1c5 crumb v2c6 crumb v2c17 13 crumb v0c6 crumb v1c6 crumb v2c7 crumb v2c18 LS:14 crumb v0c7 crumb v1c7 crumb v2c8 crumb v2c19 The 12-bit chunks are constrained with plookups and the 2-bit crumbs are constrained with degree-4 constraints of the form x(x−1)(x−2)(x−3). Note that copy denotes a plookup that is deferred to the 4th gate (i.e. Zero). This is because of the limitation that we have at most 4 lookups per row. The copies are constrained using the permutation argument. Gate types: Different rows are constrained using different CircuitGate types Row CircuitGate Purpose 0 RangeCheck0 Partially constrain v0​ 1 RangeCheck0 Partially constrain v1​ 2 RangeCheck1 Fully constrain v2​ (and trigger plookups constraints on row 3) 3 Zero Complete the constraining of v0​ and v1​ using lookups Note Each CircuitGate type corresponds to a unique polynomial and thus is assigned its own unique powers of alpha RangeCheck0 - Range check constraints This circuit gate is used to partially constrain values v0​ and v1​ Optionally, it can be used on its own as a single 64-bit range check by constraining columns 1 and 2 to zero The rest of v0​ and v1​ are constrained by the lookups in the Zero gate row This gate operates on the Curr row It uses three different types of constraints: copy - copy to another cell (12-bits) plookup - plookup (12-bits) crumb - degree-4 constraint (2-bits) Given value v the layout looks like this Column Curr 0 v 1 copy vp0 2 copy vp1 3 plookup vp2 4 plookup vp3 5 plookup vp4 6 plookup vp5 7 crumb vc0 8 crumb vc1 9 crumb vc2 10 crumb vc3 11 crumb vc4 12 crumb vc5 13 crumb vc6 14 crumb vc7 where the notation vpi and vci defined in the \"Layout\" section above. RangeCheck1 - Range check constraints This circuit gate is used to fully constrain v2​ It operates on the Curr and Next rows It uses two different types of constraints: plookup - plookup (12-bits) crumb - degree-4 constraint (2-bits) Given value v2 the layout looks like this Column Curr Next 0 v2 crumb v2c9 1 optional v12 crumb v2c10 2 crumb v2c0 crumb v2c11 3 plookup v2p0 (ignored) 4 plookup v2p1 (ignored) 5 plookup v2p2 (ignored) 6 plookup v2p3 (ignored) 7 crumb v2c1 crumb v2c12 8 crumb v2c2 crumb v2c13 9 crumb v2c3 crumb v2c14 10 crumb v2c4 crumb v2c15 11 crumb v2c5 crumb v2c16 12 crumb v2c6 crumb v2c17 13 crumb v2c7 crumb v2c18 14 crumb v2c8 crumb v2c19 where the notation v2ci and v2pi defined in the \"Layout\" section above. Foreign Field Addition These circuit gates are used to constrain that left_input +/- right_input = field_overflow * foreign_modulus + result Documentation For more details please see the Foreign Field Addition chapter. Mapping To make things clearer, the following mapping between the variable names used in the code and those of the RFC document can be helpful. left_input_lo -> a0 right_input_lo -> b0 result_lo -> r0 bound_lo -> u0 left_input_mi -> a1 right_input_mi -> b1 result_mi -> r1 bound_mi -> u1 left_input_hi -> a2 right_input_hi -> b2 result_hi -> r2 bound_hi -> u2 field_overflow -> q sign -> s carry_lo -> c0 carry_mi -> c1 bound_carry_lo -> k0 bound_carry_mi -> k1 Note: Our limbs are 88-bit long. We denote with: lo the least significant limb (in little-endian, this is from 0 to 87) mi the middle limb (in little-endian, this is from 88 to 175) hi the most significant limb (in little-endian, this is from 176 to 263) Let left_input_lo, left_input_mi, left_input_hi be 88-bit limbs of the left element Let right_input_lo, right_input_mi, right_input_hi be 88-bit limbs of the right element Let foreign_modulus_lo, foreign_modulus_mi, foreign_modulus_hi be 88-bit limbs of the foreign modulus Then the limbs of the result are result_lo = left_input_lo +/- right_input_lo - field_overflow * foreign_modulus_lo - 2^{88} * result_carry_lo result_mi = left_input_mi +/- right_input_mi - field_overflow * foreign_modulus_mi - 2^{88} * result_carry_mi + result_carry_lo result_hi = left_input_hi +/- right_input_hi - field_overflow * foreign_modulus_hi + result_carry_mi field_overflow =0 or 1 or −1 handles overflows in the field result_carry_i =−1,0,1 are auxiliary variables that handle carries between limbs Apart from the range checks of the chained inputs, we need to do an additional range check for a final bound to make sure that the result is less than the modulus, by adding 2^{3*88} - foreign_modulus to it. (This can be computed easily from the limbs of the modulus) Note that 2^{264} as limbs represents: (0, 0, 0, 1) then: The upper-bound check can be calculated as: bound_lo = result_lo - foreign_modulus_lo - bound_carry_lo * 2^{88} bound_mi = result_mi - foreign_modulus_mi - bound_carry_mi * 2^{88} + bound_carry_lo bound_hi = result_hi - foreign_modulus_hi + 2^{88} + bound_carry_mi Which is equivalent to another foreign field addition with right input 2^{264}, q = 1 and s = 1 bound_lo = result_lo + s * 0 - q * foreign_modulus_lo - bound_carry_lo * 2^{88} bound_mi = result_mi + s * 0 - q * foreign_modulus_mi - bound_carry_mi * 2^{88} + bound_carry_lo bound_hi = result_hi + s * 2^{88} - q * foreign_modulus_hi + bound_carry_mi bound_carry_i =0 or 1 or −1 are auxiliary variables that handle carries between limbs The range check of bound can be skipped until the end of the operations and result is an intermediate value that is unused elsewhere (since the final result must have had the right amount of moduli subtracted along the way, meaning a multiple of the modulus). In other words, intermediate results could potentially give a valid witness that satisfies the constraints but where the result is larger than the modulus (yet smaller than 2^{264}). The reason that we have a final bound check is to make sure that the final result (min_result) is indeed the minimum one (meaning less than the modulus). A more optimized version of these constraints is able to reduce by 2 the number of constraints and by 1 the number of witness cells needed. The idea is to condense the low and middle limbs in one longer limb of 176 bits (which fits inside our native field) and getting rid of the low carry flag. With this idea in mind, the sole carry flag we need is the one located between the middle and the high limbs. Layout The sign of the operation (whether it is an addition or a subtraction) is stored in the fourth coefficient as a value +1 (for addition) or -1 (for subtraction). The first 3 coefficients are the 3 limbs of the foreign modulus. One could lay this out as a double-width gate for chained foreign additions and a final row, e.g.: col ForeignFieldAdd chain ForeignFieldAdd final ForeignFieldAdd final Zero 0 left_input_lo (copy) result_lo (copy) min_result_lo (copy) bound_lo (copy) 1 left_input_mi (copy) result_mi (copy) min_result_mi (copy) bound_mi (copy) 2 left_input_hi (copy) result_hi (copy) min_result_hi (copy) bound_hi (copy) 3 right_input_lo (copy) 0 (check) 4 right_input_mi (copy) 0 (check) 5 right_input_hi (copy) 2^88 (check) 6 field_overflow (copy?) 1 (check) 7 carry bound_carry 8 9 10 11 12 13 14 We reuse the foreign field addition gate for the final bound check since this is an addition with a specific parameter structure. Checking that the correct right input, overflow, and overflow are used shall be done by copy constraining these values with a public input value. One could have a specific gate for just this check requiring less constrains, but the cost of adding one more selector gate outweights the savings of one row and a few constraints of difference. Integration Copy final overflow bit from public input containing value 1 * Range check the final bound Foreign Field Multiplication This gadget is used to constrain that left_input * right_input = quotient * foreign_field_modulus + remainder Documentation For more details please see the Foreign Field Multiplication chapter or the original Foreign Field Multiplication RFC Notations For clarity, we use more descriptive variable names in the code than in the RFC, which uses mathematical notations. In order to relate the two documents, the following mapping between the variable names used in the code and those of the RFC can be helpful. left_input0 => a0 right_input0 => b0 quotient0 => q0 remainder01 => r01\nleft_input1 => a1 right_input1 => b1 quotient1 => q1\nleft_input2 => a2 right_input2 => b2 quotient2 => q2 remainder2 => r2 product1_lo => p10 product1_hi_0 => p110 product1_hi_1 => p111 carry0 => v0 carry1_lo => v10 carry1_hi => v11 quotient_hi_bound => q'2 Suffixes The variable names in this code uses descriptive suffixes to convey information about the positions of the bits referred to. When a word is split into up to n parts we use: 0, 1 ... n (where n is the most significant). For example, if we split word x into three limbs, we'd name them x0, x1 and x2 or x[0], x[1] and x[2]. Continuing in this fashion, when one of those words is subsequently split in half, then we add the suffixes _lo and _hi, where hi corresponds to the most significant bits. For our running example, x1 would become x1_lo and x1_hi. If we are splitting into more than two things, then we pick meaningful names for each. So far we've explained our conventions for a splitting depth of up to 2. For splitting deeper than two, we simply cycle back to our depth 1 suffixes again. So for example, x1_lo would be split into x1_lo_0 and x1_lo_1. Parameters hi_foreign_field_modulus := high limb of foreign field modulus $f$ (stored in gate coefficient 0) neg_foreign_field_modulus := negated foreign field modulus $f'$ (stored in gate coefficients 1-3) n := the native field modulus is obtainable from F, the native field's trait bound Witness left_input := left foreign field element multiplicand $ ~\\in F_f$ right_input := right foreign field element multiplicand $ ~\\in F_f$ quotient := foreign field quotient $ ~\\in F_f$ remainder := foreign field remainder $ ~\\in F_f$ carry0 := 2 bit carry carry1_lo := low 88 bits of carry1 carry1_hi := high 3 bits of carry1 product1_lo := lowest 88 bits of middle intermediate product product1_hi_0 := lowest 88 bits of middle intermediate product's highest 88 + 2 bits product1_hi_1 := highest 2 bits of middle intermediate product quotient_hi_bound := quotient high bound for checking q2 ≤ f2 Layout The foreign field multiplication gate's rows are laid out like this col ForeignFieldMul Zero 0 left_input0 (copy) remainder01 (copy) 1 left_input1 (copy) remainder2 (copy) 2 left_input2 (copy) quotient0 (copy) 3 right_input0 (copy) quotient1 (copy) 4 right_input1 (copy) quotient2 (copy) 5 right_input2 (copy) quotient_hi_bound (copy) 6 product1_lo (copy) product1_hi_0 (copy) 7 carry1_0 (plookup) product1_hi_1 (dummy) 8 `carry1_12 (plookup) carry1_48 (plookup) 9 carry1_24 (plookup) carry1_60 (plookup) 10 carry1_36 (plookup) carry1_72 (plookup) 11 carry1_84 carry0 12 carry1_86 13 carry1_88 14 carry1_90 Rotation Rotation of a 64-bit word by a known offset Rot64 onstrains known-length rotation of 64-bit words: This circuit gate is used to constrain that a 64-bit word is rotated by $r < 64$ bits to the \"left\". The rotation is performed towards the most significant side (thus, the new LSB is fed with the old MSB). This gate operates on the Curr and Next rows. The idea is to split the rotation operation into two parts: Shift to the left Add the excess bits to the right We represent shifting with multiplication modulo $2^{64}$. That is, for each word to be rotated, we provide in the witness a quotient and a remainder, similarly to ForeignFieldMul such that the following operation holds: $$word \\cdot 2^{rot} = quotient \\cdot 2^{64} + remainder$$ Then, the remainder corresponds to the shifted word, and the quotient corresponds to the excess bits. $$word \\cdot 2^{rot} = excess \\cdot 2^{64} + shifted$$ Thus, in order to obtain the rotated word, we need to add the quotient and the remainder as follows: $$rotated = shifted + excess$$ The input word is known to be of length 64 bits. All we need for soundness is check that the shifted and excess parts of the word have the correct size as well. That means, we need to range check that: $$ \\begin{aligned} excess &< 2^{rot}\\ shifted &< 2^{64} \\end{aligned} $$ The latter can be obtained with a RangeCheck0 gate setting the two most significant limbs to zero. The former is equivalent to the following check: $$bound = excess - 2^{rot} + 2^{64} < 2^{64}$$ which is doable with the constraints in a RangeCheck0 gate. Since our current row within the Rot64 gate is almost empty, we can use it to perform the range check within the same gate. Then, using the following layout and assuming that the gate has a coefficient storing the value $2^{rot}$, which is publicly known Gate Rot64 RangeCheck0 gadgets (designer's duty) Column Curr Next ------ ------------------- ---------------- 0 copy word shifted 1 copy rotated 0 2 excess 0 3 bound_limb0 shifted_limb0 4 bound_limb1 shifted_limb1 5 bound_limb2 shifted_limb2 6 bound_limb3 shifted_limb3 7 bound_crumb0 shifted_crumb0 8 bound_crumb1 shifted_crumb1 9 bound_crumb2 shifted_crumb2 10 bound_crumb3 shifted_crumb3 11 bound_crumb4 shifted_crumb4 12 bound_crumb5 shifted_crumb5 13 bound_crumb6 shifted_crumb6 14 bound_crumb7 shifted_crumb7 In Keccak, rotations are performed over a 5x5 matrix state of w-bit words each cell. The values used to perform the rotation are fixed, public, and known in advance, according to the following table, depending on the coordinate of each cell within the 5x5 matrix state: y \\ x 0 1 2 3 4 0 0 36 3 105 210 1 1 300 10 45 66 2 190 6 171 15 253 3 28 55 153 21 120 4 91 276 231 136 78 But since we will always be using 64-bit words in our Keccak usecase ($w = 64$), we can have an equivalent table with these values modulo 64 to avoid needing multiple passes of the rotation gate (a single step would cause overflows otherwise): y \\ x 0 1 2 3 4 0 0 36 3 41 18 1 1 44 10 45 2 2 62 6 43 15 61 3 28 55 25 21 56 4 27 20 39 8 14 Since there is one value of the coordinates (x, y) where the rotation is 0 bits, we can skip that step in the gadget. This will save us one gate, and thus the whole 25-1=24 rotations will be performed in just 48 rows. Xor Xor16 - Chainable XOR constraints for words of multiples of 16 bits. This circuit gate is used to constrain that in1 xored with in2 equals out The length of in1, in2 and out must be the same and a multiple of 16bits. This gate operates on the Curr and Next rows. It uses three different types of constraints: copy - copy to another cell (32-bits) plookup - xor-table plookup (4-bits) decomposition - the constraints inside the gate The 4-bit nybbles are assumed to be laid out with 0 column being the least significant nybble. Given values in1, in2 and out, the layout looks like this: Column Curr Next 0 copy in1 copy in1' 1 copy in2 copy in2' 2 copy out copy out' 3 plookup0 in1_0 4 plookup1 in1_1 5 plookup2 in1_2 6 plookup3 in1_3 7 plookup0 in2_0 8 plookup1 in2_1 9 plookup2 in2_2 10 plookup3 in2_3 11 plookup0 out_0 12 plookup1 out_1 13 plookup2 out_2 14 plookup3 out_3 One single gate with next values of in1', in2' and out' being zero can be used to check that the original in1, in2 and out had 16-bits. We can chain this gate 4 times as follows to obtain a gadget for 64-bit words XOR: Row CircuitGate Purpose 0 Xor16 Xor 2 least significant bytes of the words 1 Xor16 Xor next 2 bytes of the words 2 Xor16 Xor next 2 bytes of the words 3 Xor16 Xor 2 most significant bytes of the words 4 Generic Zero values, can be reused as generic gate Info We could halve the number of rows of the 64-bit XOR gadget by having lookups for 8 bits at a time, but for now we will use the 4-bit XOR table that we have. Rough computations show that if we run 8 or more Keccaks in one circuit we should use the 8-bit XOR table.","breadcrumbs":"Kimchi » Gates","id":"254","title":"Gates"},"255":{"body":"Here we describe basic gadgets that we build using a combination of the gates described above. Not We implement NOT, i.e. bitwise negation, as a gadget in two different ways, needing no new gate type for it. Instead, it reuses the XOR gadget and the Generic gate. The first version of the NOT gadget reuses Xor16 by making the following observation: the bitwise NOT operation is equivalent to the bitwise XOR operation with the all one words of a certain length . In other words, $$\\neg x = x \\oplus 1^ $$ where $1^ $ denotes a bitstring of all ones of length $|x|$. Let $x_i$ be the $i$-th bit of $x$, the intuition is that if $x_i = 0$ then XOR with $1$ outputs $1$, thus negating $x_i$. Similarly, if $x_i = 1$ then XOR with 1 outputs 0, again negating $x_i$. Thus, bitwise XOR with $1^*$ is equivalent to bitwise negation (i.e. NOT). Then, if we take the XOR gadget with a second input to be the all one word of the same length, that gives us the NOT gadget. The correct length can be imposed by having a public input containing the 2^bits - 1 value and wiring it to the second input of the XOR gate. This approach needs as many rows as an XOR would need, for a single negation, but it comes with the advantage of making sure the input is of a certain length. The other approach can be more efficient if we already know the length of the inputs. For example, the input may be the input of a range check gate, or the output of a previous XOR gadget (which will be the case in our Keccak usecase). In this case, we simply perform the negation as a subtraction of the input word from the all one word (which again can be copied from a public input). This comes with the advantage of holding up to 2 word negations per row (an eight-times improvement over the XOR approach), but it requires the user to know the length of the input. NOT Layout using XOR Here we show the layout of the NOT gadget using the XOR approach. The gadget needs a row with a public input containing the all-one word of the given length. Then, a number of XORs follow, and a final Zero row is needed. In this case, the NOT gadget needs $\\ceil(\\frac{|x|}{16})$ Xor16 gates, that means one XOR row for every 16 bits of the input word. Row CircuitGate Purpose pub Generic Leading row with the public $1^*$ value i...i+n-1 Xor16 Negate every 4 nybbles of the word, from least to most significant i+n Generic Constrain that the final row is all zeros for correctness of Xor gate NOT Layout using Generic gates Here we show the layout of the NOT gadget using the Generic approach. The gadget needs a row with a public input containing the all-one word of the given length, exactly as above. Then, one Generic gate reusing the all-one word as left inputs can be used to negate up to two words per row. This approach requires that the input word is known (or constrained) to have a given length. Row CircuitGate Purpose pub Generic Leading row with the public $1^*$ value i Generic Negate one or two words of the length given by the length of the all-one word And We implement the AND gadget making use of the XOR gadget and the Generic gate. A new gate type is not needed, but we could potentially add an And16 gate type reusing the same ideas of Xor16 so as to save one final generic gate, at the cost of one additional AND lookup table that would have the same size as that of the Xor. For now, we are willing to pay this small overhead and produce AND gadget as follows: We observe that we can express bitwise addition as follows: $$A + B = (A \\oplus B) + 2 \\cdot (A \\wedge B)$$ where $\\oplus$ is the bitwise XOR operation, $\\wedge$ is the bitwise AND operation, and $+$ is the addition operation. In other words, the value of the addition is nothing but the XOR of its operands, plus the carry bit if both operands are 1. Thus, we can rewrite the above equation to obtain a definition of the AND operation as follows: $$A \\wedge B = \\frac{A + B - (A \\oplus B)}{2}$$ Let us define the following operations for better readability: a + b = sum\na x b = xor\na ^ b = and Then, we can rewrite the above equation as follows: $$ 2 \\cdot and = sum - xor $$ which can be expressed as a double generic gate. Then, our AND gadget for $n$ bytes looks as follows: $n/8$ Xor16 gates 1 (single) Generic gate to check that the final row of the XOR chain is all zeros. 1 (double) Generic gate to check sum $a + b = sum$ and the conjunction equation $2\\cdot and = sum - xor$. Finally, we connect the wires in the following positions (apart from the ones already connected for the XOR gates): Column 2 of the first Xor16 row (the output of the XOR operation) is connected to the right input of the second generic operation of the last row. Column 2 of the first generic operation of the last row is connected to the left input of the second generic operation of the last row. Meaning, the xor in a x b = xor is connected to the xor in 2 \\cdot and = sum - xor the sum in a + b = sum is connected to the sum in 2 \\cdot and = sum - xor","breadcrumbs":"Kimchi » Gadgets","id":"255","title":"Gadgets"},"256":{"body":"In this section we specify the setup that goes into creating two indexes from a circuit: A prover index , necessary for the prover to create proofs. A verifier index , necessary for the verifier to verify proofs. Note The circuit creation part is not specified in this document. It might be specified in a separate document, or we might want to specify how to create the circuit description tables. As such, the transformation of a circuit into these two indexes can be seen as a compilation step. Note that the prover still needs access to the original circuit to create proofs, as they need to execute it to create the witness (register table).","breadcrumbs":"Kimchi » Setup","id":"256","title":"Setup"},"257":{"body":"In this section we describe data that both the prover and the verifier index share. URS (Uniform Reference String) The URS is a set of parameters that is generated once, and shared between the prover and the verifier. It is used for polynomial commitments, so refer to the poly-commitment specification for more details. Note Kimchi currently generates the URS based on the circuit, and attach it to the index. So each circuit can potentially be accompanied with a different URS. On the other hand, Mina reuses the same URS for multiple circuits ( see zkapps for more details ). Domain . A domain large enough to contain the circuit and the zero-knowledge rows (used to provide zero-knowledge to the protocol). Specifically, the smallest subgroup in our field that has order greater or equal to n + ZK_ROWS, with n is the number of gates in the circuit. TODO: what if the domain is larger than the URS? Ordering of elements in the domain Note that in this specification we always assume that the first element of a domain is $1$. Shifts . As part of the permutation, we need to create PERMUTS shifts. To do that, the following logic is followed (in pseudo code): (TODO: move shift creation within the permutation section?) shifts[0] = 1 # first shift is identity for i in 0..7: # generate 7 shifts i = 7 shift, i = sample(domain, i) while shifts.contains(shift) do: shift, i = sample(domain, i) shift[i] = shift def sample(domain, i): i += 1 shift = Field(Blake2b512(to_be_bytes(i))) while is_not_quadratic_non_residue(shift) || domain.contains(shift): i += 1 shift = Field(Blake2b512(to_be_bytes(i))) return shift, i Public . This variable simply contains the number of public inputs. (TODO: actually, it's not contained in the verifier index) The compilation steps to create the common index are as follow: If the circuit is less than 2 gates, abort. Compute the number of zero-knowledge rows (zk_rows) that will be required to achieve zero-knowledge. The following constraints apply to zk_rows: The number of chunks c results in an evaluation at zeta and zeta * omega in each column for 2*c evaluations per column, so zk_rows >= 2*c + 1. The permutation argument interacts with the c chunks in parallel, so it is possible to cross-correlate between them to compromise zero knowledge. We know that there is some c >= 1 such that zk_rows = 2*c + k from the above. Thus, attempting to find the evaluation at a new point, we find that: the evaluation of every witness column in the permutation contains k unknowns; the evaluations of the permutation argument aggregation has k-1 unknowns; the permutation argument applies on all but zk_rows - 3 rows; and thus we form the equation zk_rows - 3 < 7 * k + (k - 1) to ensure that we can construct fewer equations than we have unknowns. This simplifies to k > (2 * c - 2) / 7, giving zk_rows > (16 * c - 2) / 7. We can derive c from the max_poly_size supported by the URS, and thus we find zk_rows and domain_size satisfying the fixpoint zk_rows = (16 * (domain_size / max_poly_size) + 5) / 7\ndomain_size = circuit_size + zk_rows Create a domain for the circuit. That is, compute the smallest subgroup of the field that has order greater or equal to n + zk_rows elements. Pad the circuit: add zero gates to reach the domain size. sample the PERMUTS shifts.","breadcrumbs":"Kimchi » Common Index","id":"257","title":"Common Index"},"258":{"body":"If lookup is used, the following values are added to the common index: LookupSelectors . The list of lookup selectors used. In practice, this tells you which lookup tables are used. TableIds . This is a list of table ids used by the Lookup gate. MaxJointSize . This is the maximum number of columns appearing in the lookup tables used by the lookup selectors. For example, the XOR lookup has 3 columns. To create the index, follow these steps: If no lookup is used in the circuit, do not create a lookup index Get the lookup selectors and lookup tables that are specified implicitly Concatenate explicit runtime lookup tables with the ones (implicitly) used by gates. Get the highest number of columns max_table_width that a lookup table can have. Create the concatenated table of all the fixed lookup tables. It will be of height the size of the domain, and of width the maximum width of any of the lookup tables. In addition, create an additional column to store all the tables' table IDs. For example, if you have a table with ID 0 1 2 3 5 6 7 0 0 0 and another table with ID 1 8 9 the concatenated table in a domain of size 5 looks like this: 1 2 3 5 6 7 0 0 0 8 9 0 0 0 0 with the table id vector: table id 0 0 0 1 0 To do this, for each table: Update the corresponding entries in a table id vector (of size the domain as well) with the table ID of the table. Copy the entries from the table to new rows in the corresponding columns of the concatenated table. Fill in any unused columns with 0 (to match the dummy value) Pad the end of the concatened table with the dummy value. Pad the end of the table id vector with 0s. pre-compute polynomial and evaluation form for the look up tables pre-compute polynomial and evaluation form for the table IDs, only if a table with an ID different from zero was used.","breadcrumbs":"Kimchi » Lookup Index","id":"258","title":"Lookup Index"},"259":{"body":"Both the prover and the verifier index, besides the common parts described above, are made out of pre-computations which can be used to speed up the protocol. These pre-computations are optimizations, in the context of normal proofs, but they are necessary for recursion. pub struct ProverIndex> { /// constraints system polynomials #[serde(bound = \"ConstraintSystem: Serialize + DeserializeOwned\")] pub cs: ConstraintSystem, /// The symbolic linearization of our circuit, which can compile to concrete types once certain values are learned in the protocol. #[serde(skip)] pub linearization: Linearization>, Column>, /// The mapping between powers of alpha and constraints #[serde(skip)] pub powers_of_alpha: Alphas, /// polynomial commitment keys #[serde(skip)] #[serde(bound(deserialize = \"OpeningProof::SRS: Default\"))] pub srs: Arc, /// maximal size of polynomial section pub max_poly_size: usize, #[serde(bound = \"ColumnEvaluations: Serialize + DeserializeOwned\")] pub column_evaluations: ColumnEvaluations, /// The verifier index corresponding to this prover index #[serde(skip)] pub verifier_index: Option>, /// The verifier index digest corresponding to this prover index #[serde_as(as = \"Option\")] pub verifier_index_digest: Option,\n}","breadcrumbs":"Kimchi » Prover Index","id":"259","title":"Prover Index"},"26":{"body":"The homomorphic commitment commit(x,r)=xG+rH described above is known as a Pedersen commitment. If you remove the rH term you get a non-hiding commitment, called a Pedersen hash . Both rely on the ECDLP hardness assumption. This means that, at least theoretically, you might be lucky (or have a quantum computer) and figure out that H=hG, which would allow you to find different values x′ and h′ to open the commitment. We say that pedersen commitments are computationally binding and not unconditionally binding. For example, you could express c=xG+rH alternatively as c=xG+rhG=(x+rh)G and compute a satisfying opening pair x′=rh and r′=hx​. On the other hand, Pedersen commitments are unconditionally hiding , as there is no way (even with a magic computer) to reveal what x is without knowing r. Lack of perfect binding is the reason why most of the \"proofs\" we will see later in this book are not referred to as proofs, but instead are referred to as arguments of knowledge (although we may care little about this distinction). Just remember that you need perfect binding to be called a proof. Interestingly, it is impossible to have a commitment scheme that has both perfect hiding and perfect binding. To recap, in cryptography the following distinctions are important Perfect. The property that an algorithm is statistically sound without hardness assumptions, also known as unconditional or statistical soundness. Computational. The algorithm relies on a hardness assumption or computational limitation for soundness. Thus, said another way, Pedersen commitments provide perfect hiding and computational binding.","breadcrumbs":"Commitments » Pedersen commitments","id":"26","title":"Pedersen commitments"},"260":{"body":"Same as the prover index, we have a number of pre-computations as part of the verifier index. #[serde_as]\n#[derive(Serialize, Deserialize, Debug, Clone)]\npub struct LookupVerifierIndex { pub joint_lookup_used: bool, #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub lookup_table: Vec>, #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub lookup_selectors: LookupSelectors>, /// Table IDs for the lookup values. /// This may be `None` if all lookups originate from table 0. #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub table_ids: Option>, /// Information about the specific lookups used pub lookup_info: LookupInfo, /// An optional selector polynomial for runtime tables #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub runtime_tables_selector: Option>,\n} #[serde_as]\n#[derive(Serialize, Deserialize, Debug, Clone)]\npub struct VerifierIndex> { /// evaluation domain #[serde_as(as = \"o1_utils::serialization::SerdeAs\")] pub domain: D, /// maximal size of polynomial section pub max_poly_size: usize, /// the number of randomized rows to achieve zero knowledge pub zk_rows: u64, /// polynomial commitment keys #[serde(skip)] #[serde(bound(deserialize = \"OpeningProof::SRS: Default\"))] pub srs: Arc, /// number of public inputs pub public: usize, /// number of previous evaluation challenges, for recursive proving pub prev_challenges: usize, // index polynomial commitments /// permutation commitment array #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub sigma_comm: [PolyComm; PERMUTS], /// coefficient commitment array #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub coefficients_comm: [PolyComm; COLUMNS], /// coefficient commitment array #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub generic_comm: PolyComm, // poseidon polynomial commitments /// poseidon constraint selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub psm_comm: PolyComm, // ECC arithmetic polynomial commitments /// EC addition selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub complete_add_comm: PolyComm, /// EC variable base scalar multiplication selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub mul_comm: PolyComm, /// endoscalar multiplication selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub emul_comm: PolyComm, /// endoscalar multiplication scalar computation selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub endomul_scalar_comm: PolyComm, /// RangeCheck0 polynomial commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub range_check0_comm: Option>, /// RangeCheck1 polynomial commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub range_check1_comm: Option>, /// Foreign field addition gates polynomial commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub foreign_field_add_comm: Option>, /// Foreign field multiplication gates polynomial commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub foreign_field_mul_comm: Option>, /// Xor commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub xor_comm: Option>, /// Rot commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub rot_comm: Option>, /// wire coordinate shifts #[serde_as(as = \"[o1_utils::serialization::SerdeAs; PERMUTS]\")] pub shift: [G::ScalarField; PERMUTS], /// zero-knowledge polynomial #[serde(skip)] pub permutation_vanishing_polynomial_m: OnceCell>, // TODO(mimoo): isn't this redundant with domain.d1.group_gen ? /// domain offset for zero-knowledge #[serde(skip)] pub w: OnceCell, /// endoscalar coefficient #[serde(skip)] pub endo: G::ScalarField, #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub lookup_index: Option>, #[serde(skip)] pub linearization: Linearization>, Column>, /// The mapping between powers of alpha and constraints #[serde(skip)] pub powers_of_alpha: Alphas,\n}","breadcrumbs":"Kimchi » Verifier Index","id":"260","title":"Verifier Index"},"261":{"body":"Originally, kimchi is based on an interactive protocol that was transformed into a non-interactive one using the Fiat-Shamir transform. For this reason, it can be useful to visualize the high-level interactive protocol before the transformation: sequenceDiagram participant Prover participant Verifier Note over Prover,Verifier: Prover produces commitments to secret polynomials Prover->>Verifier: public input & witness commitment Verifier->>Prover: beta & gamma Prover->>Verifier: permutation commitment opt lookup Prover->>Verifier: sorted Prover->>Verifier: aggreg end Note over Prover,Verifier: Prover produces commitment to quotient polynomial Verifier->>Prover: alpha Prover->>Verifier: quotient commitment Note over Prover,Verifier: Verifier produces an evaluation point Verifier->>Prover: zeta Note over Verifier: change of verifier (change of sponge) opt recursion Prover->>Verifier: recursion challenges end Note over Prover,Verifier: Prover provides needed evaluations for the linearization Prover->>Verifier: ft(zeta * omega) Prover->>Verifier: negated public input p(zeta) & p(zeta * omega) Prover->>Verifier: the 15 registers w_i(zeta) & w_i(zeta * omega) Prover->>Verifier: permutation poly z(zeta) & z(zeta * omega) Prover->>Verifier: the 6 sigmas s_i(zeta) & s_i(zeta * omega) Prover->>Verifier: the 15 coefficients c_i(zeta) & c_i(zeta * omega) opt lookup Prover->>Verifier: sorted(zeta) & sorted(zeta * omega) Prover->>Verifier: aggreg(zeta) & aggreg(zeta * omega) Prover->>Verifier: table(zeta) & table(zeta * omega) end Prover->>Verifier: the generic selector gen(zeta) & gen(zeta * omega) Prover->>Verifier: the poseidon selector pos(zeta) & pos(zeta * omega) Note over Prover,Verifier: Batch verification of evaluation proofs Verifier->>Prover: v, u Note over Verifier: change of verifier (change of sponge) Prover->>Verifier: aggregated evaluation proof (involves more interaction) The Fiat-Shamir transform simulates the verifier messages via a hash function that hashes the transcript of the protocol so far before outputing verifier messages. You can find these operations under the proof creation and proof verification algorithms as absorption and squeezing of values with the sponge.","breadcrumbs":"Kimchi » Proof Construction & Verification","id":"261","title":"Proof Construction & Verification"},"262":{"body":"A proof consists of the following data structures: /// Evaluations of a polynomial at 2 points\n#[serde_as]\n#[derive(Copy, Clone, Serialize, Deserialize, Default, Debug)]\n#[cfg_attr( feature = \"ocaml_types\", derive(ocaml::IntoValue, ocaml::FromValue, ocaml_gen::Struct)\n)]\n#[serde(bound( serialize = \"Vec: serde_with::SerializeAs\", deserialize = \"Vec: serde_with::DeserializeAs<'de, Evals>\"\n))]\npub struct PointEvaluations { /// Evaluation at the challenge point zeta. #[serde_as(as = \"Vec\")] pub zeta: Evals, /// Evaluation at `zeta . omega`, the product of the challenge point and the group generator. #[serde_as(as = \"Vec\")] pub zeta_omega: Evals,\n} // TODO: this should really be vectors here, perhaps create another type for chunked evaluations?\n/// Polynomial evaluations contained in a `ProverProof`.\n/// - **Chunked evaluations** `Field` is instantiated with vectors with a length\n/// that equals the length of the chunk\n/// - **Non chunked evaluations** `Field` is instantiated with a field, so they\n/// are single-sized#[serde_as]\n#[serde_as]\n#[derive(Debug, Clone, Serialize, Deserialize)]\npub struct ProofEvaluations { /// public input polynomials pub public: Option, /// witness polynomials pub w: [Evals; COLUMNS], /// permutation polynomial pub z: Evals, /// permutation polynomials /// (PERMUTS-1 evaluations because the last permutation is only used in /// commitment form) pub s: [Evals; PERMUTS - 1], /// coefficient polynomials pub coefficients: [Evals; COLUMNS], /// evaluation of the generic selector polynomial pub generic_selector: Evals, /// evaluation of the poseidon selector polynomial pub poseidon_selector: Evals, /// evaluation of the elliptic curve addition selector polynomial pub complete_add_selector: Evals, /// evaluation of the elliptic curve variable base scalar multiplication /// selector polynomial pub mul_selector: Evals, /// evaluation of the endoscalar multiplication selector polynomial pub emul_selector: Evals, /// evaluation of the endoscalar multiplication scalar computation selector /// polynomial pub endomul_scalar_selector: Evals, // Optional gates /// evaluation of the RangeCheck0 selector polynomial pub range_check0_selector: Option, /// evaluation of the RangeCheck1 selector polynomial pub range_check1_selector: Option, /// evaluation of the ForeignFieldAdd selector polynomial pub foreign_field_add_selector: Option, /// evaluation of the ForeignFieldMul selector polynomial pub foreign_field_mul_selector: Option, /// evaluation of the Xor selector polynomial pub xor_selector: Option, /// evaluation of the Rot selector polynomial pub rot_selector: Option, // lookup-related evaluations /// evaluation of lookup aggregation polynomial pub lookup_aggregation: Option, /// evaluation of lookup table polynomial pub lookup_table: Option, /// evaluation of lookup sorted polynomials pub lookup_sorted: [Option; 5], /// evaluation of runtime lookup table polynomial pub runtime_lookup_table: Option, // lookup selectors /// evaluation of the runtime lookup table selector polynomial pub runtime_lookup_table_selector: Option, /// evaluation of the Xor range check pattern selector polynomial pub xor_lookup_selector: Option, /// evaluation of the Lookup range check pattern selector polynomial pub lookup_gate_lookup_selector: Option, /// evaluation of the RangeCheck range check pattern selector polynomial pub range_check_lookup_selector: Option, /// evaluation of the ForeignFieldMul range check pattern selector /// polynomial pub foreign_field_mul_lookup_selector: Option,\n} /// Commitments linked to the lookup feature\n#[serde_as]\n#[derive(Debug, Clone, Serialize, Deserialize)]\n#[serde(bound = \"G: ark_serialize::CanonicalDeserialize + ark_serialize::CanonicalSerialize\")]\npub struct LookupCommitments { /// Commitments to the sorted lookup table polynomial (may have chunks) pub sorted: Vec>, /// Commitment to the lookup aggregation polynomial pub aggreg: PolyComm, /// Optional commitment to concatenated runtime tables pub runtime: Option>,\n} /// All the commitments that the prover creates as part of the proof.\n#[serde_as]\n#[derive(Debug, Clone, Serialize, Deserialize)]\n#[serde(bound = \"G: ark_serialize::CanonicalDeserialize + ark_serialize::CanonicalSerialize\")]\npub struct ProverCommitments { /// The commitments to the witness (execution trace) pub w_comm: [PolyComm; COLUMNS], /// The commitment to the permutation polynomial pub z_comm: PolyComm, /// The commitment to the quotient polynomial pub t_comm: PolyComm, /// Commitments related to the lookup argument pub lookup: Option>,\n} /// The proof that the prover creates from a\n/// [ProverIndex](super::prover_index::ProverIndex) and a `witness`.\n#[serde_as]\n#[derive(Debug, Clone, Serialize, Deserialize)]\n#[serde(bound = \"G: ark_serialize::CanonicalDeserialize + ark_serialize::CanonicalSerialize\")]\npub struct ProverProof { /// All the polynomial commitments required in the proof pub commitments: ProverCommitments, /// batched commitment opening proof #[serde(bound( serialize = \"OpeningProof: Serialize\", deserialize = \"OpeningProof: Deserialize<'de>\" ))] pub proof: OpeningProof, /// Two evaluations over a number of committed polynomials pub evals: ProofEvaluations>>, /// Required evaluation for [Maller's /// optimization](https://o1-labs.github.io/proof-systems/kimchi/maller_15.html#the-evaluation-of-l) #[serde_as(as = \"o1_utils::serialization::SerdeAs\")] pub ft_eval1: G::ScalarField, /// The challenges underlying the optional polynomials folded into the proof pub prev_challenges: Vec>,\n} /// A struct to store the challenges inside a `ProverProof`\n#[serde_as]\n#[derive(Debug, Clone, Deserialize, Serialize)]\n#[serde(bound = \"G: ark_serialize::CanonicalDeserialize + ark_serialize::CanonicalSerialize\")]\npub struct RecursionChallenge\nwhere G: AffineRepr,\n{ /// Vector of scalar field elements #[serde_as(as = \"Vec\")] pub chals: Vec, /// Polynomial commitment pub comm: PolyComm,\n} The following sections specify how a prover creates a proof, and how a verifier validates a number of proofs.","breadcrumbs":"Kimchi » Proof Structure","id":"262","title":"Proof Structure"},"263":{"body":"To create a proof, the prover expects: A prover index, containing a representation of the circuit (and optionaly pre-computed values to be used in the proof creation). The (filled) registers table, representing parts of the execution trace of the circuit. Note The public input is expected to be passed in the first Public rows of the registers table. The following constants are set: EVAL_POINTS = 2. This is the number of points that the prover has to evaluate their polynomials at. ($\\zeta$ and $\\zeta\\omega$ where $\\zeta$ will be deterministically generated.) ZK_ROWS = 3. This is the number of rows that will be randomized to provide zero-knowledgeness. Note that it only needs to be greater or equal to the number of evaluations (2) in the protocol. Yet, it contains one extra row to take into account the last constraint (final value of the permutation accumulator). (TODO: treat the final constraint separately so that ZK_ROWS = 2) The prover then follows the following steps to create the proof: Ensure we have room in the witness for the zero-knowledge rows. We currently expect the witness not to be of the same length as the domain, but instead be of the length of the (smaller) circuit. If we cannot add zk_rows rows to the columns of the witness before reaching the size of the domain, abort. Pad the witness columns with Zero gates to make them the same length as the domain. Then, randomize the last zk_rows of each columns. Setup the Fq-Sponge. Absorb the digest of the VerifierIndex. Absorb the commitments of the previous challenges with the Fq-sponge. Compute the negated public input polynomial as the polynomial that evaluates to $-p_i$ for the first public_input_size values of the domain, and $0$ for the rest. Commit (non-hiding) to the negated public input polynomial. Absorb the commitment to the public polynomial with the Fq-Sponge. Note: unlike the original PLONK protocol, the prover also provides evaluations of the public polynomial to help the verifier circuit. This is why we need to absorb the commitment to the public polynomial at this point. Commit to the witness columns by creating COLUMNS hidding commitments. Note: since the witness is in evaluation form, we can use the commit_evaluation optimization. Absorb the witness commitments with the Fq-Sponge. Compute the witness polynomials by interpolating each COLUMNS of the witness. As mentioned above, we commit using the evaluations form rather than the coefficients form so we can take advantage of the sparsity of the evaluations (i.e., there are many 0 entries and entries that have less-than-full-size field elemnts.) If using lookup: if using runtime table: check that all the provided runtime tables have length and IDs that match the runtime table configuration of the index we expect the given runtime tables to be sorted as configured, this makes it easier afterwards calculate the contribution to the second column of the lookup table (the runtime vector) If queries involve a lookup table with multiple columns then squeeze the Fq-Sponge to obtain the joint combiner challenge $j'$, otherwise set the joint combiner challenge $j'$ to $0$. Derive the scalar joint combiner $j$ from $j'$ using the endomorphism (TODO: specify) If multiple lookup tables are involved, set the table_id_combiner as the $j^i$ with $i$ the maximum width of any used table. Essentially, this is to add a last column of table ids to the concatenated lookup tables. Compute the dummy lookup value as the combination of the last entry of the XOR table (so (0, 0, 0)). Warning: This assumes that we always use the XOR table when using lookups. Compute the lookup table values as the combination of the lookup table entries. Compute the sorted evaluations. Randomize the last EVALS rows in each of the sorted polynomials in order to add zero-knowledge to the protocol. Commit each of the sorted polynomials. Absorb each commitments to the sorted polynomials. Sample $\\beta$ with the Fq-Sponge. Sample $\\gamma$ with the Fq-Sponge. If using lookup: Compute the lookup aggregation polynomial. Commit to the aggregation polynomial. Absorb the commitment to the aggregation polynomial with the Fq-Sponge. Compute the permutation aggregation polynomial $z$. Commit (hidding) to the permutation aggregation polynomial $z$. Absorb the permutation aggregation polynomial $z$ with the Fq-Sponge. Sample $\\alpha'$ with the Fq-Sponge. Derive $\\alpha$ from $\\alpha'$ using the endomorphism (TODO: details) TODO: instantiate alpha? Compute the quotient polynomial (the $t$ in $f = Z_H \\cdot t$). The quotient polynomial is computed by adding all these polynomials together: the combined constraints for all the gates the combined constraints for the permutation TODO: lookup the negated public polynomial and by then dividing the resulting polynomial with the vanishing polynomial $Z_H$. TODO: specify the split of the permutation polynomial into perm and bnd? commit (hiding) to the quotient polynomial $t$ Absorb the commitment of the quotient polynomial with the Fq-Sponge. Sample $\\zeta'$ with the Fq-Sponge. Derive $\\zeta$ from $\\zeta'$ using the endomorphism (TODO: specify) If lookup is used, evaluate the following polynomials at $\\zeta$ and $\\zeta \\omega$: the aggregation polynomial the sorted polynomials the table polynonial Chunk evaluate the following polynomials at both $\\zeta$ and $\\zeta \\omega$: $s_i$ $w_i$ $z$ lookup (TODO, see this issue ) generic selector poseidon selector By \"chunk evaluate\" we mean that the evaluation of each polynomial can potentially be a vector of values. This is because the index's max_poly_size parameter dictates the maximum size of a polynomial in the protocol. If a polynomial $f$ exceeds this size, it must be split into several polynomials like so: $$f(x) = f_0(x) + x^n f_1(x) + x^{2n} f_2(x) + \\cdots$$ And the evaluation of such a polynomial is the following list for $x \\in {\\zeta, \\zeta\\omega}$: $$(f_0(x), f_1(x), f_2(x), \\ldots)$$ TODO: do we want to specify more on that? It seems unnecessary except for the t polynomial (or if for some reason someone sets that to a low value) Evaluate the same polynomials without chunking them (so that each polynomial should correspond to a single value this time). Compute the ft polynomial. This is to implement Maller's optimization . construct the blinding part of the ft polynomial commitment see this section Evaluate the ft polynomial at $\\zeta\\omega$ only. Setup the Fr-Sponge Squeeze the Fq-sponge and absorb the result with the Fr-Sponge. Absorb the previous recursion challenges. Compute evaluations for the previous recursion challenges. Absorb the unique evaluation of ft: $ft(\\zeta\\omega)$. Absorb all the polynomial evaluations in $\\zeta$ and $\\zeta\\omega$: the public polynomial z generic selector poseidon selector the 15 register/witness 6 sigmas evaluations (the last one is not evaluated) Sample $v'$ with the Fr-Sponge Derive $v$ from $v'$ using the endomorphism (TODO: specify) Sample $u'$ with the Fr-Sponge Derive $u$ from $u'$ using the endomorphism (TODO: specify) Create a list of all polynomials that will require evaluations (and evaluation proofs) in the protocol. First, include the previous challenges, in case we are in a recursive prover. Then, include: the negated public polynomial the ft polynomial the permutation aggregation polynomial z polynomial the generic selector the poseidon selector the 15 registers/witness columns the 6 sigmas the optional gates optionally, the runtime table if using lookup: add the lookup sorted polynomials add the lookup aggreg polynomial add the combined table polynomial if present, add the runtime table polynomial the lookup selectors Create an aggregated evaluation proof for all of these polynomials at $\\zeta$ and $\\zeta\\omega$ using $u$ and $v$.","breadcrumbs":"Kimchi » Proof Creation","id":"263","title":"Proof Creation"},"264":{"body":"TODO: we talk about batch verification, but is there an actual batch operation? It seems like we're just verifying an aggregated opening proof We define two helper algorithms below, used in the batch verification of proofs. Fiat-Shamir argument We run the following algorithm: Setup the Fq-Sponge. This sponge mostly absorbs group Absorb the digest of the VerifierIndex. Absorb the commitments of the previous challenges with the Fq-sponge. Absorb the commitment of the public input polynomial with the Fq-Sponge. Absorb the commitments to the registers / witness columns with the Fq-Sponge. If lookup is used: If it involves queries to a multiple-column lookup table, then squeeze the Fq-Sponge to obtain the joint combiner challenge $j'$, otherwise set the joint combiner challenge $j'$ to $0$. Derive the scalar joint combiner challenge $j$ from $j'$ using the endomorphism. (TODO: specify endomorphism) absorb the commitments to the sorted polynomials. Sample the first permutation challenge $\\beta$ with the Fq-Sponge. Sample the second permutation challenge $\\gamma$ with the Fq-Sponge. If using lookup, absorb the commitment to the aggregation lookup polynomial. Absorb the commitment to the permutation trace with the Fq-Sponge. Sample the quotient challenge $\\alpha'$ with the Fq-Sponge. Derive $\\alpha$ from $\\alpha'$ using the endomorphism (TODO: details). Enforce that the length of the $t$ commitment is of size 7. Absorb the commitment to the quotient polynomial $t$ into the argument. Sample $\\zeta'$ with the Fq-Sponge. Derive $\\zeta$ from $\\zeta'$ using the endomorphism (TODO: specify). Setup the Fr-Sponge. This sponge absorbs elements from Squeeze the Fq-sponge and absorb the result with the Fr-Sponge. Absorb the previous recursion challenges. Compute evaluations for the previous recursion challenges. Evaluate the negated public polynomial (if present) at $\\zeta$ and $\\zeta\\omega$. NOTE: this works only in the case when the poly segment size is not smaller than that of the domain. Absorb the unique evaluation of ft: $ft(\\zeta\\omega)$. Absorb all the polynomial evaluations in $\\zeta$ and $\\zeta\\omega$: the public polynomial z generic selector poseidon selector the 15 register/witness 6 sigmas evaluations (the last one is not evaluated) Sample the \"polyscale\" $v'$ with the Fr-Sponge. Derive $v$ from $v'$ using the endomorphism (TODO: specify). Sample the \"evalscale\" $u'$ with the Fr-Sponge. Derive $u$ from $u'$ using the endomorphism (TODO: specify). Create a list of all polynomials that have an evaluation proof. Compute the evaluation of $ft(\\zeta)$. Partial verification For every proof we want to verify, we defer the proof opening to the very end. This allows us to potentially batch verify a number of partially verified proofs. Essentially, this steps verifies that $f(\\zeta) = t(\\zeta) * Z_H(\\zeta)$. Check the length of evaluations inside the proof. Commit to the negated public input polynomial. Run the Fiat-Shamir argument . Combine the chunked polynomials' evaluations (TODO: most likely only the quotient polynomial is chunked) with the right powers of $\\zeta^n$ and $(\\zeta * \\omega)^n$. Compute the commitment to the linearized polynomial $f$. To do this, add the constraints of all of the gates, of the permutation, and optionally of the lookup. (See the separate sections in the constraints section.) Any polynomial should be replaced by its associated commitment, contained in the verifier index or in the proof, unless a polynomial has its evaluation provided by the proof in which case the evaluation should be used in place of the commitment. Compute the (chuncked) commitment of $ft$ (see Maller's optimization ). List the polynomial commitments, and their associated evaluations, that are associated to the aggregated evaluation proof in the proof: recursion public input commitment ft commitment (chunks of it) permutation commitment index commitments that use the coefficients witness commitments coefficient commitments sigma commitments optional gate commitments lookup commitments Batch verification of proofs Below, we define the steps to verify a number of proofs (each associated to a verifier index ). You can, of course, use it to verify a single proof. If there's no proof to verify, the proof validates trivially. Ensure that all the proof's verifier index have a URS of the same length. (TODO: do they have to be the same URS though? should we check for that?) Validate each proof separately following the partial verification steps. Use the PolyCom.verify to verify the partially evaluated proofs.","breadcrumbs":"Kimchi » Proof Verification","id":"264","title":"Proof Verification"},"265":{"body":"commit_evaluation: TODO","breadcrumbs":"Kimchi » Optimizations","id":"265","title":"Optimizations"},"266":{"body":"TODO","breadcrumbs":"Kimchi » Security Considerations","id":"266","title":"Security Considerations"},"267":{"body":"Note Note that the URS is called SRS in our codebase at the moment. SRS is incorrect, as it is used in the presence of a trusted setup (which kimchi does not have). This needs to be fixed. The URS comprises of: Gs: an arbitrary ordered list of curve points that can be used to commit to a polynomial in a non-hiding way. H: a blinding curve point that can be used to add hidding to a polynomial commitment scheme. The URS is generated deterministically, and thus can be rederived if not stored. As different circuits of different size do not share the same domain, and the size of the domain is directly equal to the number of Gs curve points we use to commit to polynomials, we often want to share the same URS of maximum size across many circuits (constraint systems). Circuit associated with smaller domains will simply use a truncation of the shared URS. Note A circuit's domain is generated as the first power of 2 that is equal, or larger, to the number of gates used in the circuit (including zero-knowledge gates).","breadcrumbs":"Universal Reference String (URS) » Universal Reference String (URS)","id":"267","title":"Universal Reference String (URS)"},"268":{"body":"TODO: specify this part","breadcrumbs":"Universal Reference String (URS) » Group map","id":"268","title":"Group map"},"269":{"body":"The following pseudo-code specifies how the Gs curve points, as well as the H curve point, are generated. def create(depth): # generators Gs = [] for i in range(0, depth): digest = Blake2b512.hash(be_encode(i)) Gs.push(point_of_random_bytes(digest)) # blinding point digest = (b\"srs_misc\" || be_encode(0)) H = point_of_random_bytes(digest) # return (Gs, H) def point_of_random_bytes(random_bytes): # packing in bit-representation const N: usize = 31 let mut bits = [false; 8 * N] for i in range(0, N) { for j in range(0, 8) { bits[8 * i + j] = (random_bytes[i] >> j) & 1 == 1; } } let n = BigInt::from_bits_be(&bits); let t = G::BaseField::from_repr(n) return map.to_group(t) TODO: specify map.to_group in the previous section.","breadcrumbs":"Universal Reference String (URS) » Curve points to commit to a polynomial","id":"269","title":"Curve points to commit to a polynomial"},"27":{"body":"We can commit to several values x1​,⋯,xn​ by sending separate Pedersen commitments to all of these values as such: x1​G+r1​H,⋮xn​G+rn​H But we can instead batch/aggregate all of these commitments together as a single commitment: x1​G1​+⋯+xn​Gn​+rH with G1​,⋯,Gn​,H independent bases with unknown discrete logarithms. If you represent xs and the Gs as two vectors x=(x1​,⋯,xn​) and G=(G1​,⋯,Gn​), we can quickly write the previous statement as an inner product xG+rH Vector commitments (sometimes referred to as multi-commitments) are a powerful construction because an arbitrarily large vector can be committed with a single curve point. The naive approach to constructing an opening proof for a length n vector commitment has size O(n). It is simply the tuple (x1​,…,xn​,r). As we will see later, opening proofs for vector commitments is an interesting topic and there is a much more efficient algorithm.","breadcrumbs":"Commitments » Vector commitments","id":"27","title":"Vector commitments"},"270":{"body":"As there is no limit to the number of commitment curve points you can generate, we only provide the first three ones to serve as test vectors. TODO: specify the encoding","breadcrumbs":"Universal Reference String (URS) » URS values in practice","id":"270","title":"URS values in practice"},"271":{"body":"Gs . G0 = (x=121C4426885FD5A9701385AAF8D43E52E7660F1FC5AFC5F6468CC55312FC60F8, y=21B439C01247EA3518C5DDEB324E4CB108AF617780DDF766D96D3FD8AB028B70)\nG1 = (x=26C9349FF7FB4AB230A6F6AEF045F451FBBE9B37C43C3274E2AA4B82D131FD26, y=1996274D67EC0464C51F79CCFA1F511C2AABB666ABE67733EE8185B71B27A504)\nG2 = (x=26985F27306586711466C5B2C28754AA62FE33516D75CEF1F7751F1A169713FD, y=2E8930092FE6A18B331CE0E6E27B413AA18E76394F18A2835DA9FAE10AA3229D) H : H = (x=092060386301C999AAB4F263757836369CA27975E28BC7A8E5B2CE5B26262201, y=314FC4D83AE66A509F9D41BE6165F2606A209A9B5805EE85CE20249C5EBCBE26)","breadcrumbs":"Universal Reference String (URS) » Vesta","id":"271","title":"Vesta"},"272":{"body":"G0 = (x363D83141FD1E0540718FADBA7278ABAEEDB46D7A3F050F2CFF1DF4F300C9C30, y=034C68F4079B4F338A19BE2D7BFA44B395C65B9790DD273F361327446C778764)\nG1 = (x2CC40B77D87665244AE5EB5304E8744004C80061AD08476A0F0656C13134EA45, y=28146EC860159DB55CB5EA5B14F0AA2F8751DEDFE0DDAFD1C313B15575C4B4AC)\nG2 = (x2808BC21BEB90314377BF6130285FABE6CE4B8A4457FB25BC95EBA0083DF27E3, y=1E04E53DD6395FAB8018D7FE98F9C7FAB39C40BFBE48589626A7B8532728B002) H : H = (x221B959DACD2052AAE26193FCA36B53279866A4FBBAB0D5A2F828B5FD7778201, y=058C8F1105CAE57F4891EADC9B85C8954E5067190E155E61D66855ACE69C16C0)","breadcrumbs":"Universal Reference String (URS) » Pallas","id":"272","title":"Pallas"},"273":{"body":"Pickles is Mina’s inductive zk-SNARK composition system. It lets you construct proofs with zk-SNARKs and combine them in flexible ways to deliver incremental verifiable computation. Pickles uses a pair of amicable curves called Pasta in order to deliver incremental verifiable computation efficiently. These curves are referred to as \"tick\" and \"tock\" within the Mina source code. Tick - Vesta (a.k.a. Step), constraint domain size 218 (block and transaction proofs) Tock - Pallas (a.k.a. Wrap), constraint domain size 217 (signatures) The Tock prover does less (only performs recursive verifications and no other logic), so it requires fewer constraints and has a smaller domain size. Internally Pickles refers to Tick and Tock as Step and Wrap, respectively. Tock is used to prove the verification of a Tick proof and outputs a Tick proof. Tick is used to prove the verification of a Tock proof and outputs a Tock proof. In other words, Provetock​(Verify(Tick))=Tickproof​ Provetick​(Verify(Tock))=Tockproof​ Both Tick and Tock can verify at most 2 proofs of the opposite kind, though, theoretically more is possible. Currently, in Mina we have the following situation. Every Tock always wraps 1 Tick proof, such as Tick proofs can verify 2 Tock proofs Blockchain SNARK takes previous blockchain SNARK proof and a transaction proof Verifying two Tock transaction proofs","breadcrumbs":"Pickles » Pickles","id":"273","title":"Pickles"},"274":{"body":"vrf is appendix C of https://eprint.iacr.org/2017/573.pdf","breadcrumbs":"Consensus » Consensus","id":"274","title":"Consensus"},"28":{"body":"To construct SNARKs we use polynomial commitments. A polynomial commitment scheme for a field F (or it could even be a ring) is a way of committing to a polynomial f∈F[x] to get a commitment c, in such a way that for any α∈F, you can provide y=f(α), along with an \"opening proof\" π that proves that the polynomial committed to in c equals y when evaluated at α. In other words, it is a type of commitment C, a type of randomness R, a type of opening proof P along with algorithms commit:F[x]×R→Copen:C×F×(F[x]×R)→F×Pverify:C×(F×P)→Bool such that for any f∈F[x],r∈R,α∈F , we have c:=commit(f,r)verify(c,open(c,α,(f,r)))=true and if b=f(α) then it is not possible to compute π∈P such that verify(c,(b,π))=true In other words, if b=f(α) then every π which is feasible to compute results in verify(c,(b,π))=false. One thing that's pretty cool is that because polynomial commitment schemes let you construct zk-SNARKs, polynomial commitment schemes imply commitment schemes with arbitrary opening functionality. TODO","breadcrumbs":"Commitments » Polynomial commitments","id":"28","title":"Polynomial commitments"},"29":{"body":"All known constructions of polynomial commitment schemes are a bit complicated. The easiest to describe is called the Kate (pronounced \"kah-TAY\") scheme, also known as \"KZG\". It requires a \"prime-order group with a pairing\", which is three groups G1​,G2​,GT​ of prime order p (hence, all isomorphic cyclic groups) together with a function e:G1​×G2​→GT​ such that for any a1​,a2​∈Z, g1​∈G1​, g2​∈G2​, we have e(a1​⋅g1​,a2​⋅g2​)=a1​a2​⋅e(g1​,g2​) e is called a \"pairing\" or a \"bilinear pairing\". What this lets us do is \"multiply in the scalar\" but only once. Fix a degree bound d on the polynomials we would like to be able to commit to. The KZG scheme, will let us commit to polynomials in Fp​[x]| |<------ b | OT Prot. | | |-------> x_b +----------+ Due to a result of Impagliazzo and Rudich in this paper , it is very unlikely that OT is possible without the use of public-key cryptography. However, OT can be efficiently extended. That is, starting with a small number of base OTs, one could create many more OTs with only symmetric primitives. The seminal work of IKNP03 presented a very efficient protocol for extending OTs, requiring only black-box use of symmetric primitives and λ base OTs, where λ is the security parameter. This doc focuses on the family of protocols inspired by IKNP.","breadcrumbs":"Two Party Computation » Oblivious Transfer » Oblivious Transfer","id":"66","title":"Oblivious Transfer"},"67":{"body":"We introduce the CO15 OT as the base OT. Notation . The protocol is described over an additive group (G,G,p,+) of prime order p generated by G. Denote H:(G×G)×G→{0,1}λ as a key-derivation function to extract a λ-bit key from group elements. Note that H here is different from the tweakable hash function in garbled circuit. In the CO15 protocol, the sender holds n 128-bit strings {(Xi0​,Xi1​)}i∈[n]​ and the receiver holds n-bit string b∈{0,1}n. The protocol is as follows. The sender samples y←Fp​, and computes S=y⋅G and T=y⋅S. The sender sends S to the receiver, who aborts if S∈/G. For i∈[n], the receiver samples xi​←Fp​, and computes: Ri​=bi​S+xi​G The receiver sends {Ri​}i∈[n]​ to the sender, who aborts if Ri​∈/G. For i∈[n], the sender computes ki0​=H(S,Ri​,yRi​), ki1​=H(S,Ri​,yRi​−T) and sends {(ci0​=ki0​⊕Xi0​,ci1​=ki1​⊕Xi1​)}i∈[n]​ to the receiver. For i∈[n], the receiver computes ki​=H(S,Ri​,xi​S), and outputs cibi​​⊕ki​ Correctness . The receiver always computes the hash of xi​S=(xi​y)G. The sender sends the hashes of yRi​ and yRi​−T. If bi​=0, ki0​ is the hash of yRi​=(yxi​)G, then the receive will get Xi0​. if bi​=1, ki1​ is the hash of yRi​−T=y(S+xi​G)−T=(yxi​)G, then the receiver will get Xi1​. Security . We refer the security analysis to CO15 .","breadcrumbs":"Two Party Computation » Oblivious Transfer » Base OT » Base OT","id":"67","title":"Base OT"},"68":{"body":"To meet the large-scale demand of OTs, OT extensions can be used. An OT extension protocol works by running a small number of base OTs that are used as a base for obtaining many OTs via the use of cheap symmetric cryptographic operations only. This is conceptually similar to hybrid encryption. In this subsection, we focus on the maliciously secure OT extension presented in KOS15 . To be compatible with the garbled circuit optimizations, Correlated OT (COT) is considered here. COT is a variant of OT, where the sender chooses a secret string Δ and the receiver has a choice bit b. After the protocol, then sender obtains two strings X0​=X and X1​=X⊕Δ and the receiver obtains Xb​, where X is a random string. Sender Receiver +----------+ Delta ------->| |<------ b | COT Prot.| x_0 = x, x_1 = x + Delta <-------| |-------> x_b +----------+ Notation . Let λ be the security parameter, we use λ=128 in zkOracles. Identify F2λ​ with the finite field F2λ​. Use \"⋅\" for multiplication in F2λ​ and \"⋆\" for the component-wise product in F2λ​. Given a matrix A, denote its rows by subindices ai​ and its columns by superindices ak. Use v[i] to denote the i-th entry of v. The KOS15 OT protocol to generate ℓ(≫λ) OTs is described as follows. Initialize The receiver samples λ pairs of λ-bit seeds {(k0i​,k1i​)}i=1λ​. The sender choose a uniformly random λ-bit string Δ. The two parties calls the base OT with inputs Δ and {(k0i​,k1i​)}i=1λ​. Note that the sender acts as the receiver in the base OT, and the receiver acts as the sender in the base OT. The sender obtains kΔi​i​ for i∈[λ]. Extend The receiver takes as input the choice bits x1​,...,xℓ​, defines ℓ′=ℓ+λ+s, where s is the statistic parameter, and we set s=40 in zkOracles. Let x=x1​∥x2​∥...∥xℓ​∥x′∈F2ℓ′​, with x′∈F2ℓ′−ℓ​ uniformly chosen. The receiver defines vectors x1​,...,xℓ′​ as xi​=x[i]⋅(1,...,1)∈F2λ​ for i∈[ℓ′]. The receiver expands k0i​ and k1i​ with a pseudo random generator (PRG), letting t0i​=PRG(k0i​)∈F2ℓ′​ and t1i​=PRG(k1i​)∈F2ℓ′​ for i∈[λ] The sender expands kΔi​i​ with the same PRG and gets tΔi​i​=PRG(kΔi​i​) for i∈[λ] The receiver computes ui=t0i​⊕t1i​⊕x∈F2ℓ′​ for i∈[λ] and sends them to the sender. The sender computes qi=Δi​⋅ui⊕tΔi​i​=t0i​⊕Δi​⋅x∈F2ℓ′​ for i∈[λ] Let qj​ be the j-th row of the ℓ′×λ matrix Q=[q1∣q2∣⋯∣qλ], and similarly let tj​ be the j-th row of T=[t01​∣t02​∣⋯∣t0λ​]. Note that qj​=tj​⊕(xj​⋆Δ)=tj​⊕(x[j]⋅Δ) for j∈[ℓ′] Correlation Check The sender and receiver run a πRAND​(F2λℓ′​) protocol to obtain random elements χ1​,...,χℓ′​. πRAND​(F2λℓ′​) will be described later. The receiver computes x=j=1∑ℓ′​x[j]⋅χj​ and t=j=1∑ℓ′​tj​⋅χj​ and sends them to the sender. The sender computes q=j=1∑ℓ′​qj​⋅χj​ and checks t=q⊕x⋅Δ. If the check fails, abort. Output The sender computes vj​=H(j,qj​)⊕H(j,qj​⊕Δ)⊕Δ for j∈[ℓ] and sends them to the receiver. H is a tweakable hash function as used in garbled circuit. The sender outputs (H(j,qj​), H(j,qj​)⊕Δ) For j∈[ℓ], the receiver outputs the following: If xj​=0, outputs H(j,tj​); If xj​=1, outputs H(j,tj​)⊕vj​ This ends the description of the KOS15 protocol. We note that a very recent work pointed out that there is a flaw in the security proof of the main lemma of KOS15 and provided a fix, which is less efficient. They also pointed out that this the original KOS15 protocol is still secure in practice. Therefore, we choose the original KOS15 protocol in zkOracles in this version. We begin to describe the πRAND​(F2λℓ​) protocol as follows. The sender and receiver locally choose uniformly 128-bit random seeds s0​ and s1​, respectively. The sender and receiver compute commitments of the seeds by choosing 128-bit random strings r0​,r1​ and generate c0​=SHA256(s0​∥r0​) and c1​=SHA256(s1​∥r1​), respectively. The sender sends c0​ to the receiver, and the receiver sends c1​ to the sender. On receiving the commitments, the sender and receiver open the commitments by sending (s0​,r0​) and (s1​,r1​) to each other, respectively. The sender checks c1​ by re-computing it with s1​,r1​, and the receiver checks c0​ by re-computing it with s0​,r0​. If checks fail, abort. The sender and receiver locally compute s=s0​⊕s1​ and generate random elements with a PRG as χ1​,...,χℓ​←PRG(s)","breadcrumbs":"Two Party Computation » Oblivious Transfer » OT Extension » OT Extension","id":"68","title":"OT Extension"},"69":{"body":"With garbled circuit and oblivious transfer, we are ready to describe the well-known Yao's method to construct secure two-party computation protocols for any polynomial-size function. Given any function F(x,y), where x is the private input of Alice, and y is the private input of Bob. Let x=x1​∥x2​∥⋯∥xℓ1​​ and y=y1​∥y2​∥⋯∥yℓ2​​, where xi​ and yi​ are bits. Let Alice be the garbler and Bob be the evaluator, Yao's protocol is described as follows. Alice and Bob run a COT protocol to generate ℓ2​ OTs, where Alice is the sender and Bob is the receiver. Alice obtains (Y1​,Y1​⊕Δ),...,(Yℓ2​​,Yℓ2​​⊕Δ), and Bob obtains (Y1​⊕y1​⋅Δ,...,Yℓ2​​⊕yℓ2​​⋅Δ) Alice chooses uniformly random labels X1​,X2​,...,Xℓ1​​. Alice uses (X1​,...,Xℓ1​​,Y1​,...,Yℓ2​​) and global Δ to generate the garbled circuit of F and sends GC(F) to Bob. Alice also sends the decoding information to Bob. Alice encodes her inputs and sends X1​⊕x1​⋅Δ,...,Xℓ1​​⊕xℓ1​​⋅Δ to Bob. With the encoded X labels and Y labels, Bob evaluates GC(F) to get the output labels, and then decodes these output labels with decoding information and gets the output bits.","breadcrumbs":"Two Party Computation » Full Protocol » Full Protocol","id":"69","title":"Full Protocol"},"7":{"body":"When cryptographers talk about \"groups\", they usually mean a \"computational group\", which is a group equipped with efficient algorithms for computing the group operation and inversion. This is different because a group in the mathematical sense need not have its operations be computable at all.","breadcrumbs":"Groups » More things to know","id":"7","title":"More things to know"},"70":{"body":"Many modern proof systems (and I think all that are in use) are constructed according to the following recipe. You start out with a class of computations. You devise a way to arithmetize those computations. That is, to express your computation as a statement about polynomials. More specifically, you describe what is often called an \"algebraic interactive oracle proof\" (AIOP) that encodes your computation. An AIOP is a protocol describing an interaction between a prover and a verifier, in which the prover sends the verifier some \"polynomial oracles\" (basically a black box function that given a point evaluates a polynomial at that point), the verifier sends the prover random challenges, and at the end, the verifier queries the prover's polynomials at points of its choosing and makes a decision as to whether it has been satisfied by the proof. An AIOP is an imagined interaction between parties. It is an abstract description of the protocol that will be \"compiled\" into a SNARK. There are several \"non-realistic\" aspects about it. One is that the prover sends the verifier black-box polynomials that the verifier can evaluate. These polynomials have degree comparable to the size of the computation being verified. If we implemented these \"polynomial oracles\" by having the prover really send the O(n) size polynomials (say by sending all their coefficients), then we would not have a zk-SNARK at all, since the verifier would have to read this linearly sized polynomial so we would lose succinctness, and the polynomials would not be black-box functions, so we may lose zero-knowledge. Instead, when we concretely instantiate the AIOP, we have the prover send constant-sized, hiding polynomial commitments . Then, in the phase of the AIOP where the verifier queries the polynomials, the prover sends an opening proof for the polynomial commitments which the verifier can check, thus simulating the activity of evaluating the prover's polynomials on your own. So this is the next step of making a SNARK: instantiating the AIOP with a polynomial commitment scheme of one's choosing. There are several choices here and these affect the properties of the SNARK you are constructing, as the SNARK will inherit efficiency and setup properties of the polynomial commitment scheme used. An AIOP describes an interactive protocol between the verifier and the prover. In reality, typically, we also want our proofs to be non-interactive. This is accomplished by what is called the Fiat--Shamir transformation . The basic idea is this: all that the verifier is doing is sampling random values to send to the prover. Instead, to generate a \"random\" value, the prover simulates the verifier by hashing its messages. The resulting hash is used as the \"random\" challenge. At this point we have a fully non-interactive proof. Let's review our steps. Start with a computation. Translate the computation into a statement about polynomials and design a corresponding AIOP. Compile the AIOP into an interactive protocol by having the prover send hiding polynomial commitments instead of polynomial oracles. Get rid of the verifier-interaction by replacing it with a hash function. I.e., apply the Fiat--Shamir transform.","breadcrumbs":"Proof Systems » Proof Systems Design Overview","id":"70","title":"Proof Systems Design Overview"},"71":{"body":"Finally we can discuss zk-SNARKs. In these notes right now I will not start with intuitive motivations for SNARKs because I can't think of them right now, but maybe later. Let f:A×W→Bool be a program (in some yet to be specified programming language). For a:A, zk-SNARKs let you produce proofs for statements of the form I know w:W such that f(a,w)=true in such a way that no information about w is revealed by the proof (this is the zero-knowledge or ZK part of them). In other words, zk-SNARKs allow you to prove execution of programs, where you are also allowed to hide a selected portion (i.e., the W portion) of the data involved in the computation. We could also phrase zk-SNARKs in terms of computing functions f:A×W→B, which seems more general, but actually this would be taken care of by the previous sense where we can only use boolean-valued functions, by proving against the function g:(A×B)×W→Bool defined by g((a,b),w):=b=?​f(a,w). SNARK stands for Succinct Non-interactive ARguments of Knowledge . If it also satisfies zero-knowledge, then it is called a zk-SNARK. Preprocessing SNARKs allow the verifier to precompute some encodings of the relation to run independently of the instance length. Definitions of these schemes owe the succinctness property to the fact that both the verifier and the proof length are (at most) polylogarithmic in the size of the computation (the witness length). Recent schemes such as Plonk, go beyond that definition and provide zkSNARKs with constant proof length. Unlike the verifier in preprocessing SNARKs, the prover can't be faster than linear, as it has to read the circuit at least.","breadcrumbs":"Proof Systems » zk-SNARKs » zk-SNARKs","id":"71","title":"zk-SNARKs"},"72":{"body":"We will define a notion of computation (a programming language basically) for which it is easy to construct zk-SNARKs.","breadcrumbs":"Proof Systems » zk-SNARKs » Expressing computations with \"iterating over trace tables\"","id":"72","title":"Expressing computations with \"iterating over trace tables\""},"73":{"body":"The proof system in Mina is a variant of PlonK . To understand PlonK, you can refer to our series of videos on the scheme. In this section we explain our variant, called kimchi . kimchi is not formally a zk-SNARK, as it is not succinct in the proof size. zk-SNARKs must have a log(n) proof size where n is the number of gates in the circuit. In kimchi, due to the Bootleproof polynomial commitment approach, our proofs are log(d) where d is the maximum degree of the committed polynomials (in the order of the number of rows). In practice, our proofs are in the order of dozens of kilobytes. Note that kimchi is a zk-SNARK in terms of verification complexity (log(n)) and the proving complexity is quasilinear (O(nlogn)) – recall that the prover cannot be faster than linear. Essentially what PlonK allows you to do is to, given a program with inputs and outputs, take a snapshot of its execution. Then, you can remove parts of the inputs, or outputs, or execution trace, and still prove to someone that the execution was performed correctly for the remaining inputs and outputs of the snapshot. There are a lot of steps involved in the transformation of \"I know an input to this program such that when executed with these other inputs it gives this output\" to \"here's a sequence of operations you can execute to convince yourself of that\". At the end of this chapter, you will understand that the protocol boils down to filling a number of tables (illustrated in the diagram below): tables that specify the circuit, and tables that describes an execution trace of the circuit (given some secret and public inputs). kimchi","breadcrumbs":"Overview » Overview","id":"73","title":"Overview"},"74":{"body":"Size: number of rows Columns: number of variables per rows Cell: a pair (row, column) Witness: the values assigned in all the cells Gate: polynomials that act on the variables in a row Selector vector: a vector of values 1 or 0 (not true for constant vector I think) that toggles gates and variables in a row Gadget: a series of contiguous rows with some specific gates set (via selector vectors)","breadcrumbs":"Overview » Glossary » Glossary","id":"74","title":"Glossary"},"75":{"body":"PlonK needs a domain to encode the circuit on (for each point we can encode a row/constraint/gate). We choose a domain H such that it is a multiplicative subgroup of the scalar field of our curve.","breadcrumbs":"Domain » Domain","id":"75","title":"Domain"},"76":{"body":"Furthermore, as FFT (used for interpolation, see the section on FFTs ) works best in domains of size 2k for some k, we choose H to be a subgroup of order 2k. Since the pasta curves both have scalar fields of prime orders p and q, their multiplicative subgroup is of order p−1 and q−1 respectively (without the zero element). The pasta curves were generated specifically so that both p−1 and q−1 are multiples of 232. We say that they have 2-adicity of 32. Looking at the definition of one of the pasta fields in Rust you can see that it is defined specifically for a trait related to FFTs: impl FftParameters for FqParameters { type BigInt = BigInteger; const TWO_ADICITY: u32 = 32; #[rustfmt::skip] const TWO_ADIC_ROOT_OF_UNITY: BigInteger = BigInteger([ 0x218077428c9942de, 0xcc49578921b60494, 0xac2e5d27b2efbee2, 0xb79fa897f2db056 ]); The 2-adicity of 32 means that there's a multiplicative subgroup of size 232 that exists in the field. The code above also defines a generator g for it, such that g232=1 and gi=1 for i∈[[1,232−1]] (so it's a primitive 232-th root of unity). Lagrange's theorem tells us that if we have a group of order n, then we'll have subgroups with orders dividing n. So in our case, we have subgroups with all the powers of 2, up to the 32-th power of 2. To find any of these groups, it is pretty straight forward as well. Notice that: let h=g2, then h231=g232=1 and so h generates a subgroup of order 31 let t=g22, then t230=g232=1 and so t generates a subgroup of order 30 and so on... In arkworks you can see how this is implemented: let size = n.next_power_of_two() as u64;\nlet log_size_of_group = ark_std::log2(usize::try_from(size).expect(\"too large\"));\nomega = Self::TWO_ADIC_ROOT_OF_UNITY;\nfor _ in log_size_of_group..Self::TWO_ADICITY { omega.square_in_place();\n} this allows you to easily find subgroups of different sizes of powers of 2, which is useful in zero-knowledge proof systems as FFT optimizations apply well on domains that are powers of 2. You can read more about that in the mina book .","breadcrumbs":"Domain » 2-adicity","id":"76","title":"2-adicity"},"77":{"body":"For each curve, we generate a domain H as the set H=1,ω,ω2,⋯,ωn−1. As we work in a multiplicative subgroup, the polynomial that vanishes in all of these points can be written and efficiently calculated as ZH​(x)=x∣H∣−1. This is because, by definition, ω∣H∣=1. If x∈H, then x=ωi for some i, and we have: x∣H∣=(ωi)∣H∣=(ω∣H∣)i=1i=1 This optimization is important, as without it calculating the vanishing polynomial would take a number of operations linear in ∣H∣ (which is not doable in a verifier circuit, for recursion). With the optimization, it takes us a logarithmic number of operation (using exponentiation by squaring ) to calculate the vanishing polynomial.","breadcrumbs":"Domain » Efficient computation of the vanishing polynomial","id":"77","title":"Efficient computation of the vanishing polynomial"},"78":{"body":"What's a lagrange base? Li​(x)=1 if x=gi, 0 otherwise.","breadcrumbs":"Lagrange Basis in Multiplicative Subgroups » Lagrange basis in multiplicative subgroups","id":"78","title":"Lagrange basis in multiplicative subgroups"},"79":{"body":"Arkworks has the formula to construct a lagrange base: Evaluate all Lagrange polynomials at τ to get the lagrange coefficients. Define the following as H: The coset we are in, with generator g and offset h m: The size of the coset H ZH​: The vanishing polynomial for H. ZH​(x)=∏i∈[m]​(x−h⋅gi)=xm−hm vi​: A sequence of values, where v0​=m∗hm−11​, and vi+1​=g⋅vi​ We then compute Li,H​(τ) as Li,H​(τ)=ZH​(τ)⋅τ−hgivi​​ However, if τ in H, both the numerator and denominator equal 0 when i corresponds to the value tau equals, and the coefficient is 0 everywhere else. We handle this case separately, and we can easily detect by checking if the vanishing poly is 0. following this, for h=1 we have: L0​(x)=m(x−1)ZH​(x)​ L1​(x)=m(x−g)ZH​(x)g​ L2​(x)=m(x−g2)ZH​(x)g2​ and so on","breadcrumbs":"Lagrange Basis in Multiplicative Subgroups » What's the formula?","id":"79","title":"What's the formula?"},"8":{"body":"Implement a type JacobianPoint and functions for computing the group operation Familiarize yourself with the types and traits in ark_ec. Specifically, todo Implement fn decompress(c: (F, bool)) -> (F, F)","breadcrumbs":"Groups » Exercises","id":"8","title":"Exercises"},"80":{"body":"https://en.wikipedia.org/wiki/Lagrange_polynomial#Barycentric_form","breadcrumbs":"Lagrange Basis in Multiplicative Subgroups » What's the logic here?","id":"80","title":"What's the logic here?"},"81":{"body":"So far we've talked about an interactive protocol between a prover and a verifier. The zero-knowledge proof was also in the honest verifier zero-knowedlge (HVZK) model, which is problematic. In practice, we want to remove the interaction and have the prover produce a proof by themselves, that anyone can verify.","breadcrumbs":"Non-Interactivity via Fiat-Shamir » Non-Interactivity via Fiat-Shamir","id":"81","title":"Non-Interactivity via Fiat-Shamir"},"82":{"body":"public-coin protocols are protocols were the messages of the verifier are simply random messages. This is important as our technique to transform an interactive protocol to a non-interactive protocol works on public-coin protocols.","breadcrumbs":"Non-Interactivity via Fiat-Shamir » Public-coin protocols","id":"82","title":"Public-coin protocols"},"83":{"body":"The whole idea is to replace the verifier by a random oracle, which in practice is a hash function. Note that by doing this, we remove potential leaks that can happen when the verifier acts dishonestly. Initially the Fiat-Shamir transformation was only applied to sigma protocols, named after the greek letter Σ due to its shape resembling the direction of messages (prover sends a commit to a verifier, verifier sends a challenge to a prover, prover sends the final proof to a verifier). A Z would have made more sense but here we are.","breadcrumbs":"Non-Interactivity via Fiat-Shamir » Fiat-Shamir trick","id":"83","title":"Fiat-Shamir trick"},"84":{"body":"As our protocol has more than three moves, where every even move is a challenge from the verifier, we need to generalize Fiat-Shamir. This is simple: every verifier move can be replaced by a hash of the transcript (every message sent and received so far) to obtain a challenge.","breadcrumbs":"Non-Interactivity via Fiat-Shamir » Generalization of Fiat-Shamir","id":"84","title":"Generalization of Fiat-Shamir"},"85":{"body":"While we use a hash function for that, a different construction called the duplex construction is particularly useful in such situations as they allow to continuously absorb the transcript and produce challenges, while automatically authenticating the fact that they produced a challenge. Merlin is a standardization of such a construction using the Strobe protocol framework (a framework to make use of a duplex construction). Note that the more recent Xoodyak (part of NIST's lightweight competition) could have been used for this as well. Note also that Mina uses none of these standards, instead it simply uses Poseidon (see section on poseidon).","breadcrumbs":"Non-Interactivity via Fiat-Shamir » In practice: a duplex construction as Merlin","id":"85","title":"In practice: a duplex construction as Merlin"},"86":{"body":"Plookup allows us to check if witness values belong to a look up table. This is usually useful for reducing the number of constraints needed for bit-wise operations. So in the rest of this document we'll use the XOR table as an example. 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For the Rust documentation, see here . Note that this book is a work in progress, not necessarily reflecting the current state of Mina. Authors: Izaak Meckler, Vanishree Rao, Matthew Ryan, Anaïs Querol, Joseph Spadavecchia, David Wong, Xiang Xie In memory of Vitaly Zelov.","breadcrumbs":"Introduction » Introduction","id":"0","title":"Introduction"},"1":{"body":"This document is intended primarily to communicate mathematical ideas to programmers with some experience of math. To that end, we will often be ambivalent about the difference between sets and types (in whatever programming language, but usually I am imagining some idealized form of Rust or OCaml). So for example we may write A is a type A is a set A is a type and these should all be interpreted the same way, assuming it makes sense in the context. Similarly, we may write either of the following, and potentially mean the same thing: a:A a∈A a : A We use → for function types ↦ for defining functions Also, we usually assume functions are computed by a program (in whatever sense of \"program\" makes sense in the given context). So if we say \"let f:A→B\", we typically mean that A and B are types in some programming language which makes sense in the context (or maybe that they are sets, again depending on context) f is actually a program computing a function of the type A→B (again, in whatever sense of program makes sense in context)","breadcrumbs":"Terminology » Terminology","id":"1","title":"Terminology"},"10":{"body":"The purpose of this document is to give the reader mathematical, cryptographic, and programming context sufficient to become an effective practitioner of zero-knowledge proofs and ZK-SNARKs specifically.","breadcrumbs":"Fields » Intro","id":"10","title":"Intro"},"100":{"body":"Unlike the latest version of vanilla PlonK that implements the final check using a polynomial opening (via Maller's optimization), we implement it manually. (That is to say, Izaak implemented Maller's optimization for 5-wires.) But the check is not exactly f(ζ)=ZH​(ζ)t(ζ). This note describes how and why the implementation deviates a little.","breadcrumbs":"Final Check » Understanding the implementation of the f(ζ)=ZH​(ζ)t(ζ) check","id":"100","title":"Understanding the implementation of the f(ζ)=ZH​(ζ)t(ζ) check"},"101":{"body":"If you look at how the evaluation of f(z) is computed on the prover side (or the commitment of f is computed on the verifier side), you can see that f is missing two things: the public input part some terms in the permutation What is it missing in the permutation part? Let's look more closely. This is what we have: −1⋅⋅⋅⋅⋅⋅⋅⋅​z(ζω)⋅zkpm(ζ)⋅αPERM0(w[0](ζ)+β⋅σ[0](ζ)+γ)(w[1](ζ)+β⋅σ[1](ζ)+γ)(w[2](ζ)+β⋅σ[2](ζ)+γ)(w[3](ζ)+β⋅σ[3](ζ)+γ)(w[4](ζ)+β⋅σ[4](ζ)+γ)(w[5](ζ)+β⋅σ[5](ζ)+γ)β⋅σ[6](x)​​ In comparison, here is the list of the stuff we needed to have: the two sides of the coin: z(ζ)⋅⋅⋅⋅⋅⋅⋅⋅​zkpm(ζ)⋅αPERM0(w[0](ζ)+β⋅shift[0](ζ)+γ)(w[1](ζ)+β⋅shift[1](ζ)+γ)(w[2](ζ)+β⋅shift[2](ζ)+γ)(w[3](ζ)+β⋅shift[3](ζ)+γ)(w[4](ζ)+β⋅shift[4](ζ)+γ)(w[5](ζ)+β⋅shift[5](ζ)+γ)(w[6](ζ)+β⋅shift[6](ζ)+γ)​​ with shift[0]=1 and −1⋅⋅⋅⋅⋅⋅⋅⋅​z(ζω)⋅zkpm(ζ)⋅αPERM0(w[0](ζ)+β⋅σ[0](ζ)+γ)(w[1](ζ)+β⋅σ[1](ζ)+γ)(w[2](ζ)+β⋅σ[2](ζ)+γ)(w[3](ζ)+β⋅σ[3](ζ)+γ)(w[4](ζ)+β⋅σ[4](ζ)+γ)(w[5](ζ)+β⋅σ[5](ζ)+γ)(w[6](ζ)+β⋅σ[6](ζ)+γ)​​ the initialization: (z(ζ)−1)⋅L1​(ζ)⋅αPERM1 and the end of the accumulator: (z(ζ)−1)⋅Ln−k​(ζ)⋅αPERM2 You can read more about why it looks like that in this post . We can see clearly that the permutation stuff we have in f is clearly lacking. It's just the subtraction part (equation 2), and even that is missing some terms. It is missing exactly this: −1 ⋅⋅⋅⋅⋅⋅⋅⋅​z(ζω)⋅zkpm(ζ)⋅αPERM0(w[0](ζ)+β⋅σ[0](ζ)+γ)(w[1](ζ)+β⋅σ[1](ζ)+γ)(w[2](ζ)+β⋅σ[2](ζ)+γ)(w[3](ζ)+β⋅σ[3](ζ)+γ)(w[4](ζ)+β⋅σ[4](ζ)+γ)(w[5](ζ)+β⋅σ[5](ζ)+γ)(w[6](ζ)+γ)​​ So at the end, when we have to check for the identity f(ζ)=ZH​(ζ)t(ζ) we'll actually have to check something like this (I colored the missing parts on the left hand side of the equation): f(ζ)​+pub(ζ)+z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+βζ+γ)⋅(w[1](ζ)+β⋅shift[0](ζ)+γ)⋅(w[2](ζ)+β⋅shift[1](ζ)+γ)⋅(w[3](ζ)+β⋅shift[2](ζ)+γ)⋅(w[4](ζ)+β⋅shift[3](ζ)+γ)⋅(w[5](ζ)+β⋅shift[4](ζ)+γ)⋅(w[6](ζ)+β⋅shift[5](ζ)+γ)−z(ζω)⋅zkpm(ζ)⋅αPERM0⋅⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅⋅(w[6](ζ)+γ)+(z(ζ)−1)⋅L1​(ζ)⋅αPERM1+(z(ζ)−1)⋅Ln−k​(ζ)⋅αPERM2=t(ζ)(ζn−1)​​ This is not exactly what happens in the code. But if we move things around a bit, we end up with what's implemented. I highlight what changes in each steps. First, we just move things around: f(ζ)​+pub(ζ)+z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅shift[0]ζ+γ)⋅(w[1](ζ)+β⋅shift[1]ζ+γ)⋅(w[2](ζ)+β⋅shift[2]ζ+γ)⋅(w[3](ζ)+β⋅shift[3]ζ+γ)⋅(w[4](ζ)+β⋅shift[4]ζ+γ)⋅(w[5](ζ)+β⋅shift[5]ζ+γ)⋅(w[6](ζ)+β⋅shift[6]ζ+γ)−z(ζω)⋅zkpm(ζ)⋅αPERM0⋅⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅(w[6](ζ)+γ)−t(ζ)(ζn−1)=(1−z(ζ))L1​(ζ)αPERM1+(1−z(ζ))Ln−k​(ζ)αPERM2​​ here are the actual lagrange basis calculated with the formula here , oh and we actually use L0​ in the code, not L1​, so let's change that as well: f(ζ)++−−=​pub(ζ)z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅shift[0]ζ+γ)⋅(w[1](ζ)+β⋅shift[1]ζ+γ)⋅(w[2](ζ)+β⋅shift[2]ζ+γ)⋅(w[3](ζ)+β⋅shift[3]ζ+γ)⋅(w[4](ζ)+β⋅shift[4]ζ+γ)⋅(w[5](ζ)+β⋅shift[5]ζ+γ)⋅(w[6](ζ)+β⋅shift[6]ζ+γ)+z(ζω)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅(w[6](ζ)+γ)+t(ζ)(ζn−1)(1−z(ζ))[n(ζ−1)(ζn−1)​αPERM1+n(ζ−ωn−k)ωn−k(ζn−1)​αPERM2]​​ finally we extract some terms from the lagrange basis: ​[f(ζ)+pub(ζ)+z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅shift[0]ζ+γ)⋅(w[1](ζ)+β⋅shift[1]ζ+γ)⋅(w[2](ζ)+β⋅shift[2]ζ+γ)⋅(w[3](ζ)+β⋅shift[3]ζ+γ)⋅(w[4](ζ)+β⋅shift[4]ζ+γ)⋅(w[5](ζ)+β⋅shift[5]ζ+γ)⋅(w[6](ζ)+β⋅shift[6]ζ+γ)+−z(ζω)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅(w[6](ζ)+γ)−t(ζ)(ζn−1)]⋅(ζ−1)(ζ−ωn−k)=(1−z(ζ))[n(ζn−1)(ζ−ωn−k)​αPERM1+nωn−k(ζn−1)(ζ−1)​αPERM2]​​​ with αPERM0=α17,αPERM1=α18,αPERM2=α19 Why do we do things this way? Most likely to reduce Also, about the code: the division by n in the αPERM1 and the αPERM2 terms is missing (why?) the multiplication by wn−k in the αPERM2 term is missing (why?) As these terms are constants, and do not prevent the division by ZH​ to form t, t also omits them. In other word, they cancel one another.","breadcrumbs":"Final Check » What's f and what's missing in the final equation?","id":"101","title":"What's f and what's missing in the final equation?"},"102":{"body":"In verifier.rs you can see the following code (cleaned to remove anything not important) // compute the witness polynomials $w_0, \\cdots, w_14$ and the permutation polynomial $z$ in evaluation forms on different domains\nlet lagrange = index.cs.evaluate(&w, &z); // compute parts of the permutation stuff that is included in t\nlet (perm, bnd) = index.cs.perm_quot(&lagrange, &oracles, &z, &alpha[range::PERM])?; // divide contributions with vanishing polynomial\nlet (mut t, res) = (perm + some_other_stuff).divide_by_vanishing_poly() // add the other permutation stuff\nt += &bnd; // t is evaluated at zeta and sent... Notice that bnd is not divided by the vanishing polynomial . Also what's perm? Let's answer the latter TESTREMOVEME: perm(x)= −​aPERM0⋅zkpl(x)⋅[z(x)⋅(w[0](x)+γ+x⋅β⋅shift[0])⋅(w[1](x)+γ+x⋅β⋅shift[1])⋅…z(x⋅w)⋅(w[0](x)+γ+σ[0]⋅β) ⋅(w[1](x)+γ+σ[1]⋅β)⋅…]​​ and bnd is: bnd(x)=aPERM1⋅x−1z(x)−1​+aPERM2⋅x−sid[n−k]z(x)−1​ you can see that some terms are missing in bnd, specifically the numerator xn−1. Well, it would have been cancelled by the division by the vanishing polynomial, and this is the reason why we didn't divide that term by ZH​(x)=xn−1. Also, note that the same constant terms that were missing in f are missing in t.","breadcrumbs":"Final Check » What about t?","id":"102","title":"What about t?"},"103":{"body":"This document proposes a protocol change for kimchi .","breadcrumbs":"Maller's Optimization for Kimchi » Maller's optimization for Kimchi","id":"103","title":"Maller's optimization for Kimchi"},"104":{"body":"See the section on Maller's optimization for background.","breadcrumbs":"Maller's Optimization for Kimchi » What is Maller's optimization?","id":"104","title":"What is Maller's optimization?"},"105":{"body":"We want the verifier to form commitment to the following polynomial: L(x)=f(x)−ZH​(ζ)⋅t(x) They could do this like so: com(L)=com(f)−ZH​(ζ)⋅com(t) Since they already know f, they can produce com(f), the only thing they need is com(t). So the protocol looks like that: maller 15 1 In the rest of this document, we review the details and considerations needed to implement this change in kimchi .","breadcrumbs":"Maller's Optimization for Kimchi » Overview","id":"105","title":"Overview"},"106":{"body":"There's one challenge that prevents us from directly using this approach: com(t) is typically sent and received in several commitments (called chunks or segments throughout the codebase). As t is the largest polynomial, usually exceeding the size of the SRS (which is by default set to be the size of the domain).","breadcrumbs":"Maller's Optimization for Kimchi » How to deal with a chunked t?","id":"106","title":"How to deal with a chunked t?"},"107":{"body":"Thus, the verifier will have to produce split commitments of L and combine them with powers of ζn to verify an evaluation proof. Let's define L as: L=L0​+xnL1​+x2nL1​+⋯ where every Li​ is of degree n−1 at most. Then we have that com(L)=com(L0​)+com(xn⋅L1​)+com(x2n⋅L2​)+⋯ Which the verifier can't produce because of the powers of xn, but we can linearize it as we already know which x we're going to evaluate that polynomial with: com(L~)=com(L0​)+ζn⋅com(L1​)+ζ2n⋅com(L2​)+⋯","breadcrumbs":"Maller's Optimization for Kimchi » The verifier side","id":"107","title":"The verifier side"},"108":{"body":"This means that the prover will produce evaluation proofs on the following linearized polynomial: L~(x)=1⋅L0​(x)+ζn⋅L1​(x)+ζ2n⋅L2​(x)+⋯ which is the same as L(x) only if evaluated at ζ. As the previous section pointed out, we will need L (ζω) and L (ζω)=L(ζω).","breadcrumbs":"Maller's Optimization for Kimchi » The prover side","id":"108","title":"The prover side"},"109":{"body":"To create an evaluation proof, the prover also needs to produce the blinding factor rL​ associated with the chunked commitment of: L =f ​−(ζn−1)t~ To compute it, there are two rules to follow: when two commitment are added together, their associated blinding factors get added as well: com(a)+com(b)⟹ra​+rb​ when scaling a commitment, its blinding factor gets scalled too: n⋅com(a)⟹n⋅ra​ As such, if we know rf​ and rt​, we can compute: rL ​=rf ​​+(ζn−1)⋅rt~​ The prover knows the blinding factor of the commitment to t, as they committed to it. But what about f? They never commit to it really, and the verifier recreates it from scratch using: The commitments we sent them . In the linearization process, the verifier actually gets rid of most prover commitments, except for the commitment to the last sigma commitment Sσ6​. (TODO: link to the relevant part in the spec) As this commitment is public, it is not blinded. The public commitments . Think commitment to selector polynomials or the public input polynomial. These commitments are not blinded, and thus do not impact the calculation of the blinding factor. The evaluations we sent them . Instead of using commitments to the wires when recreating f, the verifier uses the (verified) evaluations of these in ζ. If we scale our commitment com(z) with any of these scalars, we will have to do the same with rz​. Thus, the blinding factor of L is (ζn−1)⋅rt ​.","breadcrumbs":"Maller's Optimization for Kimchi » Evaluation proof and blinding factors","id":"109","title":"Evaluation proof and blinding factors"},"11":{"body":"In this section we'll discuss the fundamental objects used in the construction of most ZK-SNARKs in use today. These objects are used extensively and without ceremony, so it's important to understand them very well. If you find you don't understand something at first, don't worry: practice is often the best teacher and in using these objects you will find they will become like the bicycle you've had for years: like an extension of yourself that you use without a thought.","breadcrumbs":"Fields » Some fundamental mathematical objects","id":"11","title":"Some fundamental mathematical objects"},"110":{"body":"The prover actually does not send a commitment to the full f polynomial. As described in the last check section . The verifier will have to compute the evaluation of L~(ζ) because it won't be zero. It should be equal to the following: ​f ​(ζ)−t (ζ)(ζn−1)=(ζ−1)(ζ−ωn−k)1−z(ζ)​[n(ζn−1)(ζ−ωn−k)​αPERM1+nωn−k(ζn−1)(ζ−1)​αPERM2]−pub(ζ)−z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅shift[0]ζ+γ)⋅(w[1](ζ)+β⋅shift[1]ζ+γ)⋅(w[2](ζ)+β⋅shift[2]ζ+γ)⋅(w[3](ζ)+β⋅shift[3]ζ+γ)⋅(w[4](ζ)+β⋅shift[4]ζ+γ)⋅(w[5](ζ)+β⋅shift[5]ζ+γ)⋅(w[6](ζ)+β⋅shift[6]ζ+γ)+z(ζω)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅(w[6](ζ)+γ)+​​ Because we use the inner product polynomial commitment , we also need: L (ζω)=f ​(ζω)−ZH​(ζ)⋅t~(ζω) Notice the ZH​(ζ). That evaluation must be sent as part of the proof as well.","breadcrumbs":"Maller's Optimization for Kimchi » The evaluation of L~","id":"110","title":"The evaluation of L~"},"111":{"body":"Now here's how we need to modify the current protocol: The evaluations f(ζ),f(ζω),t(ζ),t(ζω) (and their corresponding evaluation proofs) don't have to be part of the protocol anymore. The prover must still send the chunked commitments to t. The prover must create a linearized polynomial L by creating a linearized polynomial f ​ and a linearized polynomial t and computing: L =f ​+(ζn−1)⋅t While the verifier can compute the evaluation of L (ζ) by themselves, they don't know the evaluation of L (ζω), so the prover needs to send that. The verifier must recreate com(L ), the commitment to L , themselves so that they can verify the evaluation proofs of both L (ζ) and L (ζω). The evaluation of L~(ζω) must be absorbed in both sponges (Fq and Fr). maller 15 2 The proposal is implemented in #150 with the following details: the L~ polynomial is called ft. the evaluation of L~(ζ) is called ft_eval0. the evaluation L~(ζω) is called ft_eval1.","breadcrumbs":"Maller's Optimization for Kimchi » The actual protocol changes","id":"111","title":"The actual protocol changes"},"112":{"body":"In 2020, Plookup showed how to create lookup proofs. Proofs that some witness values are part of a lookup table . Two years later, an independent team published plonkup showing how to integrate Plookup into PlonK. This document specifies how we integrate Plookup in kimchi. It assumes that the reader understands the basics behind Plookup.","breadcrumbs":"Lookup Tables » Plookup in Kimchi","id":"112","title":"Plookup in Kimchi"},"113":{"body":"We integrate Plookup in kimchi with the following differences: we snake-ify the sorted table instead of wrapping it around (see later) we allow fixed-ahead-of-time linear combinations of columns of the queries we make we implemented different tables, like RangeCheck and XOR. we allow several lookups (or queries) to be performed within the same row zero-knowledgeness is added in a specific way (see later) The following document explains the protocol in more detail","breadcrumbs":"Lookup Tables » Overview","id":"113","title":"Overview"},"114":{"body":"As per the Plookup paper, the prover will have to compute three vectors: f, the (secret) query vector , containing the witness values that the prover wants to prove are part of the lookup table. t, the (public) lookup table . s, the (secret) concatenation of f and t, sorted by t (where elements are listed in the order they are listed in t). Essentially, Plookup proves that all the elements in f are indeed in the lookup table t if and only if the following multisets are equal: {(1+β)f,diff(t)} diff(sorted(f,t)) where diff is a new set derived by applying a \"randomized difference\" between every successive pairs of a vector, and (f,t) is the set union of f et t. More precisely, for a set S={s0​,s1​,⋯,sn​}, diff(S) is defined as the set {s0​+βs1​,s1​+βs2​,⋯,sn−1​+βsn​}. For example, with: f={5,4,1,5} t={1,4,5} we have: sorted(f,t)={1,1,4,4,5,5,5} {(1+β)f,diff(t)}={(1+β)5,(1+β)4,(1+β)1,(1+β)5,1+β4,4+β5} diff(sorted(f,t))={1+β1,1+β4,4+β4,4+β5,5+β5,5+β5} Note: This assumes that the lookup table is a single column. You will see in the next section how to address lookup tables with more than one column. The equality between the multisets can be proved with the permutation argument of PlonK, which would look like enforcing constraints on the following accumulator: init: acc0​=1 final: accn​=1 for every 0 RAM loading, disk -> RAM loading, and RAM r/w. This gives the maximum flexibility for circuit designers, and each primitive has uses for projects that we are pursuing. This design has been refined over several iterations to reduce the number of components exposed in the proof and the amount of computation that the verifier requires. The largest remaining parts represent the data itself and the method for combining them, where simplifying assumptions have been made to minimize their impact while still satisfying the design constraints.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » Why is this design the best in the space of possible designs?","id":"138","title":"Why is this design the best in the space of possible designs?"},"139":{"body":"Other designs building upon the lookup primitive reduce on some subset of this design, or on relaxing a design constraint that would remove some useful / required functionality.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What other designs have been considered and what is the rationale for not choosing them?","id":"139","title":"What other designs have been considered and what is the rationale for not choosing them?"},"14":{"body":"For a finite field Fp​ where p fits in n bits (i.e., p<2n) we have Addition, subtraction: O(n) Multiplication O(n2) Division O(n2) I believe, in practice significantly slower than multiplication. Actually, on a practical level, it's more accurate to model the complexity in terms of the number of limbs rather than the number of bits where a limb is 64 bits. Asymptotically it makes no difference but concretely it's better to think about the number of limbs for the most part. As a result you can see it's the smaller n is the better, especially with respect to multiplication, which dominates performance considerations for implementations of zk-SNARKs, since they are dominated by elliptic curve operations that consist of field operations. While still in development, Mina used to use a field of 753 bits or 12 limbs and now uses a field of 255 bits or 4 limbs. As a result, field multiplication became automatically sped up by a factor of 122/42=9, so you can see it's very useful to try to shrink the field size.","breadcrumbs":"Fields » Algorithmics of prime order fields","id":"14","title":"Algorithmics of prime order fields"},"140":{"body":"We are currently unable to expose certain useful functionality to users of Snapps / SnarkyJS, and are missing the ability to use that same functionality for important parts of the zkOracles ecosystem (such as parsing / scraping). We also are currently forced to hard-code any constraints that we need for particular uses in the verifier circuit, and users are unable to remove the parts they do not need and to replace them with other useful constraints that their circuits might benefit from. The Mina transaction model is unable to support a bytecode interpreter without this or similar extensions.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What is the impact of not doing this?","id":"140","title":"What is the impact of not doing this?"},"141":{"body":"(Elided)","breadcrumbs":"Lookup Tables » Extended Lookup Tables » Prior art","id":"141","title":"Prior art"},"142":{"body":"","breadcrumbs":"Lookup Tables » Extended Lookup Tables » Unresolved questions","id":"142","title":"Unresolved questions"},"143":{"body":"The maximum number of tables / entries of each kind that we want to support in the 'standard' protocol. Validation (or rejection) of the simplifying assumptions. Confirmation of the design constraints, or simplifications thereto. Necessity / utility of the permutation argument.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What parts of the design do you expect to resolve through the RFC process before this gets merged?","id":"143","title":"What parts of the design do you expect to resolve through the RFC process before this gets merged?"},"144":{"body":"None, this RFC has been developed in part from an initial design and iteration on implementations of parts of that design.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What parts of the design do you expect to resolve through the implementation of this feature before merge?","id":"144","title":"What parts of the design do you expect to resolve through the implementation of this feature before merge?"},"145":{"body":"Bytecode design and implementation details. Mina transaction interface for interacting with these features. Snarky / SnarkyJS interface for interacting with these features.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What related issues do you consider out of scope for this RFC that could be addressed in the future independently of the solution that comes out of this RFC?","id":"145","title":"What related issues do you consider out of scope for this RFC that could be addressed in the future independently of the solution that comes out of this RFC?"},"146":{"body":"This section explains how to design and add a custom constraint to our proof-systems library. PLONK is an AIOP. That is, it is a protocol in which the prover sends polynomials as messages and the verifier sends random challenges, and then evaluates the prover's polynomials and performs some final checks on the outputs. PLONK is very flexible. It can be customized with constraints specific to computations of interest. For example, in Mina, we use a PLONK configuration called kimchi that has custom constraints for poseidon hashing, doing elliptic curve operations, and more. A \"PLONK configuration\" specifies The set of types of constraints that you would like to be able to enforce. We will describe below how these types of constraints are specified. A number of \"eq-able\" columns W A number of \"advice\" columns A Under such configuration, a circuit is specified by A number of rows n A vector cs of constraint-types of length n. I.e., a vector that specifies, for each row, which types of constraints should be enforced on that row. A vector eqs : Vec<(Position, Position)> of equalities to enforce, where struct Position { row: usize, column: usize }. E.g., if the pair (Position { row: 0, col: 8 }, Position { row: 10, col: 2 }) is in eqs, then the circuit is saying the entries in those two positions should be equal, or in other words that they refer to the same value. This is where the distinction between \"eq-able\" and \"advice\" columns comes in. The column field of a position in the eqs array can only refer to one of the first W columns. Equalities cannot be enforced on entries in the A columns after that. Then, given such a circuit, PLONK lets you produce proofs for the statement I know W + A \"column vectors\" of field elements vs: [Vec; W + A] such that for each row index i < n, the constraint of type cs[i] holds on the values [vs[0][i], ..., vs[W+A - 1][i], vs[0][i+1], ..., vs[W+A - 1][i+1] and all the equalities in eqs hold. I.e., for (p1, p2) in eqs we have vs[p1.col][p1.row] == vs[p2.col][p2.row]. So, a constraint can check the values in two adjacent rows. Mathematically speaking, a constraint is a multivariate polynomial over the variables cCurr,i​,…,vCurr,W+A−1​,vNext,0​,…,vNext,W+A−1​. In other words, there is one variable corresponding to the value of each column in the \"current row\" and one variable correspond to the value of each column in the \"next row\". In Rust, vr,i​ is written E::cell(Column::Witness(i), r). So, for example, the variable vNext,3​ is written E::cell(Column::Witness(3), CurrOrNext::Next). let w = |i| v(Column::Witness(i)); Let's","breadcrumbs":"Custom Constraints » Specifying a constraint","id":"146","title":"Specifying a constraint"},"147":{"body":"The art in proof systems comes from knowing how to design a PLONK configuration to ensure maximal efficiency for the sorts of computations you are trying to prove. That is, how to choose the numbers of columns W and A, and how to define the set of constraint types. Let's describe the trade-offs involved here. The majority of the proving time for the PLONK prover is in committing to the W + A column polynomials, which have length equal to the number of rows n committing to the \"permutation accumulator polynomial, which has length n. committing to the quotient polynomial, which reduces to computing max(k, W) MSMs of size n, where k is the max degree of a constraint. performing the commitment opening proof, which is mostly dependent on the number of rows n. So all in all, the proving time is approximately equal to the time to perform W + A + 1 + max(k - 1, W) MSMs of size n, plus the cost of an opening proof for polynomials of degree n - 1. and maybe computing the combined constraint polynomial, which has degree k * n where k is the maximum degree of a constraint Increasing W and A increase proof size, and they potentially impact the prover-time as the prover must compute polynomial commitments to each column, and computing a polynomial commitment corresponds to doing one MSM (multi-scalar multiplication, also called a multi-exponentiation.) However, often increasing the number of columns allows you to decrease the number of rows required for a given computation. For example, if you can perform one Poseidon hash in 36 rows with 5 total columns, then you can also perform it in 12 (= 36 / 3) rows with 15 (= 5 * 3) total columns. Decreasing the number of rows (even while keeping the total number of table entries the same) is desirable because it reduces the cost of the polynomial commitment opening proof, which is dominated by a factor linear in the number of rows, and barely depends on the number of columns. Increasing the number of columns also increases verifier time, as the verifier must perform one scalar-multiplication and one hash per column. Proof length is also affected by a larger number of columns, as more polynomials need to be committed and sent along to the verifier. There is typically some interplay between these","breadcrumbs":"Custom Constraints » Defining a PLONK configuration","id":"147","title":"Defining a PLONK configuration"},"148":{"body":"The goal of this section is to draw the connections between the practical aspect of designing custom gates and the theory behind it. First, suppose we had the following layout of gates. This means that our circuit has four gates: a (single) generic gate, a (non-optimized) complete add, and then two different Example gates that we just make up for the example; ExamplePairs and ExampleTriples. row GateType:: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 Generic l r o 1 CompleteAdd x1 y1 x2 y2 x3 y3 inf same_x s inf_z x21_inv 2 ExamplePairs a e i o u y 3 ExampleTriples a b c d e f g h i j k l m n o The most important aspect to take into account when you are designing a new custom gate is writing the actual description of the new functionality. For this, we are using the Expression framework that allows us to write equations generically by representing each column of the table (wire) as a variable. Then we can write operations between them and define the gate constraints . From a mathematical perspective, these will have the shape of multivariate polynomials of as many variables as columns in the table (currently 15 for Kimchi) that is constrained to equal 0. In our example, we would define equations that represent the operations carried out by each of the gates. In the following equations we use the variable names that we wrote in the cells above, and we don't imply that the same name in a different row correspond to the same value (we are not imposing equalities between gates yet): Generic: l ⋅cl​+ r ⋅cr​+ o ⋅co​+ l ⋅ r ⋅cm​+cc​=0 CompleteAdd: x21_inv ⋅( x2 − x1 )−(1− same_x)=0 same_x ⋅( x2 − x1 )=0 same_x ⋅(2⋅ s ⋅ y1 −3⋅ x12)+(1− same_x )⋅((x2 − x1 )⋅ s − y2 + y1 )=0 x1 + x2 + x3 − s2=0 s ⋅( x1 − x3 )− y1 − y3=0 ( y2 − y1 )⋅( same_x − inf )=0 ( y2 − y1 )⋅ inf_z − inf =0 ExamplePairs: a − e =0 i − o =0 u − y =0 ExampleTriples: a ⋅ b ⋅ c =0 d ⋅ e ⋅ f =0 g ⋅ h ⋅ i =0 j ⋅ k ⋅ l =0 m ⋅ n ⋅ o =0 Nonetheless, when we write these constraints in proof-systems, we use a shared framework that helps us use the columns in each row as placeholders. For example, the i-th column of the current row Curr is represented as: E::::cell(Column::Witness(i), CurrOrNext::Curr) where the E type is a shorthand for Expr. More succinctly, we refer to the above simply as: witness_curr(i) For readability, let's just refer to them here as w0..w14. Then, the above constraints would look like the following: Generic: w0 ⋅cl​+ w1 ⋅cr​+ w2 ⋅co​+ w0 ⋅ w1 ⋅cm​+cc​=0 CompleteAdd: w10 ⋅( w2 − w0 )−(1− w7)=0 w7 ⋅( w2 − w0 )=0 w7 ⋅(2⋅ w8 ⋅ w1 −3⋅ w02)+(1− w7 )⋅((w2 − w0 )⋅ w8 − w3 + w1 )=0 w0 + w2 + w4 − w82=0 w8 ⋅( w0 − w4 )− w1 − w4=0 ( w3 − w1 )⋅( w7 − w6 )=0 ( w3 − w1 )⋅ w9 − w6 =0 ExamplePairs: w0 − w1 =0 w2 − w3 =0 w4 − w5 =0 ExampleTriples: w0 ⋅ w1 ⋅ w2 =0 w3 ⋅ w4 ⋅ w5 =0 w6 ⋅ w7 ⋅ w8 =0 w9 ⋅ w10 ⋅ w11 =0 w12 ⋅ w13 ⋅ w14 =0 Looking at the list of equations, each of them look like polynomials in the variables w0..w14, constrained to zero. But so far, the type of equations we had studied were univariate (i.e. f(X) rather than f(X0​,...,X14​)). But this will not be a problem, since these multivariate equations will soon become univariate with a simple trick: interpolation . Right now, we are describing the gate constraints row-wise (i.e. horizontally ). But as we said before, our domain will be determined by the number of rows; we will transform these equations to a vertical and univariate behaviour. The strategy to follow will be obtaining witness polynomials that evaluate to the cells of the execution trace. First , note that so far we didn't mention any values for the w0..w14 terms, but instead they are placeholders for what is yet to come . And what is this you may ask: the witness of the relation . Put it another way, the witness will be the instantiation of the cells of the table that will make all of the above constraints hold. And where do we get these values from? Well, the prover will know or will compute them . Second , we count the number of rows that we ended up with: here it's four (conveniently, a power of two). What we want to do is to create as many witness polynomials as the number of columns in the execution trace. These polynomials, that we can call w0​(X),...,w14​(X), must be design to evaluate to the cells of the trace. Formally we define each wj​(X) as: G=:wj​(gi)=trace[i,j] Meaning that wi​(x) returns the i-th column of the x-th row of the execution trace. And how can we create such polynomials? We will use interpolation, which can be efficiently computed over power-of-two-sized groups, as it is the case in the example. Recalling Lagrangian terms , they act like linearly independent selectors over a set, in particular: Let G be a multiplicative group formed by the powers of a generator g. Given any element x in the group, the i-th Lagrangian term evaluates to 1 iff x=gi and 0 iff x=gi. ∀ x∈G:LgiG​(x)=1⟺x=θ and LgiG​(x)=0⟺x=θ Given that our group has size 4, the number of rows (ignoring ZK for now), G will be the set {1,g,g2,g3}. This means that we can build such 15 witness polynomials as: ∀j∈[0..15):wj​(X)=i=0∑3​trace[i,j]⋅LgiG​(X) Then, the above constraints will become the following equations using our 15 univariate witness polynomials: Generic: w0​(X)⋅cl​+w1​(X)⋅cr​+w2​(X)⋅co​+w0​(X)⋅w1​(X)⋅cm​+cc​=0 CompleteAdd: w10​(X)⋅(w2​(X)−w0​(X))−(1−w7​(X))=0 w7​(X)⋅(w2​(X)−w0​(X))=0 w7​(X)⋅(2⋅w8​(X)⋅w1​(X)−3⋅w0​(X)2)+(1−w7​(X))⋅((w2​(X)−w0​(X))⋅w8​(X)−w3​(X)+w1​(X))=0 w0​(X)+w2​(X)+w4​(X)−w8​(X)2=0 w8​(X)⋅(w0​(X)−w4​(X))−w1​(X)−w4​(X)=0 (w3​(X)−w1​(X))⋅(w7​(X)−w6​(X))=0 (w3​(X)−w1​(X))⋅w9​(X)−w6​(X)=0 ExamplePairs: w0​(X)−w1​(X)=0 w2​(X)−w3​(X)=0 w4​(X)−w5​(X)=0 ExampleTriples: w0​(X)⋅w1​(X)⋅w2​(X)=0 w3​(X)⋅w4​(X)⋅w5​(X)=0 w6​(X)⋅w7​(X)⋅w8​(X)=0 w9​(X)⋅w10​(X)⋅w11​(X)=0 w12​(X)⋅w13​(X)⋅w14​(X)=0 These are the 16 constraints that will need to be satisfied by a correct witness for the relation. We already know how to check many equations at once (combining them with powers of alpha), and how to check that the above holds for the full domain G (by providing the quotient polynomial of the huge equation after dividing by the vanishing polynomial on G). All there's left to do is making a correspondence between the constraints that need to be satisfied, and the particular row in the execution trace to which they apply. This implies, that before even aggregating the 16 constraints into a single one, we need to impose this restriction. In proof-systems, we represent this restriction with the GateType selector. In particular, we would multiply each of the constraints above by the Expr terms: Index(GateType::YourType), which will be transformed (again) into mutually exclusive selector polynomials. This means, that each row can only have one of these polynomials being nonzero (equal to 1), whereas the evaluation of all of the other selector polynomials for the given row should be 0. These polynomials are known by the verifier, who could check that this is indeed the case. But what will these polynomials look like? I am sure the above paragraph is familiar for you already. Once again, we will be able to build them using interpolation with our 4 Lagrangian terms for G. In our example: generic(X) will equal 1 when X=g0 and 0 otherwise; add(X) will equal 1 when X=g1 and 0 otherwise; pairs(X) will equal 1 when X=g2 and 0 otherwise; and triples(X) will equal 1 when X=g3 and 0 otherwise. These polynomials are, in fact, a description of the circuit. Note that we managed to make constraints for different gates already independent from each other (thanks to the mutually exclusivity). Nonetheless, granting independence within the gate is still needed. Here's where we will be using the powers of alpha: within a gate the powers cannot be repeated, but they can be reused among gates. Then, we will need 7 different powers of alpha for this stage of the protocol, as this is the largest number of different constraints within the same gate. Putting all of the above together, this means that our LONG and single constraint will look like: gates(X)= generic(X)⋅ α0⋅(w0​(X)⋅cl​+w1​(X)⋅cr​+w2​(X)⋅co​+w0​(X)⋅w1​(X)⋅cm​+cc​) +add(X)⋅( α0⋅(w10​(X)⋅(w2​(X)−w0​(X))−(1−w7​(X))) +α1⋅(w7​(X)⋅(w2​(X)−w0​(X))) +α2⋅(w7​(X)⋅(2⋅w8​(X)⋅w1​(X)−3⋅w0​(X)2)+(1−w7​(X))⋅((w2​(X)−w0​(X))⋅w8​(X)−w3​(X)+w1​(X))) +α3⋅(w0​(X)+w2​(X)+w4​(X)−w8​(X)2) +α4⋅(w8​(X)⋅(w0​(X)−w4​(X))−w1​(X)−w4​(X)) +α5⋅((w3​(X)−w1​(X))⋅(w7​(X)−w6​(X))) +α6⋅((w3​(X)−w1​(X))⋅w9​(X)−w6​(X))) +pairs(X)⋅( α0⋅(w0​(X)−w1​(X)) +α1⋅(w2​(X)−w3​(X)) +α2⋅(w4​(X)−w5​(X))) +triples(X)⋅( α0⋅(w0​(X)⋅w1​(X)⋅w2​(X)) +α1⋅(w3​(X)⋅w4​(X)⋅w5​(X)) +α2⋅(w6​(X)⋅w7​(X)⋅w8​(X)) +α3⋅(w9​(X)⋅w10​(X)⋅w11​(X)) +α4⋅(w12​(X)⋅w13​(X)⋅w14​(X))) Finally, providing q(X)=gates(X)/vG​(X) and performing the check on a random X∈F \\ G would give the verifier the assurance of the following: The prover knows a polynomial gates(X) that equals zero on any x∈{1,g,g2,g3}. Nonetheless, it would still remain to verify that gates(X) actually corresponds to the encoding of the actual constraints. Meaning, checking that this polynomial encodes the column witnesses, and the agreed circuit. So instead of providing just gates(X) (actually, a commitment to it), the prover can send commitments to each of th 15 witness polynomials, so that the verifier can reconstruct the huge constraint using the encodings of the circuit (which is known ahead).","breadcrumbs":"Custom Gates » Custom gates","id":"148","title":"Custom gates"},"149":{"body":"This document is meant to explain the foreign field addition gate in Kimchi.","breadcrumbs":"Custom Gates » Foreign Field Addition » Foreign Field Addition RFC","id":"149","title":"Foreign Field Addition RFC"},"15":{"body":"The next object to know about is polynomials. A polynomial is a syntactic object that also defines a function. Specifically, let R be a field (or more generally it could even be a ring, which is like a field but without necessarily having division, like the integers Z). And pick a variable name like x. Then R[x] (pronounced, \"R adjoin x\" or \"polynomials in x over R\" or \"polynomials in x with R coefficients\") is the set of syntactic expressions of the form a0​+a1​x+a2​x2+⋯+ad​xd where each ai​∈R and d is any natural number. If we wanted to be very formal about it, we could define R[x] to be the set of lists of elements of R (the lists of coefficients), and the above is just a suggestive way of writing that list. An important fact about polynomials over R is that they can be interpreted as functions R→R. In other words, there is a function eval:R[x]→(R→R) defined by eval(a0​+a1​x+⋯+ad​xd)=(y:R)↦a0​+a1​y+⋯+ad​yd where x is just a variable name without a value assigned to it and eval maps it to a field element y and evaluates the function as the inner product between the list of coefficients and the powers of y. It is important to remember that polynomials are different than the functions that they compute. Polynomials are really just lists of coefficients. Over some fields, there are polynomials p1 and p2 such that eval(p1)=eval(p2), but p1=p2. For example, consider in F2​[x] the polynomials x and x2. These map to the same function F2​→F2​ (meaning for x∈R=F2​={0,1} it holds that x=x2), but are distinct polynomials. Some definitions and notation If f is a polynomial in R[x] and a∈R, we write f(a) for eval(f)(a) If f is a polynomial, the degree of f is the largest d for which the coefficient of xd is non-zero in f. For example, the degree of x3+2x10 is 10. We will use the notation R[x]n. In order to deal with this, we will divide foreign field elements into limbs that fit in our native field. We want to be compatible with the foreign field multiplication gate, and thus the parameters we will be using are the following: 3 limbs of 88 bits each (for a total of 264 bits) So f<2264, concretely: The modulus f of the foreign field Ff​ is 256 bit The modulus of our the native field Fn​ is 255 bit In other words, using 3 limbs of 88 bits each allows us to represent any foreign field element in the range [0,2264) for foreign field addition, but only up to 2259 for foreign field multiplication. Thus, with the current configuration of our limbs, our foreign field must be smaller than 2259 (because 2264⋅2255>22592+2259, more on this in Foreign Field Multiplication or the original FFmul RFC .","breadcrumbs":"Custom Gates » Foreign Field Addition » Overview","id":"150","title":"Overview"},"151":{"body":"Let's take a closer look at what we have, if we split our variables in limbs (using little endian) bits 0..............87|88...........175|176...........263 a = (-------a0-------|-------a1-------|-------a2-------)\n+/-\nb = (-------b0-------|-------b1-------|-------b2-------)\n=\nr = (-------r0-------|-------r1-------|-------r2-------) mod(f) We will perform the addition in 3 steps, one per limb. Now, when we split long additions in this way, we must be careful with carry bits between limbs. Also, even if a and b are foreign field elements, it could be the case that a+b is already larger than the modulus (in this case a+b could be at most 2f−2). But it could also be the case that the subtraction produces an underflow because a > >\n= c_0 c_1 c_2\nq = -1 | 0 | 1\n·\nf = (-------f0-------|-------f1-------|-------f2-------)\n+\nr = (-------r0-------|-------r1-------|-------r2-------) Our witness computation is currently using the BigUint library, which takes care of all of the intermediate carry limbs itself so we do not have to address all casuistries ourselves (such as propagating the low carry flag to the high limb in case the middle limb is all zeros).","breadcrumbs":"Custom Gates » Foreign Field Addition » Splitting the addition","id":"151","title":"Splitting the addition"},"152":{"body":"Last but not least, we should perform some range checks to make sure that the result r is contained in Ff​. This is important because there could be other values of the result which still fit in <2264 but are larger than f, and we must make sure that the final result is the minimum one (that we will be referring to as rmin​ in the following). Ideally, we would like to reuse some gates that we already have. In particular, we can perform range checks for 0≤X<23ℓ=23⋅88. But we want to check that 0≤rmin​ >\n= k_0 k_1\nf\n+\nu = (-------u0-------|-------u1-------|-------u2-------) Following the steps above, and representing this equation in limb form, we have: u0​u1​u2​​=r0​+0−f0​−k0​⋅288=r1​+0−f1​−k1​⋅288+k0​=r2​+288−f2​+k1​​ But now we also have to check that 0≤r. But this is implicitly covered by r being a field element of at most 264 bits (range check). When we have a chain of additions ai​+bi​=ri​ with ai​=ri−1​, we could apply the field membership check naively to every intermediate ri​, however it is sufficient to apply it only once at the end of the computations for rn​, and keep intermediate ri​∈[0,2264), in a \"raw\" form. Generally speaking, treating intermediate values lazily helps to save a lot of computation in many different FF addition and multiplication scenarios.","breadcrumbs":"Custom Gates » Foreign Field Addition » Upper bound check for foreign field membership","id":"152","title":"Upper bound check for foreign field membership"},"153":{"body":"Mathematically speaking, a subtraction within a field is no more than an addition over that field. Negative elements are not different from \"positive\" elements in finite fields (or in any modular arithmetic). Our witness computation code computes negative sums by adding the modulus to the result. To give a general example, the element −e within a field Fm​ of order m and e > > c_0 c_1 0 Inspired by the halving approach in foreign field multiplication, an optimized version of the above gate results in a reduction by 2 in the number of constraints and by 1 in the number of witness cells needed. The main idea is to condense the claims about the low and middle limbs in one single larger limb of 176 bits, which fit in our native field. This way, we can get rid of the low carry flag, its corresponding carry check, and the three result checks become just two. bits 0..............87|88...........175|176...........263\nr = (-------r0------- -------r1-------|-------r2-------)\n=\na = (-------a0------- -------a1-------|-------a2-------)\n+\ns = 1 | -1\n·\nb = (-------b0------- -------b1-------|-------b2-------)\n-\nq = -1 | 0 | 1\n·\nf = (-------f0------- -------f1-------|-------f2-------) > > c 0 These are the new equations: rbot​rtop​​=(a0​+288⋅a1​)+s⋅(b0​+288⋅b1​)−q⋅(f0​+288⋅f1​)−c⋅2176=a2​+s⋅b2​−q⋅f2​+c​ with rtop​=r2​ and c=c1​.","breadcrumbs":"Custom Gates » Foreign Field Addition » Optimization","id":"154","title":"Optimization"},"155":{"body":"The full foreign field addition/subtraction gadget will be composed of: 1 public input row containing the value 1; n rows with ForeignFieldAdd gate type: for the actual n chained additions or subtractions; 1 ForeignFieldAdd row for the bound addition; 1 Zero row for the final bound addition. 1 RangeCheck gadget for the first left input of the chain a1​:=a; Then, n of the following set of rows: 1 RangeCheck gadget for the i-th right input bi​; 1 RangeCheck gadget for the i-th result which will correspond to the (i+1)-th left input of the chain ri​=ai+1​. 1 final RangeCheck gadget for the bound check u. A total of 20 rows with 15 columns in Kimchi for 1 addition. All ranges below are inclusive. Row(s) Gate type(s) Witness 0 PublicInput 1 1..n ForeignFieldAdd +/- n+1 ForeignFieldAdd bound n+2 Zero bound n+3..n+6 multi-range-check left n+7+8i..n+10+8i multi-range-check right n+11+8i..n+14+8i multi-range-check r 9n+7..9n+10 multi-range-check u This mechanism can chain foreign field additions together. Initially, there are n foreign field addition gates, followed by a foreign field addition gate for the bound addition (whose current row corresponds to the next row of the last foreign field addition gate), and an auxiliary Zero row that holds the upper bound. At the end, an initial left input range check is performed, which is followed by a n pairs of range check gates for the right input and intermediate result (which become the left input for the next iteration). After the chained inputs checks, a final range check on the bound takes place. For example, chaining the following set of 3 instructions would result in a full gadget with 37 rows: add(add(add(a,b),c),d) Row(s) Gate type(s) Witness 0 PublicInput 1 1 ForeignFieldAdd a+b 2 ForeignFieldAdd (a+b)+c 3 ForeignFieldAdd ((a+b)+c)+d 4 ForeignFieldAdd bound 5 Zero bound 6..9 multi-range-check a 10..13 multi-range-check b 14..17 multi-range-check a+b 18..21 multi-range-check c 22..25 multi-range-check a+b+c 26..29 multi-range-check d 30..33 multi-range-check a+b+c+d 34..37 multi-range-check bound Nonetheless, such an exhaustive set of checks are not necessary for completeness nor soundness. In particular, only the very final range check for the bound is required. Thus, a shorter gadget that is equally valid and takes (8⋅n+4) fewer rows could be possible if we can assume that the inputs of each addition are correct foreign field elements. It would follow the next layout (with inclusive ranges): Row(s) Gate type(s) Witness 0 public input row for soundness of bound overflow flag 1 1..n ForeignFieldAdd n+1 ForeignFieldAdd n+2 Zero n+3..n+6 multi-range-check for bound u Otherwise, we would need range checks for each new input of the chain, but none for intermediate results; implying 4⋅n fewer rows. Row(s) Gate type(s) Witness 0 public input row for soundness of bound overflow flag 1 1..n ForeignFieldAdd ai​+bi​ n+1 ForeignFieldAdd n+2 Zero n+3..n+6 multi-range-check for first left input a1​ n+7+4i..n+10+4i multi-range-check for i-th right input bi​ 5n+7..5n+10 multi-range-check for bound u For more details see the Bound Addition section in Foreign Field Multiplication or the original Foreign Field Multiplication RFC .","breadcrumbs":"Custom Gates » Foreign Field Addition » Gadget","id":"155","title":"Gadget"},"156":{"body":"For the full mode of tests of this gate, we need to perform 4 range checks to assert that the limbs of a,b,r,u have a correct size, meaning they fit in 288 (and thus, range-checking a,b,r,u for 2264). Because each of these elements is split into 3 limbs, we will have to use 3 copy constraints between the RangeCheck gates and the ForeignFieldAdd rows (per element). That amounts to 12 copy constraints. Recalling that Kimchi only allows for the first 7 columns of each row to host a copy constraint, we necessarily have to use 2 rows for the actual addition gate. The layout of these two rows is the following: Curr Next ... Final Column ForeignFieldAdd ForeignFieldAdd Zero 0 a0​ (copy) r0​ (copy) u0​ (copy) 1 a1​ (copy) r1​ (copy) u1​ (copy) 2 a2​ (copy) r2​ (copy) u2​ (copy) 3 b0​ (copy) 4 b1​ (copy) 5 b2​ (copy) 6 q 7 c 8 9 10 11 12 13 14","breadcrumbs":"Custom Gates » Foreign Field Addition » Layout","id":"156","title":"Layout"},"157":{"body":"So far, we have pointed out the following sets of constraints: Main addition 0=a0​+b0​−r0​−q⋅f0​−288⋅c0​ 0=a1​+b1​−r1​−q⋅f1​+c0​−288⋅c1​ 0=a2​+b2​−r2​−q⋅f2​+c1​ Carry checks 0=c0​⋅(c0​+1)⋅(c0​−1) 0=c1​⋅(c0​+1)⋅(c1​−1) Sign checks TODO: decide if we really want to keep this check or leave it implicit as it is a coefficient value 0=(s+1)⋅(s−1) Overflow checks 0=q⋅(q−s)","breadcrumbs":"Custom Gates » Foreign Field Addition » Constraints","id":"157","title":"Constraints"},"158":{"body":"When we use this gate as part of a larger chained gadget, we should optimize the number of range check rows to avoid redundant checks. In particular, if the result of an addition becomes one input of another addition, there is no need to check twice that the limbs of that term have the right length. The sign is now part of the coefficients of the gate. This will allow us to remove the sign check constraint and release one witness cell element. But more importantly, it brings soundness to the gate as it is now possible to wire the 1 public input to the overflow witness of the final bound check of every addition chain.","breadcrumbs":"Custom Gates » Foreign Field Addition » Optimizations","id":"158","title":"Optimizations"},"159":{"body":"This document is an original RFC explaining how we constrain foreign field multiplication (i.e. non-native field multiplication) in the Kimchi proof system. For a more recent RFC, see FFmul RFC . Changelog Author(s) Date Details Joseph Spadavecchia and Anais Querol October 2022 Design of foreign field multiplication in Kimchi","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Foreign Field Multiplication","id":"159","title":"Foreign Field Multiplication"},"16":{"body":"An important fact for zk-SNARKs about polynomials is that we can use them to encode arrays of field elements. In other words, there is a way of converting between polynomials of degree d and arrays of field elements of length d+1. This is important for a few reasons It will allow us to translate statements about arrays of field elements into statements about polynomials, which will let us construct zk-SNARKs. It will allow us to perform certain operations on polynomials more efficiently. So let's get to defining and proving this connection between arrays of field elements. The first thing to understand is that we won't be talking about arrays directly in this section. Instead, we'll be talking about functions A→F where A is a finite subset of F. The idea is, if A has size n, and we put an ordering on A, then we can think of a function A→F as the same thing as an immutable array of length n, since such a function can be thought of as returning the value of an array at the input position. With this understanding in hand, we can start describing the \"fundamental theorem of polynomials\". If A⊆F has size d+1, this theorem will define an isomorphism between functions A→F and F[x]≤d​, the set of polynomials of degree at most d. Now let's start defining this isomorphism. One direction of it is very easy to define. It is none other than the evaluation map, restricted to A: evalA​evalA​​:F[x]≤d​→(A→F)(c0​+⋯+cd​xd)=a↦i≤d∑​ci​ai​ We would now like to construct an inverse to this map. What would that mean? It would be a function that takes as input a function φ:A→F (remember, basically an array of length ∣A∣), and returns a polynomial f which agrees with φ on the set A. In other words, we want to construct a polynomial that interpolates between the points (a,φ(a)) for a∈A. Our strategy for constructing this polynomial will be straightforward. For each a∈A, we will construct a polynomial fa​ that is equal to φ(a) when evaluated at a, and equal to 0 when evaluated anywhere else in the set A. Then, our final polynomial will be f:=∑a∈A​fa​. Then, when f is evaluated at a0​∈A, only the fa0​​ term will be non-zero (and equal to φ(a0​) as desired), all the other terms will disappear. Constructing the interpolation map requires a lemma. Lemma. (construction of vanishing polynomials) Let S⊆F. Then there is a polynomial vS​∈F[x] of degree ∣S∣ such that vS​ evaluates to 0 on S, and is non-zero off of S. vS​ is called the vanishing polynomial of S. Proof. Let vS​=∏s∈S​(x−s). Clearly vS​ has degree ∣S∣. If t∈S, then vS​(t)=0 since x−t is one of the terms in the product defining vS​, so when we evaluate at t, t−t=0 is one of the terms. If t∈/S, then all the terms in the product are non-zero, and thus vS​(t) is non-zero as well. □ Now we can define the inverse map. Define interpA​interpA​​:(A→F)→F[x]≤d​(f)=a∈A∑​vA∖{a}​(a)f(a)​⋅vA∖{a}​​ Since each vA∖{a}​ has degree d, this polynomial has degree d. Now we have, for any b∈A, evalA​(interpA​(f))(b)​=a∈A∑​vA∖{a}​(a)f(a)​⋅vA∖{a}​(b)=vA∖{b}​(b)f(b)​⋅vA∖{b}​(b)+a=b∑​vA∖{a}​(a)f(a)​⋅vA∖{a}​(b)=f(b)+a=b∑​vA∖{a}​(a)f(a)​⋅0=f(b)​ Thus evalA​∘interpA​ is the identity. So we have successfully devised a way of interpolating a set of d+1 points with a polynomial of degree d. What remains to be seen is that these two maps are inverse in the other direction. That is, that for any polynomial f∈F[x], we have interpA​(evalA​(f))=f This says that if we interpolate the function that has the values of f on A, we get back f itself. This essentially says that there is only one way to interpolate a set of d+1 points with a degree d polynomial. So let's prove it by proving that statement. Lemma: polynomials that agree on enough points are equal. Let F be a field and suppose f,g∈F[x] have degree at most d. Let A⊆F have size d+1. Suppose for all a∈A, f(a)=g(a). Then f=g as polynomials. Proof. Define h:=f−g. Our strategy will be to show that h is the zero polynomial. This would then imply that f=g as polynomials. Note that h vanishes on all of A since f and g are equal on A. To that end, let a∈A. Then we can apply the polynomial division algorithm to divide h by x−a and obtain polynomials qa​,ra​ such that h=qa​(x−a)+ra​ and ra​ has degree less than 1. I.e., ra​ is a constant in F. Now, 0=h(a)=qa​(a)⋅0+ra​=ra​ so ra​=0 and thus h=qa​(x−a). Note that qa​ is 0 on all elements b∈A with b=a since h(b)=qa​(b)(b−a)=0, but b−a=0. Thus, if we iterate the above, enumerating the elements of A as a0​,…,ad​, we find that h​=(x−a0​)qa0​​=(x−a0​)(x−a1​)qa1​​=…=(x−a0​)(x−a1​)…(x−ad​)qad​​​ Now, if qad​​ is not the zero polynomial, then h will have degree at least d+1 since it would have as a factor the d+1 linear terms (x−ai​). But since f,g both have degree at most d and h=f−g, h has degree at most d as well. Thus, qad​​=0, which means h=0 as well. This gives us the desired conclusion, that f=g as polynomials. □ Now we can easily show that interpolating the evaluations of a polynomial yields that polynomial itself. Let f∈F[x]≤d​. Then interpA​(evalA​(f)) is a polynomial of degree at most d that agrees with f on A, a set of size d+1. Thus, by the lemma, they are equal as polynomials. So indeed interpA​(evalA​(f))=f for all f∈F[x]. So far we have proven that interpA​ and evalA​ give an isomorphism of sets (i.e., a bijection) between the sets A→F and F[x]≤d​. But we can take this theorem a bit further. The set of functions A→F can be given the structure of a ring, where addition and multiplication happen pointwise. I.e., for f,g:A→F we define f+g:=a↦f(a)+g(a) and f⋅g:=a↦f(a)⋅g(a). Then we can strengthen our theorem to say Fundamental theorem of polynomials (final version) Let d∈N and let A⊆F with ∣A∣=d+1. With evalA​:F[x]≤d​→(A→F)interpA​:(A→F)→F[x]≤d​ defined as above, these two functions define an isomorphism of rings. That is is, they are mutually inverse and each one respects addition, subtraction and multiplication. The fundamental theorem of polynomials is very important when it comes to computing operations on polynomials. As we will see in the next section , the theorem will help us to compute the product of degree n polynomials in time O(nlogn), whereas the naive algorithm takes time O(n2). To put this in perspective, if n=216, n2 is 4096 times larger than nlogn and the gap only gets bigger as n grows.","breadcrumbs":"Polynomials » Fundamental theorem of polynomials","id":"16","title":"Fundamental theorem of polynomials"},"160":{"body":"This gate constrains a⋅b=cmodf where a,b,c∈Ff​, a foreign field with modulus f, using the native field Fn​ with prime modulus n.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Overview","id":"160","title":"Overview"},"161":{"body":"In foreign field multiplication the foreign field modulus f could be bigger or smaller than the native field modulus n. When the foreign field modulus is bigger, then we need to emulate foreign field multiplication by splitting the foreign field elements up into limbs that fit into the native field element size. When the foreign modulus is smaller everything can be constrained either in the native field or split up into limbs. Since our projected use cases are when the foreign field modulus is bigger (more on this below) we optimize our design around this scenario. For this case, not only must we split foreign field elements up into limbs that fit within the native field, but we must also operate in a space bigger than the foreign field. This is because we check the multiplication of two foreign field elements by constraining its quotient and remainder. That is, renaming c to r, we must constrain a⋅b=q⋅f+r, where the maximum size of q and r is f−1 and so we have a⋅b​≤q(f−1)​​⋅f+r(f−1)​​≤f2−1.​ Thus, the maximum size of the multiplication is quadratic in the size of foreign field. Naïve approach Naïvely, this implies that we have elements of size f2−1 that must split them up into limbs of size at most n−1. For example, if the foreign field modulus is 256 and the native field modulus is 255 bits, then we'd need log2​((2256)2−1)≈512 bits and, thus, require 512/255≈3 native limbs. However, each limb cannot consume all 255 bits of the native field element because we need space to perform arithmetic on the limbs themselves while constraining the foreign field multiplication. Therefore, we need to choose a limb size that leaves space for performing these computations. Later in this document (see the section entitled \"Choosing the limb configuration\") we determine the optimal number of limbs that reduces the number of rows and gates required to constrain foreign field multiplication. This results in ℓ=88 bits as our optimal limb size. In the section about intermediate products we place some upperbounds on the number of bits required when constraining foreign field multiplication with limbs of size ℓ thereby proving that the computations can fit within the native field size. Observe that by combining the naïve approach above with a limb size of 88 bits, we would require 512/88≈6 limbs for representing foreign field elements. Each limb is stored in a witness cell (a native field element). However, since each limb is necessarily smaller than the native field element size, it must be copied to the range-check gate to constrain its value. Since Kimchi only supports 7 copyable witness cells per row, this means that only one foreign field element can be stored per row. This means a single foreign field multiplication would consume at least 4 rows (just for the operands, quotient and remainder). This is not ideal because we want to limit the number of rows for improved performance. Chinese remainder theorem Fortunately, we can do much better than this, however, by leveraging the chinese remainder theorem (CRT for short) as we will now show. The idea is to reduce the number of bits required by constraining our multiplication modulo two coprime moduli: 2t and n. By constraining the multiplication with both moduli the CRT guarantees that the constraints hold with the bigger modulus 2t⋅n. The advantage of this approach is that constraining with the native modulus n is very fast, allowing us to speed up our non-native computations. This practically reduces the costs to constraining with limbs over 2t, where t is something much smaller than 512. For this to work, we must select a value for t that is big enough. That is, we select t such that 2t⋅n>f2−1. As described later on in this document, by doing so we reduce the number of bits required for our use cases to 264. With 88 bit limbs this means that each foreign field element only consumes 3 witness elements, and that means the foreign field multiplication gate now only consumes 2 rows. The section entitled \"Choosing t\" describes this in more detail. Overall approach Bringing it all together, our approach is to verify that a⋅b=q⋅f+r​​ over Z+. In order to do this efficiently we use the CRT, which means that the equation holds mod M=2t⋅n. For the equation to hold over the integers we must also check that each side of the equation is less than 2t⋅n. The overall steps are therefore Apply CRT to equation (1) Check validity with binary modulus mod2t Check validity with native modulus modn Check each side of equation (1) is less than M a⋅b<2t⋅n q⋅f+r<2t⋅n This then implies that a⋅b=cmodf. where c=r. That is, a multiplied with b is equal to c where a,b,c∈Ff​. There is a lot more to each of these steps. That is the subject of the rest of this document. Other strategies Within our overall approach, aside from the CRT, we also use a number of other strategies to reduce the number and degree of constraints. Avoiding borrows and carries Intermediate product elimination Combining multiplications The rest of this document describes each part in detail.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Approach","id":"161","title":"Approach"},"162":{"body":"This section describes important parameters that we require and how they are computed. Native field modulus n Foreign field modulus f Binary modulus 2t CRT modulus 2t⋅n Limb size in bits ℓ Choosing t Under the hood, we constrain a⋅b=q⋅f+rmod2t⋅n. Since this is a multiplication in the foreign field f, the maximum size of q and r are f−1, so this means a⋅b​≤(f−1)⋅f+(f−1)≤f2−1.​ Therefore, we need the modulus 2t⋅n such that 2t⋅n>f2−1, which is the same as saying, given f and n, we must select t such that 2t⋅nt​≥f2≥2log2​(f)−log2​(n).​ Thus, we have an lower bound on t. Instead of dynamically selecting t for every n and f combination, we fix a t that will work for the different selections of n and f relevant to our use cases. To guide us, from above we know that tmin​=2log2​(f)−log2​(n) and we know the field moduli for our immediate use cases. vesta = 2^254 + 45560315531506369815346746415080538113 (255 bits)\npallas = 2^254 + 45560315531419706090280762371685220353 (255 bits)\nsecp256k1 = 2^256 - 2^32 - 2^9 - 2^8 - 2^7 - 2^6 - 2^4 - 1 (256 bits) So we can create a table n f tmin​ vesta secp256k1 258 pallas secp256k1 258 vesta pallas 255 pallas vesta 255 and know that to cover our use cases we need t≥258. Next, given our native modulus n and t, we can compute the maximum foreign field modulus supported . Actually, we compute the maximum supported bit length of the foreign field modulus f=2m. 2t⋅nt+log2​(n)​≥f2≥(2m)2=22m>log2​(22m)=2m​ So we get mmax​=2t+log2​(n)​. With t=258,n=255 we have mmax​​=2258+255​=256.5,​ which is not enough space to handle anything larger than 256 bit moduli. Instead, we will use t=264, giving mmax​=259 bits. The above formula is useful for checking the maximum number of bits supported of the foreign field modulus, but it is not useful for computing the maximum foreign field modulus itself (because 2mmax​ is too coarse). For these checks, we can compute our maximum foreign field modulus more precisely with maxmod​=⌊2t⋅n​⌋. The max prime foreign field modulus satisfying the above inequality for both Pallas and Vesta is 926336713898529563388567880069503262826888842373627227613104999999999999999607. Choosing the limb configuration Choosing the right limb size and the right number of limbs is a careful balance between the number of constraints (i.e. the polynomial degree) and the witness length (i.e. the number of rows). Because one limiting factor that we have in Kimchi is the 12-bit maximum for range check lookups, we could be tempted to introduce 12-bit limbs. However, this would mean having many more limbs, which would consume more witness elements and require significantly more rows. It would also increase the polynomial degree by increasing the number of constraints required for the intermediate products (more on this later). We need to find a balance between the number of limbs and the size of the limbs. The limb configuration is dependent on the value of t and our maximum foreign modulus (as described in the previous section). The larger the maximum foreign modulus, the more witness rows we will require to constrain the computation. In particular, each limb needs to be constrained by the range check gate and, thus, must be in a copyable (i.e. permuteable) witness cell. We have at most 7 copyable cells per row and gates can operate on at most 2 rows, meaning that we have an upperbound of at most 14 limbs per gate (or 7 limbs per row). As stated above, we want the foreign field modulus to fit in as few rows as possible and we need to constrain operands a,b, the quotient q and remainder r. Each of these will require cells for each limb. Thus, the number of cells required for these is cells=4⋅limbs It is highly advantageous for performance to constrain foreign field multiplication with the minimal number of gates. This not only helps limit the number of rows, but also to keep the gate selector polynomial small. Because of all of this, we aim to constrain foreign field multiplication with a single gate (spanning at most 2 rows). As mentioned above, we have a maximum of 14 permuteable cells per gate, so we can compute the maximum number of limbs that fit within a single gate like this. limbsmax​​=⌊cells/4⌋=⌊14/4⌋=3​ Thus, the maximum number of limbs possible in a single gate configuration is 3. Using limbsmax​=3 and t=264 that covers our use cases (see the previous section), we are finally able to derive our limbs size ℓ​=limbsmax​t​=264/3=88​ Therefore, our limb configuration is: Limb size ℓ=88 bits Number of limbs is 3","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Parameter selection","id":"162","title":"Parameter selection"},"163":{"body":"When we constrain a⋅b−q⋅f=rmod2t we want to be as efficient as possible. Observe that the expansion of a⋅b−q⋅f into limbs would also have subtractions between limbs, requiring our constraints to account for borrowing. Dealing with this would create undesired complexity and increase the degree of our constraints. In order to avoid the possibility of subtractions we instead use a⋅b+q⋅f′ where f′​=−fmod2t=2t−f​ The negated modulus f′ becomes part of our gate coefficients and is not constrained because it is publicly auditable. Using the substitution of the negated modulus, we now must constrain a⋅b+q⋅f′=rmod2t. Observe that f′<2t since f<2t and that f′>f when f<2t−1.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Avoiding borrows","id":"163","title":"Avoiding borrows"},"164":{"body":"This section explains how we expand our constraints into limbs and then eliminate a number of extra terms. We must constrain a⋅b+q⋅f′=rmod2t on the limbs, rather than as a whole. As described above, each foreign field element x is split into three 88-bit limbs: x0​,x1​,x2​, where x0​ contains the least significant bits and x2​ contains the most significant bits and so on. Expanding the right-hand side into limbs we have ​(a0​+a1​⋅2ℓ+a2​⋅22ℓ)⋅(b0​+b1​⋅2ℓ+b2​⋅22ℓ)+(q0​+q1​⋅2ℓ+q2​⋅22ℓ)⋅(f0′​+f1′​⋅2ℓ+f2′​⋅22ℓ)= a0​⋅b0​+a0​⋅b1​⋅2ℓ+a0​⋅b2​⋅22ℓ +a1​⋅b0​⋅2ℓ+a1​⋅b1​⋅22ℓ+a1​⋅b2​⋅23ℓ +a2​⋅b0​⋅22ℓ+a2​⋅b1​⋅23ℓ+a2​⋅b2​⋅24ℓ+ q0​⋅f0′​+q0​⋅f1′​⋅2ℓ+q0​⋅f2′​⋅22ℓ +q1​⋅f0′​⋅2ℓ+q1​⋅f1′​⋅22ℓ+q1​⋅f2′​⋅23ℓ +q2​⋅f0′​⋅22ℓ+q2​⋅f1′​⋅23ℓ+q2​⋅f2′​⋅24ℓ=a0​⋅b0​+q0​⋅f0′​+2ℓ⋅(a0​⋅b1​+a1​⋅b0​+q0​⋅f1′​+q1​⋅f0′​)+22ℓ⋅(a0​⋅b2​+a2​⋅b0​+q0​⋅f2′​+q2​⋅f0′​+a1​⋅b1​+q1​⋅f1′​)+23ℓ⋅(a1​⋅b2​+a2​⋅b1​+q1​⋅f2′​+q2​⋅f1′​)+24ℓ⋅(a2​⋅b2​+q2​⋅f2′​)​ Since t=3ℓ, the terms scaled by 23ℓ and 24ℓ are a multiple of the binary modulus and, thus, congruent to zero mod2t. They can be eliminated and we don't need to compute them. So we are left with 3 intermediate products that we call p0​,p1​,p2​: Term Scale Product p0​ 1 a0​b0​+q0​f0′​ p1​ 2ℓ a0​b1​+a1​b0​+q0​f1′​+q1​f0′​ p2​ 22ℓ a0​b2​+a2​b0​+q0​f2′​+q2​f0′​+a1​b1​+q1​f1′​ So far, we have introduced these checked computations to our constraints Computation of p0​,p1​,p2​","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Intermediate products","id":"164","title":"Intermediate products"},"165":{"body":"Let's call p:=ab+qf′mod2t. Remember that our goal is to constrain that p−r=0mod2t (recall that any more significant bits than the 264th are ignored in mod2t). Decomposing that claim into limbs, that means 22ℓ(p2​−r2​)+2ℓ(p1​−r1​)+p0​−r0​=0mod2t.​​ We face two challenges Since p0​,p1​,p2​ are at least 2ℓ bits each, the right side of the equation above does not fit in Fn​ The subtraction of the remainder's limbs r0​ and r1​ could require borrowing For the moment, let's not worry about the possibility of borrows and instead focus on the first problem.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Constraining mod2t","id":"165","title":"Constraining mod2t"},"166":{"body":"The first problem is that our native field is too small to constrain 22ℓ(p2​−r2​)+2ℓ(p1​−r1​)+p0​−r0​=0mod2t. We could split this up by multiplying a⋅b and q⋅f′ separately and create constraints that carefully track borrows and carries between limbs. However, a more efficient approach is combined the whole computation together and accumulate all the carries and borrows in order to reduce their number. The trick is to assume a space large enough to hold the computation, view the outcome in binary and then split it up into chunks that fit in the native modulus. To this end, it helps to know how many bits these intermediate products require. On the left side of the equation, p0​ is at most 2ℓ+1 bits. We can compute this by substituting the maximum possible binary values (all bits set to 1) into p0​=a0​b0​+q0​f0′​ like this maxbits(p0​)​=log2​(a0​(2ℓ−1)​​b0​(2ℓ−1)​​+q0​(2ℓ−1)​​f0′​(2ℓ−1)​​)=log2​(2(22ℓ−2ℓ+1+1))=log2​(22ℓ+1−2ℓ+2+2).​ So p0​ fits in 2ℓ+1 bits. Similarly, p1​ needs at most 2ℓ+2 bits and p2​ is at most 2ℓ+3 bits. Term p0​ p1​ p2​ Bits 2ℓ+1 2ℓ+2 2ℓ+3 The diagram below shows the right hand side of the zero-sum equality from equation (2). That is, the value p−r. Let's look at how the different bits of p0​,p1​,p2​,r0​,r1​ and r2​ impact it. 0 L 2L 3L 4L\n|-------------|-------------|-------------|-------------|-------------| :\n|--------------p0-----------:-| 2L + 1 : |-------------:-p1------------| 2L + 2 p10➚ : p11➚ |----------------p2-------------| 2L + 3 :\n|-----r0------| : : |-----r1------| : |-----r2------|\n\\__________________________/ \\______________________________/ ≈ h0 ≈ h1 Within our native field modulus we can fit up to 2ℓ+δfk quickly overflows our CRT modulus 2t⋅n. For example, assuming our maximum foreign modulus of f=2259 and either of Kimchi's native moduli (i.e. Pallas or Vesta), fk>2t⋅n for k>2. That is, an overflow is possible for a chain of greater than 1 foreign field multiplication. Thus, we must check a,b,q and r are less than f for each multiplication. Fortunately, in many situations the input operands may already be checked either as inputs or as outputs of previous operations, so they may not be required for each multiplication operation. Thus, the q′ and r′ checks (or equivalently q and r) are our main focus because they must be done for every multiplication. Compute bound q′=q+f′ with addition gate (2 rows) Compute bound r′=r+f′ with addition gate (2 rows) Range check q′<2t (4 rows) Range check r′<2t (4 rows) This costs 12 rows per multiplication. In a subsequent section, we will reduce it to 8 rows.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Cost of bound checks","id":"174","title":"Cost of bound checks"},"175":{"body":"Due to the limited number of permutable cells per gate, we do not have enough cells for copy constraining q′ and r′ (or q and r) to their respective range check gadgets. To address this issue, we must decompose q′ into 2 limbs instead of 3, like so q′=q01′​+22ℓ⋅q2′​ and q01′​=q0′​+2ℓ⋅q1′​ Thus, q′ is decomposed into two limbs q01′​ (at most 2ℓ bits) and q2′​ (at most ℓ bits). Note that q′ must be range checked by a multi-range-check gadget. To do this the multi-range-check gadget must Store a copy of the limbs q0′​,q1′​ and q2′​ in its witness Range check that they are ℓ bit each Constrain that q01′​=q0′​+2ℓ⋅q1′​ (this is done with a special mode of the multi-range-check gadget) Have copy constraints for q01′​ and q2′​ as outputs of the ForeignFieldMul gate and inputs to the multi-range-check gadget Note that since the foreign field multiplication gate computes q′ from q which is already in the witness and q01′​ and q2′​ have copy constraints to a multi-range-check gadget that fully constrains their decomposition from q′, then the ForeignFieldMul gate does not need to store an additional copy of q0′​ and q1′​.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 2-limb decomposition","id":"175","title":"2-limb decomposition"},"176":{"body":"Since the q2ℓ where 2ℓ−1 is the maximum possible size of x1​ (before it overflows) and ϕ0​ is the overflow bit from the addition of the least significant limbs x0​ and f0′​. This means >2ℓ−ϕ0​−f1′​ We cannot allow x to overflow the foreign field, so we also have >x1​<(f−x0​)/22ℓ> Thus, >2ℓ−ϕ0​−f1′​<(f−x0​)/22ℓ=f/22ℓ−x0​/22ℓ> Since x0​/22ℓ=ϕ0​ we have >2ℓ−ϕ0​−f1′​ so >2ℓ−f1′​ Notice that f/22ℓ=f1​. Now we have >2ℓ−f1′​⟺>f1′​>2ℓ−f1​> However, this is a contradiction with the definition of our negated foreign field modulus limb f1′​=2ℓ−f1​. ■ We have proven that qcarry2′​ is always zero, so that allows use to simplify our constraints. We now have q0​∈[0,2ℓ) multi-range-check q1​∈[0,2ℓ) multi-range-check q01′​=q0′​+2ℓ⋅q1′​ multi-range-check qcarry01′​∈[0,2) ForeignFieldMul 22ℓ⋅qcarry01′​=s01​−q01′​ ForeignFieldMul q2​∈[0,2ℓ) multi-range-check q2′​=s2​+w01​ ForeignFieldMul In other words, we have eliminated constraint (7) and removed qcarry2′​ from the witness. Since we already needed to range-check q or q′, the total number of new constraints added is 4: 3 added to ForeignFieldMul and 1 added to multi-range-check gadget for constraining the decomposition of q01′​. This saves 2 rows per multiplication.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » An optimization","id":"176","title":"An optimization"},"177":{"body":"Due to the limited number of cells accessible to gates, we are not able to chain multiplications into multiplications. We can chain foreign field additions into foreign field multiplications, but currently do not support chaining multiplications into additions (though there is a way to do it).","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Chaining multiplications","id":"177","title":"Chaining multiplications"},"178":{"body":"Now we collect all of the checks that are required to constrain foreign field multiplication","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Constraining the computation","id":"178","title":"Constraining the computation"},"179":{"body":"If the a operand has not been constrained to [0,f) by any previous foreign field operations, then we constrain it like this Compute bound a′=a+f′ with addition gate (2 rows) ForeignFieldAdd Range check a′∈[0,2t) (4 rows) multi-range-check","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 1. Range constrain a","id":"179","title":"1. Range constrain a"},"18":{"body":"There are three common representations for polynomials used in computer implementations. Dense coefficient form. A degree d polynomial is represented as a vector of length d+1 of all the coefficients. Entry i of the vector corresponds to the coefficient ai​. This corresponds to the DensePolynomial type in arkworks. Sometimes this is also described as writing the polynomial \"in the monomial basis\", because it amounts to writing the polynomial as a linear combination of the monomials xi. Sparse coefficient form. If a polynomial does not have very many non-zero coefficients, the above representation is wasteful. In sparse coefficient form, you represent a polynomial as a vector (or potentially a hash-map) of pairs (usize, F) where F is the type of coefficients. The polynomial corresponding to the list [(i_0, b_0), ..., (i_n, b_n)] is b0​xi0​+⋯+bn​xin​ Evaluation form. We fix an index set A⊆F, with A={a0​,…,ad​}, and represent a polynomial f∈F[x]≤d​ as the vector [f(a_0), ..., f(a_d)]. By the fundamental theorem of polynomials, this is a valid way of representing the polynomials, since the coefficients form can be obtained by using the interpA​ function. The evaluation form is very important. The reason is that multiplying two polynomials in the evaluation form takes time O(n). You just multiply the two vectors entry-wise. By contrast, the coefficient forms naively require time O(n2) to multiply. Now, there is a trick. For certain sets A⊆F, we can efficiently translate between the dense coefficients form and the evaluation form. That is, for certain A, the functions interpA​ and evalA​ can be computed more efficiently than O(n2).","breadcrumbs":"Polynomials » Computer representation","id":"18","title":"Computer representation"},"180":{"body":"If the b operand has not been constrained to [0,f) by any previous foreign field operations, then we constrain it like this Compute bound b′=b+f′ with addition gate (2 rows) ForeignFieldAdd Range check b′∈[0,2t) (4 rows) multi-range-check","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 2. Range constrain b","id":"180","title":"2. Range constrain b"},"181":{"body":"The quotient q is constrained to [0,f) for each multiplication as part of the multiplication gate Compute bound q′=q+f′ with ForeignFieldMul constraints Range check q′∈[0,2t) (4 rows) multi-range-check Range check q≥0 ForeignFieldMul (implicit by storing q in witness)","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 3. Range constrain q","id":"181","title":"3. Range constrain q"},"182":{"body":"The remainder r is constrained to [0,f) for each multiplication using an external addition gate. Compute bound r′=r+f′ with addition gate (2 rows) ForeignFieldAdd Range check r′∈[0,2t) (4 rows) multi-range-check","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 4. Range constrain r","id":"182","title":"4. Range constrain r"},"183":{"body":"Compute and constrain the intermediate products p0​,p1​ and p2​ as: p0​=a0​⋅b0​+q0​⋅f0′​ ForeignFieldMul p1​=a0​⋅b1​+a1​⋅b0​+q0​⋅f1′​+q1​⋅f0′​ ForeignFieldMul p2​=a0​⋅b2​+a2​⋅b0​+a1​⋅b1​+q0​⋅f2′​+q2​⋅f0′​+q1​⋅f1′​ ForeignFieldMul where each of them is about 2ℓ-length elements.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 5. Compute intermediate products","id":"183","title":"5. Compute intermediate products"},"184":{"body":"Compute and constrain the native modulus values, which are used to check the constraints modulo n in order to apply the CRT an​=22ℓ⋅a2​+2ℓ⋅a1​+a0​modn bn​=22ℓ⋅b2​+2ℓ⋅b1​+b0​modn qn​=22ℓ⋅q2​+2ℓ⋅q1​+q0​modn rn​=22ℓ⋅r2​+2ℓ⋅r1​+r0​modn fn​=22ℓ⋅f2​+2ℓ⋅f1​+f0​modn","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 6. Native modulus checked computations","id":"184","title":"6. Native modulus checked computations"},"185":{"body":"Check that p1​=2ℓ⋅p11​+p10​: p1​=2ℓ⋅p11​+p10​ ForeignFieldMul Range check p10​∈[0,2ℓ) multi-range-check Range check p11​∈[0,2ℓ+2) p11​=p111​⋅2ℓ+p110​ ForeignFieldMul Range check p110​∈[0,2ℓ) multi-range-check Range check p111​∈[0,22) with a degree-4 constraint ForeignFieldMul","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 7. Decompose middle intermediate product","id":"185","title":"7. Decompose middle intermediate product"},"186":{"body":"Now we have to constrain the zero sum (p0​−r0​)+288(p1​−r1​)+2176(p2​−r2​)=0mod2t We constrain the first and the second halves as v0​⋅22ℓ=p0​+2ℓ⋅p10​−r0​−2ℓ⋅r1​ ForeignFieldMul v1​⋅2ℓ=(p111​⋅2ℓ+p110​)+p2​−r2​+v0​ ForeignFieldMul And some more range checks Check that v0​∈[0,22) with a degree-4 constraint ForeignFieldMul Check that v1​∈[0,2ℓ+3) Check v1​=v11​⋅288+v10​ ForeignFieldMul Check v11​∈[0,23] ForeignFieldMul Check v10​<2ℓ with range constraint multi-range-check To check that v11​∈[0,23) (i.e. that v11​ is at most 3 bits long) we first range-check v11​∈[0,212) with a 12-bit plookup. This means there can be no higher bits set beyond the 12-bits of v11​. Next, we scale v11​ by 29 in order to move the highest 12−3=9 bits beyond the 12th bit. Finally, we perform a 12-bit plookup on the resulting value. That is, we have Check v11​∈[0,212) with a 12-bit plookup (to prevent any overflow) Check scaledv11​​=29⋅v11​ Check scaledv11​​ is a 12-bit value with a 12-bit plookup Kimchi's plookup implementation is extremely flexible and supports optional scaling of the lookup target value as part of the lookup operation. Thus, we do not require two witness elements, two lookup columns, nor the scaledv11​​=29⋅v11​ custom constraint. Instead we can just store v11​ in the witness and define this column as a \"joint lookup\" comprised of one 12-bit plookup on the original cell value and another 12-bit plookup on the cell value scaled by 29, thus, yielding a 3-bit check. This eliminates one plookup column and reduces the total number of constraints.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 8. Zero sum for multiplication","id":"186","title":"8. Zero sum for multiplication"},"187":{"body":"Using the checked native modulus computations we constrain that an​⋅bn​−qn​⋅fn​−rn​=0modn.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 9. Native modulus constraint","id":"187","title":"9. Native modulus constraint"},"188":{"body":"Compute and constrain the intermediate sums s01​ and s2​ as: s01​=q01​+f01′​ s2​=q2​+f2′​ q01​=q0​+2ℓ⋅q1​ f01′​=f0′​+2ℓ⋅f1′​","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 10. Compute intermediate sums","id":"188","title":"10. Compute intermediate sums"},"189":{"body":"Check that q01′​=q0′​+2ℓ⋅q1′​. Done by (3) above with the multi-range-check on q′ q01′​=q0′​+2ℓ⋅q1′​ Range check q0′​∈[0,2ℓ) Range check q1′​∈[0,2ℓ)","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 11. Decompose the lower quotient bound","id":"189","title":"11. Decompose the lower quotient bound"},"19":{"body":"The algorithm that allows us to multiply polynomials in O(nlogn) is called the Cooley-Tukey fast Fourier transform , or FFT for short. It was discovered by Gauss 160 years earlier, but then separately rediscovered and publicized by Cooley-Tukey. The heart of Cooley-Tukey FFT is actually about converting between coefficient and evaluation representations, rather than the multiplication itself. Given polynomials p and q in dense coefficient representation, it works like this. Convert p and q from coefficient to evaluation form in O(nlogn) using Cooley-Tukey FFT Compute r=p∗q in evaluation form by multiplying their points pairwise in O(n) Convert r back to coefficient form in O(nlogn) using the inverse Cooley-Tukey FFT The key observation is that we can choose any n distinct evaluation points to represent any degree n−1 polynomial. The Cooley-Tukey FFT works by selecting points that yield an efficient FFT algorithm. These points are fixed and work for any polynomials of a given degree. The next section describes the Cooley-Tukey FFT in detail.","breadcrumbs":"Polynomials » Multiplying Polynomials » Multiplying polynomials","id":"19","title":"Multiplying polynomials"},"190":{"body":"We constrain that s01​−q01′​+22ℓ⋅(s2​−q2′​)=0mod2t by constraining the two halves 22ℓ⋅qcarry01′​=s01​−q01′​ 2ℓ⋅qcarry2′​=s2​+qcarry01′​−q2′​ We also need a couple of small range checks Check that qcarry01′​ is boolean ForeignFieldMul Check that q2′​ is boolean ForeignFieldMul","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 12. Zero sum for quotient bound addition","id":"190","title":"12. Zero sum for quotient bound addition"},"191":{"body":"Based on the constraints above, we need the following 12 values copied from the range check gates. a0, a1, a2, b0, b1, b2, q0, q1, q2, r0, r1, r2 Since we need 12 copied values for the constraints, they must span 2 rows. The q>s01​>>>​=q01​+f01′​=q0​+2ℓ⋅q1​+f0′​+2ℓ⋅f1′​=q0​+f0′​+2ℓ⋅(q1′​+f1′​)=q0′​+2ℓ⋅q1′​>​> where q0′​=q0​+f0′​ and q1′​=q1​+f1′​ can be done with checked computations. Next, for applying the CRT we compute an​=22ℓ⋅a2​+2ℓ⋅a1​+a0​modn bn​=22ℓ⋅b2​+2ℓ⋅b1​+b0​modn qn​=22ℓ⋅q2​+2ℓ⋅q1​+q0​modn rn​=22ℓ⋅r2​+2ℓ⋅r1​+r0​modn fn​=22ℓ⋅f2​+2ℓ⋅f1​+f0​modn","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Checked computations","id":"192","title":"Checked computations"},"193":{"body":"In total we require the following checks p111​∈[0,22) v10​∈[0,2ℓ) multi-range-check p110​∈[0,2ℓ) multi-range-check p10​∈[0,2ℓ) multi-range-check p11​=2ℓ⋅p111​+p110​ p1​=2ℓ⋅p11​+p10​ v0​∈[0,22) 22ℓ⋅v0​=p0​+2ℓ⋅p10​−r0​−2ℓ⋅r1​ v11​∈[0,23) v1​=2ℓ⋅v11​+v10​ 2ℓ⋅v1​=v0​+p11​+p2​−r2​ an​⋅bn​−qn​⋅fn​=rn​ q0′​∈[0,2ℓ) multi-range-check q1′​∈[0,2ℓ) multi-range-check q2′​∈[0,2ℓ) multi-range-check q01′​=q0′​+2ℓ⋅q1′​ multi-range-check qcarry01′​∈[0,2) 22ℓ⋅qcarry01′​=s01​−q01′​ q2′​=s2​+qcarry01′​","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Checks","id":"193","title":"Checks"},"194":{"body":"These checks can be condensed into the minimal number of constraints as follows. First, we have the range-check corresponding to (1) as a degree-4 constraint C1: p111​⋅(p111​−1)⋅(p111​−2)⋅(p111​−3) Checks (2) - (4) are all handled by a single multi-range-check gadget. C2: multi-range-check v10​,p10​,p110​ Next (5) and (6) can be combined into C3: 2ℓ⋅(2ℓ⋅p111​+p110​)+p10​=p1​ Now we have the range-check corresponding to (7) as another degree-4 constraint C4: v0​⋅(v0​−1)⋅(v0​−2)⋅(v0​−3) Next we have check (8) C5: 22ℓ⋅v0​=p0​+2ℓ⋅p10​−r0​−2ℓ⋅r1​ Up next, check (9) is a 3-bit range check C6: Plookup v11​ (12-bit plookup scaled by 29) Now checks (10) and (11) can be combined into C7: 2ℓ⋅(v11​⋅2ℓ+v10​)=p2​+p11​+v0​−r2​ Next, for our use of the CRT, we must constrain that a⋅b=q⋅f+rmodn. Thus, check (12) is C8: an​⋅bn​−qn​⋅fn​=rn​ Next we must constrain the quotient bound addition. Checks (13) - (16) are all combined into multi-range-check gadget C9: multi-range-check q0′​,q1′​,q2′​ and q01′​=q0′​+2ℓ⋅q1′​. Check (17) is a carry bit boolean check C10: qcarry01′​⋅(qcarry01′​−1) Next, check (18) is C11: 22ℓ⋅qcarry10′​=s01​−q01′​ Finally, check (19) is C12: q2′​=s2​+qcarry01′​ The Zero gate has no constraints and is just used to hold values required by the ForeignFieldMul gate.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Constraints","id":"194","title":"Constraints"},"195":{"body":"The following checks must be done with other gates to assure the soundness of the foreign field multiplication Range check input multi-range-check a multi-range-check b Range check witness data multi-range-check q′ and check q01′​=q0′​+2ℓq1′​ multi-range-check ℓ-bit limbs: p10​,p110​,p111​ Range check output multi-range-check either r or r′ Compute and constrain r′ the bound on r ForeignFieldAdd r+2t=1⋅f+r′ Copy constraints must connect the above witness cells to their respective input cells within the corresponding external check gates witnesses.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » External checks","id":"195","title":"External checks"},"196":{"body":"The Keccak gadget is comprised of 3 circuit gates (Xor16, Rot64, and Zero) Keccak works with 64-bit words. The state is represented using 5×5 matrix of 64 bit words. Each compression step of Keccak consists of 24 rounds. Let us denote the state matrix with A (indexing elements as A[x,y]), from which we derive further states as follows in each round. Each round then consists of the following 5 steps: C[x]D[x]E[x,y]B[y,2x+3y]F[x,y]Fp[0,0]​=A[x,0]⊕A[x,1]⊕A[x,2]⊕A[x,3]⊕A[x,4]=C[x−1]⊕ROT(C[x+1],1)=A[x,y]⊕D[x]=ROT(E[x,y],ρ[x,y])=B[x,y]⊕((NOT B[x+1,y])AND B[x+2,y])=F[0,0]⊕RC​​ for 0≤x,y≤4 and ρ[x,y] is the rotation offset defined for Keccak. The values are in the table below extracted from the Keccak reference https://keccak.team/files/Keccak-reference-3.0.pdf x = 3 x = 4 x = 0 x = 1 x = 2 y = 2 155 231 3 10 171 y = 1 55 276 36 300 6 y = 0 28 91 0 1 190 y = 4 120 78 210 66 253 y = 3 21 136 105 45 15","breadcrumbs":"Custom Gates » Keccak » Keccak Gate","id":"196","title":"Keccak Gate"},"197":{"body":"The atomic operations are XOR, ROT, NOT, AND. In the sections below, we will describe the gates for these operations. Below are some common approaches followed in their design. To fit within 15 wires, we first decompose each word into its lower and upper 32-bit components. A gate for an atomic operation works with those 32-bit components at a time. Before we describe the specific gate design approaches, below are some constraints in the Kimchi framework that dictated those approaches. only 4 lookups per row only first 7 columns are available to the permutation polynomial","breadcrumbs":"Custom Gates » Keccak » Design Approach:","id":"197","title":"Design Approach:"},"198":{"body":"It is clear from the definition of the rotation gate that its constraints are complete (meaning that honest instances always satisfy the constraints). It is left to be proven that the proposal is sound. In this section, we will give a proof that as soon as we perform the range checks on the excess and shifted parts of the input, only one possible assignment satisfies the constraints. This means that there is no dishonest instance that can make the constraints pass. We will also give an example where one could find wrong rotation witnesses that would satisfy the constraints if we did not check the range.","breadcrumbs":"Custom Gates » Keccak » Rot64","id":"198","title":"Rot64"},"199":{"body":"First of all, we will illustrate the necessity of range-checks with a simple example. For the sake of readability, we will use some toy field lengths. In particular, let us assume that our words have 4 bits, meaning all of the elements between 0x0 and 0xF. Next, we will be using the native field F32​. As we will later see, this choice of field lengths is not enough to perform any 4-bit rotation, since the operations in the constraints would overflow the native field. Nonetheless, it will be sufficient for our example where we will only rotate by 1 bit. Assume we want to rotate the word 0b1101 (meaning 13) by 1 bit to the left. This gives us the rotated word 0b1011 (meaning 11). The excess part of the word is 0b1, whereas the shifted part corresponds to 0b1010. We recall the constraints for the rotation gate: word⋅2rotrotated​=excess⋅2len+shifted=excess+shifted​ Applied to our example, this results in the following equations: 13⋅211​=excess⋅16+shifted=excess+shifted​ We can easily check that the proposed values of the shifted 0b1010=10 and the excess 0b1=1 values satisfy the above constraint because 26=1⋅16+10 and 11=1+10. Now, the question is: can we find another value for excess and shifted, such that their addition results in an incorrect rotated word? The answer to this question is yes, due to diophantine equations . We basically want to find x,y such that 26=x⋅16+y( mod 32). The solution to this equation is: ∀k∈[0…31]:xy​=k ∧=26−16⋅k​ We chose these word and field lengths to better understand the behaviour of the solution. Here, we can see two \"classes\" of evaluations. If we choose an even k, then y will have the following shape: 26−16⋅(2⋅n)⟺26−32n≡32​26 Meaning, if x=2n then y=26. If on the other hand, we chose an odd k, then y will have the following shape instead: 26−16⋅(2⋅n+1)⟺26−32n−16≡32​26−16=10 Meaning, if x=2n+1 then y=10. Thus, possible solutions to the diophantine equation are: xy​=0,1,2,3,4,5…=26,10,26,10,26,10…​ Note that our valid witness is part of that set of solutions, meaning x=1 and y=10. Of course, we can also find another dishonest instantiation such as x=0 and y=26. Perhaps one could think that we do not need to worry about this case, because the resulting rotation word would be 0+26=26, and if we later use that result as an input to a subsequent gate such as XOR, the value 26 would not fit and at some point the constraint system would complain. Nonetheless, we still have other solutions to worry about, such as (x=3,y=10) or (x=5,y=10), since they would result in a rotated word that would fit in the word length of 4 bits, yet would be incorrect (not equal to 11). All of the above incorrect solutions differ in one thing: they have different bit lengths. This means that we need to range check the values for the excess and shifted witnesses to make sure they have the correct length.","breadcrumbs":"Custom Gates » Keccak » Necessity of range checks","id":"199","title":"Necessity of range checks"},"2":{"body":"Groups are an algebraic structure which are used in most widely deployed signature schemes are used in most widely deployed zk-SNARKs generalize the notion of a collection of symmetries of an object generalize certain aspects of numbers I will not touch on the last two too much. First let's do a quick definition and then some examples. Definition. A group is a set G with a binary operation ∗:G×G→G, an identity e∈G for that operation, and an inverse map ()−1:G→G such that ∗ is associative: x∗(y∗z)=(x∗y)∗z for all x,y,z. e is an identity for ∗. I.e., e∗x=x∗e=x for all x. ()−1 is an inverse map for ∗ with identity e. I.e., x∗x−1=x−1∗x=e for all x. So basically, an invertible binary operation. Definition in hand, we can see some examples: The integers Z with +, 0 as identity, and negation as inverse. For any natural number n, the integers mod n. This can be thought of as the set of numbers {0,…,n−1} with the operation being + followed by taking remainder mod n. It can also be thought of as the group of rotations of an n-gon. For any set X, the set of invertible functions X→X (a.k.a permutations) with function composition as the operation, the identity function as the identity, and function inverse as the inverse operation. The set of translations of a plane R×R with composition of translations (since the composition of translations is a translation). The identity is translating by (0,0) and inverse is reversing the translation. This is equivalent to the group R×R with coordinate-wise addition as the operation and coordinate-wise negation as the inverse. The set of rotations of a sphere with composition as the operation, doing nothing as the identity, and performing the rotation in reverse as the inverse map. For any field F, the set F×:=F∖{0}, with field multiplication as the operation, 1 as the identity, and x↦1/x as the inverse map. This is called the group of units of F. Sometimes, instead of x∗y we simply write xy when ∗ is clear from context.","breadcrumbs":"Groups » Groups","id":"2","title":"Groups"},"20":{"body":"This section describes how the Cooley-Tukey fast Fourier transform works. As we learned in the previous section, the key is to select evaluation points that yield an efficient FFT algorithm. Specifically, say we have ω∈F such that ωn=1, and ωr=1 for any 00, then we will split f into two polynomials, recursively call FFT on them, and reconstruct the result from the recursive calls. To that end, define f0​ to be the polynomial whose coefficients are all the even-index coefficients of f and f1​ the polynomial whose coefficients are all the odd-index coefficients of f. In terms of the array representation, this just means splitting out every other entry into two arrays. So that can be done in time O(n). Write f=∑i<2k​ci​xi, so that f0​=∑i<2k−1​c2i​xi and f1​=∑i<2k−1​c2i+1​xi. Then f(x)​=i<2k∑​ci​xi=i<2k−1∑​c2i​x2i+i<2k−1∑​c2i+1​x2i+1=i<2k−1∑​c2i​(x2)i+i<2k−1∑​c2i+1​x⋅(x2)i=i<2k−1∑​c2i​(x2)i+xi<2k−1∑​c2i+1​(x2)i=f0​(x2)+xf1​(x2)​ Now, notice that if ω is a 2kth root of unity, then ω2 is a 2k−1th root of unity. Thus we can recurse with FFT(k−1,ω2,f0​) and similarly for f1​. Let [e0,0​,…,e0,2k−1−1​][e1,0​,…,e1,2k−1−1​]​=FFT(k−1,ω2,f0​)=FFT(k−1,ω2,f1​)​ By assumption ei,j​=fi​((ω2)j). So, for any j we have f(ωj)​=f0​((ω2)j)+ωjf1​((ω2)j)​ Now, since j may be larger than 2k−1−1, we need to reduce it mod 2k−1, relying on the fact that if τ is an nth root of unity then τj=τjmodn since τn=1. Thus, (ω2)j=(ω2)jmod2k−1 and so we have f(ωj)​=f0​((ω2)jmod2k−1)+ωjf1​((ω2)jmod2k−1)=e0,jmod2k−1​+ωje1,jmod2k−1​​ We can compute the array W=[1,ω,…,ω2k−1] in time O(n) (since each entry is the previous entry times ω). Then we can compute each entry of the output in O(1) as f(ωj)​=e0,jmod2k−1​+W[j]⋅e1,jmod2k−1​​ There are n such entries, so this takes time O(n). This concludes the recursive definition of the algorithm FFT(k,ω,f). Algorithm: computing evalAk​​ Input f=[c0​,…,c2k−1​] the coefficients of polynomial f(x)=∑i<2k​ci​xi Compute W←[1,ω,ω2,...,ω2k−1] FFT(k,ω,f)→[f(1),f(ω),f(ω2)…,f(ω2k−1)] if k==0 return f else Compute f0​=[c0​,c2​,...,c2k−2​] the even coefficients of f, corresponding to f0​(x)=∑i<2k−1​c2i​xi Compute f1​=[c1​,c3​,...,c2k−1​] the odd coefficients of f, corresponding to f1​(x)=∑i<2k−1​c2i+1​xi e0​←FFT(k−1,ω2,f0​) e1​←FFT(k−1,ω2,f1​) for j∈[0,2k−1] Fj​←e0,jmod2k−1​+W[j]⋅e1,jmod2k−1​ return F Now let's analyze the time complexity. Let T(n) be the complexity on an instance of size n (that is, for n=2k). Looking back at what we have done, we have done O(n) for computing f0​ and f1​ two recursive calls, each of size n/2 O(n) for computing the powers of ω O(n) for combining the results of the recursive calls In total, this is O(n)+2T(n/2). Solving this recurrence yields T(n)=O(n)⋅logn=O(nlogn). Basically, there are logn recursions before we hit the base case, and each step takes time O(n). □ Now, in practice there are ways to describe this algorithm non-recursively that have better concrete performance, but that's out of scope for this document. Read the code if you are interested. Using the FFT algorithm to compute interpAk​​ So far we have a fast way to compute evalAk​​(f) all at once, where Ak​ is the set of powers of a 2kth root of unity ω. For convenience let n=2k. Now we want to go the other way and compute a polynomial given an array of evaluations. Specifically, n evaluations [f(x0​),f(x1​),...,f(xn−1​)] uniquely define a degree n−1 polynomial. This can be written as a system of n equations f(x0​)f(x1​)⋮f(xn−1​)​=c0​+c1​x0​+…+cn−1​x0n−1​=c0​+c1​x1​+…+cn−1​x1n−1​=c0​+c1​xn−1​+…+cn−1​xn−1n−1​,​ which can be rewritten as a matrix vector product. ​f(x0​)f(x1​)⋮f(xn−1​)​​=​11⋮1​x0​x1​⋮xn−1​​⋯⋯⋱⋯​x0n−1​x1n−1​⋮xn−1n−1​​​×​c0​c1​⋮cn−1​​​ This n×n matrix is a Vandermonde matrix and it just so happens that square Vandermonde matrices are invertible, iff the xi​ are unique. Since we purposely selected our xi​ to be the powers of ω, a primitive n-th root of unity, by definition xi​=ωi are unique. Therefore, to compute the polynomial given the corresponding array of evaluations (i.e. interpolation) we can solve for the polynomial's coefficients using the inverse of the matrix. ​c0​c1​⋮cn−1​​​=​11⋮1​1ω⋮ωn−1​⋯⋯⋱⋯​1n−1ωn−1⋮ω(n−1)(n−1)​​−1×​f(1)f(ω)⋮f(ωn−1)​​ All we need now is the inverse of this matrix, which is slightly complicated to compute. I'm going to skip it for now, but if you have the details please make a pull request. Substituting in the inverse matrix we obtain the equation for interpolation. ​c0​c1​⋮cn−1​​​=n1​​11⋮1​1ω−1⋮ω−(n−1)​⋯⋯⋱⋯​1n−1ω−(n−1)⋮ω−(n−1)(n−1)​​×​f(1)f(ω)⋮f(ωn−1)​​ Observe that this equation is nearly identical to the original equation for evaluation, except with the following substitution. ωi⇒n1​ω−1i Consequently and perhaps surprisingly, we can reuse the FFT algorithm evalAk​​ in order to compute the inverse-- interpAk​​. So, suppose we have an array [a0​,…,an−1​] of field elements (which you can think of as a function Ak​→F) and we want to compute the coefficients of a polynomial f with f(ωi)=ai​. To this end, define a polynomial g by g=∑j264, then b′ cannot be smaller than 264 as it was assumed. CONTRADICTION. b=b′ and a=a′: then 264(a−a′)≡n​0 and this can only happen if a′=a+kn. But since n>2r, then a′ cannot be smaller than 2r as it was assumed. CONTRADICTION. a=a′ and b=b′: then we have something like 264α+β≡n​0. This means β≡n​−264α=k⋅n−264α for any k. According to the assumption, both 0≤a<2r and 0≤a′<2r. This means, the difference α:=(a−a′) lies anywhere in between the following interval: 1−2r≤α≤2r−1 We plug in this interval to the above equation to obtain the following interval for β: k⋅n−264(1−2r)≤β≤k⋅n−264(2r−1) We look at this interval from both sides of the inequality: β≥kn−264+264+r and β≤kn+264−264+r and we wonder if kn−264+264+r≤kn+264−264+r is at all possible. We rewrite as follows: 264+r−2642⋅264+r264+r​≤264−264+r≤2⋅264≤264​ But this can only happen if r≤0, which is impossible since we assume 0p. We have a proof system (Kimchi) over Fp​ and Fq​, where commitments to public inputs are: Pp​=⟨e,H⟩∈Ep​(Fq​)Pq​=⟨e,G⟩∈Eq​(Fp​) Respectively. See Pasta Curves for more details. When referring to the Fq​-side we mean the proof system for circuit over the field Fq​.","breadcrumbs":"Deferred Computation » Deferred Computation","id":"217","title":"Deferred Computation"},"218":{"body":"In pickles-rs we have the notion of \"passing\" a variable (including the transcript) from one side to the other. e.g. when a field element α∈Fp​ needs to be used as a scalar on Fq​. This document explains what goes on \"under the hood\". Let us start by understanding why: Let v∈Fq​ be a scalar which we want to use to do both: Field arithmetic in Fq​ Scalar operations on Eq​(Fp​) In order to do so efficiently, we need to split these operations across two circuits (and therefore proofs) because: Emulating Fq​ arithmetic in Fp​ is very expensive, e.g. computing v⋅w requires O(log(q)2) multiplications over Fp​: 100's of gates for a single multiplication. Since Eq​(Fp​)⊆Fp​×Fp​ we cannot compute [v]⋅H∈Eq​(Fp​) over Fq​ efficiently, because, like before, emulating Fp​ arithmetic in Fq​ is very expensive...","breadcrumbs":"Deferred Computation » Public Inputs / Why Passing","id":"218","title":"Public Inputs / Why Passing"},"219":{"body":"The solution is to \"pass\" a value v between the two proofs, in other words to have two values v ∈Fp​ and v∈Fq​ which are equal as integers i.e. lift(v)=lift(v )∈Z: they represent \"the same number\". A naive first attempt would be to simply add v~∈Fp​ to the witness on the Fp​-side, however this has two problems: Insufficient Field Size: pq; always the case for any cycle of curves, p is only ≈q​ smaller than q, by Hasse. Now \"v \" is \"represented\" by the two values h ,l~∈Fp​. Note that no decomposition is nessary if the \"original value\" v was in Fp​, since Fq​ is big enough to hold the lift of any element in Fp​. Problem 2: Compute Commitment to the Public Input of other side To solve the binding issue we will add l,h to the public inputs on the Fp​-side, for simplicity we will describe the case where l,h are the only public inputs in the Fp​-side, which means that the commitment Pp​∈Ep​(Fq​) to the public inputs on the Fp​ side is: Pp​=[h]⋅Gh​+[l]⋅Gl​∈Ep​(Fq​) At this point it is important to note that Ep​ is defined over Fq​! Which means that we can compute Pp​∈Ep​(Fq​) efficiently on the Fq​-side! Therefore to enforce the binding, we: Add a sub-circuit which checks: Pp​=[h]⋅Gh​+[l]⋅Gl​∈Ep​(Fq​),∣h∣=⌊logp⌋∣l∣=1 Add Pp​=(x,y)∈Fq​×Fq​ to the public input on the Fq​-side.","breadcrumbs":"Deferred Computation » Solution","id":"219","title":"Solution"},"22":{"body":"Implement types DensePolynomial and Evaluations that wrap a Vec and implement the FFT algorithms described above for converting between them Familiarize yourself with the types and functions provided by ark_poly","breadcrumbs":"Polynomials » Fast Fourier Transform » Exercises","id":"22","title":"Exercises"},"220":{"body":"At this point the statement of the proof in Fq​-side is: the Fq​-proof is sound, condition on providing an opening of Pp​ that satisifies the Fp​-relation. At this point you can stop and verify the proof (in the case of a \"step proof\" you would), by recomputing Pp​ outside the circuit while checking the Fp​-relation manually \"in the clear\". However, when recursing (e.g. wrapping in a \"wrap proof\") we need to \"ingest\" this public input Pp​; after all, to avoid blowup in the proof size everything (proofs/accumulators/public inputs etc.) must eventually become part of the witness and every computation covered by a circuit... To this end, the wrap proof is a proof for the Fp​-relation with the public input Pp​ which additionally verifies the Fq​-proof. The next \"step\" proof then verifies this wrap proof which means that Pp​ then becomes part of the witness!","breadcrumbs":"Deferred Computation » We recurse onwards...","id":"220","title":"We recurse onwards..."},"221":{"body":"We can arbitrarily choose which side should compute the public input of the other, in pickles we let \"wrap\" compute the commitment to the public input.","breadcrumbs":"Deferred Computation » In Pickles","id":"221","title":"In Pickles"},"222":{"body":"Enforces that the public input of the proof verified on the Fr side is equal to the Fp input computed on Fr side.","breadcrumbs":"Deferred Computation » Enforcing Equality","id":"222","title":"Enforcing Equality"},"223":{"body":"This section contains a series of diagrams giving an overview of implementation of pickles, closely following the structures and abstractions in the code. However, they are very technical and are primarily intended for developer use. The primary source of all the images in this section is pickles_structure.drawio file, and if one edits it one must re-generate the .svg renders by doing \"export -> selection\". The legend of the diagrams is quite straightforward: The black boxes are data structures that have names and labels following the implementation. MFNStep/MFNWrap is an abbreviation from MessagesForNextStep and MessagesForNextWrap that is used for brevity. Most other datatypes are exactly the same as in the codebase. The blue boxes are computations. Sometimes, when the computation is trivial or only vaguely indicated, it is denoted as a text sign directly on an arrow. Arrows are blue by default and denote moving a piece of data from one place to another with no (or very little) change. Light blue arrows are denoting witness query that is implemented through the handler mechanism. The \"chicken foot\" connector means that this arrow accesses just one field in an array: such an arrow could connect e.g. a input field of type old_a: A in a structure Vec<(A,B)> to an output new_a: A, which just means that we are inside a for loop and this computation is done for all the elemnts in the vector/array. Colour of the field is sometimes introduced and denotes how many steps ago was this piece of data created. The absense of the colour means either that (1) the data structure contains different subfields of different origin, or that (2) it was not coloured but it could be. The colours are assigned according to the following convention:","breadcrumbs":"Technical Diagrams » Pickles Technical Diagrams","id":"223","title":"Pickles Technical Diagrams"},"224":{"body":"The following is the diagram that explains the Wrap computation. The left half of it corresponds to the wrap.ml file and the general logic of proof creation, while the right part of it is wrap_main.ml/wrap_verifier.ml and explains in-circuit computation. This diagram explains the deferred_values computation which is in the heart of wrap.ml represented as a blue box in the middle of the left pane of the main Wrap diagram. Deferred values for Wrap are computed as follows:","breadcrumbs":"Technical Diagrams » Wrap Computatiton and Deferred Values","id":"224","title":"Wrap Computatiton and Deferred Values"},"225":{"body":"The following is the diagram that explains the Step computation, similarly to Wrap. The left half of it corresponds to the general logic in step.ml, and the right part of it is step_main.ml and explains in-circuit computation. We provide no deferred_values computation diagram for Step, but it is very conceptually similar to the one already presented for Wrap.","breadcrumbs":"Technical Diagrams » Step Computation","id":"225","title":"Step Computation"},"226":{"body":"Poseidon is a hash function that can efficiently run in a zk circuit. (See poseidon-hash.info ) It is based on the sponge function , with a state composed of field elements and a permutation based on field element operation (addition and exponentiation). The permutation contains an S-box (exponentiation of a group element), adding constants to the state, and matrix multiplication of the state (multiplications and additions) with an MDS matrix . Since a field element is around 255-bit, a single field element is enough as the capacity of the sponge to provide around 116-bit security. Note We might want to piggy back on the zcash poseidon spec at some point (perhaps by making this an extension of the zcash poseidon spec).","breadcrumbs":"Poseidon hash » Poseidon hash function","id":"226","title":"Poseidon hash function"},"227":{"body":"We define a base sponge, and a scalar sponge. Both must be instantiated when verifying a proof (this is due to recursion-support baked in Kimchi). External users of kimchi (or pickles) are most likely to interact with a wrap proof (see the pickles specification ). As such, the sponges they need to instantiate are most likely to be instantiated with: Poseidon-Fp for base sponge Poseidon-Fq for the scalar sponge","breadcrumbs":"Poseidon hash » APIs","id":"227","title":"APIs"},"228":{"body":"new(params) -> BaseSponge. Creates a new base sponge. BaseSponge.absorb(field_element). Absorbs a field element by calling the underlying sponge absorb function. BaseSponge.absorb_point(point). Absorbs an elliptic curve point. If the point is the point at infinity, absorb two zeros. Otherwise, absorb the x and y coordinates with two calls to absorb. BaseSponge.absorb_scalar(field_element_of_scalar_field). Absorbs a scalar. If the scalar field is smaller than the base field (e.g. Fp is smaller than Fq), then the scalar is casted to a field element of the base field and absorbed via absorb. Otherwise, the value is split between its least significant bit and the rest. Then both values are casted to field elements of the base field and absorbed via absorb (the high bits first, then the low bit). BaseSponge.digest() -> field_element. The squeeze function of the underlying sponge function is called and the first field element is returned. BaseSponge.digest_scalar() -> field_element_of_scalar_field. BaseSponge.challenge // TODO: specify. BaseSponge.challenge_fq // TODO: specify.","breadcrumbs":"Poseidon hash » Base Sponge","id":"228","title":"Base Sponge"},"229":{"body":"new(params) -> ScalarSponge ScalarSponge.absorb(scalar_field_element) ScalarSponge.digest() -> scalar_field_element ScalarSponge.challenge // TODO: specify","breadcrumbs":"Poseidon hash » Scalar Sponge","id":"229","title":"Scalar Sponge"},"23":{"body":"A \"commitment scheme\" is a cryptographic scheme that lets us provide a \"commitment\" to a given piece of data, in such a way that we can later \"open\" that commitment. I recommend checking out section 2.4.1 of David's book Real-World Cryptography. There are two properties we typically expect of our commitment schemes: Hiding. A commitment to a piece of data should not reveal anything about that data. Binding. There should be no ambiguity about what data is committed to. That is, it should not be possible for the committer to open the commitment to a piece of data other than the one they originally committed to. There are various kinds of commitments that allow opening in different ways, revealing only part of the data, or some function of the data. Sometimes it is even useful to elide the hiding property entirely--- so-called non-hiding commitments.","breadcrumbs":"Commitments » Commitments","id":"23","title":"Commitments"},"230":{"body":"Note that every operation is done in the field of the sponge. In this section we define the high-level algorithm behind the permutation and the sponge. The permutation is never used directly by users, it is used only by the sponge function.","breadcrumbs":"Poseidon hash » Algorithms","id":"230","title":"Algorithms"},"231":{"body":"In pseudo-code: def sbox(field_element): # modular exponentiation return field_element^7 # apply MDS matrix\ndef apply_mds(state): n = [0, 0, 0] n[0] = state[0] * mds[0][0] + state[1] * mds[0][1] + state[2] * mds[0][2] n[1] = state[0] * mds[1][0] + state[1] * mds[1][1] + state[2] * mds[1][2] n[2] = state[0] * mds[2][0] + state[1] * mds[2][1] + state[2] * mds[2][2] return n # a round in the permutation\ndef apply_round(round, state): # sbox state[0] = sbox(state[0]) state[1] = sbox(state[1]) state[2] = sbox(state[2]) # apply MDS matrix state = apply_mds(state) # add round constant state[0] += round_constants[round][0] state[1] += round_constants[round][1] state[2] += round_constants[round][2] # the permutation\ndef permutation(state): round_offset = 0 if ARK_INITIAL: constant = round_constants[0] state[0] += constant[0] state[1] += constant[1] state[2] += constant[2] round_offset = 1 for round in range(round_offset, ROUNDS + round_offset): apply_round(round, state)","breadcrumbs":"Poseidon hash » Permutation","id":"231","title":"Permutation"},"232":{"body":"In pseudo-code: def new(): return { \"state\": [0] * RATE, # `RATE` field elements set to 0 \"mode\": \"absorbing\", \"offset\": 0, } def absorb(sponge, field_element): # if we're changing mode, reset the offset if sponge.mode == \"squeezing\": sponge.mode = \"absorbing\" sponge.offset = 0 # we reached the end of the rate, permute elif sponge.offset == RATE: sponge.state = permutation(sponge.state) sponge.offset = 0 # absorb by adding to the state sponge.state[sponge.offset] += field_element sponge.offset += 1 def squeeze(sponge): # permute when changing mode or when we already squeezed everything if sponge.mode == \"absorbing\" or sponge.offset == RATE: sponge.mode = \"squeezing\" sponge.state = permutation(sponge.state) sponge.offset = 0 result = sponge.state[sponge.offset] sponge.offset += 1 return result","breadcrumbs":"Poseidon hash » Sponge","id":"232","title":"Sponge"},"233":{"body":"We instantiate two versions of Poseidon, one for the field Fp, and one for the field Fq (see the pasta specification ). Both share the following sponge configuration: capacity 1 rate: 2 and the following permutation configuration: number of full rounds: 55 sbox: 7 ARK_INITIAL: false","breadcrumbs":"Poseidon hash » Instantiations","id":"233","title":"Instantiations"},"234":{"body":"You can find the MDS matrix and round constants we use in /poseidon/src/pasta/fp_kimchi.rs .","breadcrumbs":"Poseidon hash » Poseidon-Fp","id":"234","title":"Poseidon-Fp"},"235":{"body":"You can find the MDS matrix and round constants we use in /poseidon/src/pasta/fp_kimchi.rs .","breadcrumbs":"Poseidon hash » Poseidon-Fq","id":"235","title":"Poseidon-Fq"},"236":{"body":"We have test vectors contained in /poseidon/tests/test_vectors/kimchi.json .","breadcrumbs":"Poseidon hash » Test vectors","id":"236","title":"Test vectors"},"237":{"body":"our ocaml implementation: https://github.com/minaprotocol/mina/blob/develop/src/lib/random_oracle/random_oracle.mli relies on random_oracle_input: https://github.com/minaprotocol/mina/blob/develop/src/lib/random_oracle_input/random_oracle_input.ml is instantiated with two types of fields: https://github.com/minaprotocol/mina/blob/develop/src/nonconsensus/snark_params/snark_params_nonconsensus.ml","breadcrumbs":"Poseidon hash » Pointers to the OCaml code","id":"237","title":"Pointers to the OCaml code"},"238":{"body":"Our polynomial commitment scheme is a bootleproof-type of commitment scheme, which is a mix of: The polynomial commitment scheme described in appendix A.1 of the PCD paper . The zero-knowledge opening described in section 3.1 of the HALO paper .","breadcrumbs":"Polynomial Commitment » Kimchi Polynomial Commitment","id":"238","title":"Kimchi Polynomial Commitment"},"239":{"body":"The two curves pallas and vesta (pa(llas ve)sta) created by the Zcash team. Each curve's scalar field is the other curve's base field, which is practical for recursion (see Pickles). supporting evidence mina parameters: https://github.com/o1-labs/proof-systems/tree/master/curves/src/pasta arkworks ark-pallas , pallas and vesta Note that in general Fq refers to the base field (in which the curve is defined over), while Fr refers to the scalar field (defined by the order of the curve). But in our code, because of the cycles: Fp refers to the base field of Pallas, and the scalar field of Vesta Fq refers to the base field of Vesta, and the scalar field of Pallas Note that ∣Fq​∣>∣Fp​∣. In pickles: Vesta is also referred to as the step curve, or the tick curve. Pallas is also referred to as the wrap curve, or the tock curve.","breadcrumbs":"Pasta Curves » Pasta Curves","id":"239","title":"Pasta Curves"},"24":{"body":"The simplest kind of a commitment is one in which opening reveals all of the underlying data. Let's give a simple construction of such a scheme for committing to field elements. Suppose we have Fp​ a prime order field, with p being rather large (say on the order of 2256). A hash function h:List(Fp​)→Fp​. Then we define commit(x,r)=h([x,r])open(c)=(x,r)verify(c,x,r)=true/false The argument r of the commit function is data that must only be known to the committer (until the commitment is opened). When a committer wants to commit to a field element x, they sample a random \"blinder\" r∈Fp​ and hash it together with x to form the commitment. To open the commitment c, they simply provide the committed value together with the blinder. Alternatively, if the verifier already knows x, they can just provide r, i.e. open(x,c)=r. Finally, given the commitment and the opening, we can verify whether the input was the value originally committed to using the verify function. If the hash function is collision-resistant, then this function is binding (because there's no way the committer could find another preimage of h([x,r])). If the hash function is one-way, then this commitment is also hiding (assuming x is revealed only as part of the opening). Commitments are often used in protocols between provers and verifiers. The following illustration provides an example with a prover named Peggy and a verifier named Victor. Peggyinputblinder​ }→commit​ commitment→open​ ​ ⟶⋮ ⟶⋮ ​Victorcommitment input,blinder commitmentinputblinder​ }→verify​→true/false​ ​ Here Peggy commits to an input using a blinder, obtains the commitment and sends it to Victor. The interlocutors continue their protocol, but eventually to convince Victor of her claims, Peggy must send the opening proof to her earlier commitment. Victor verifies the opening (i.e. the input and blinder) against the commitment. If the verification fails, then Victor knows that Peggy was trying to trick him, otherwise Victor has sufficient assurances that Peggy was telling the truth.","breadcrumbs":"Commitments » Simple commitments","id":"24","title":"Simple commitments"},"240":{"body":"curve equation: y2=x3+5 base field: 28948022309329048855892746252171976963363056481941560715954676764349967630337 scalar field: 28948022309329048855892746252171976963363056481941647379679742748393362948097 mina generator: (1,12418654782883325593414442427049395787963493412651469444558597405572177144507) arkworks generator: (−1,2) endo: endo_q = 2D33357CB532458ED3552A23A8554E5005270D29D19FC7D27B7FD22F0201B547\nendo_r = 397E65A7D7C1AD71AEE24B27E308F0A61259527EC1D4752E619D1840AF55F1B1 You can use sage to test this: Fp = GF(28948022309329048855892746252171976963363056481941560715954676764349967630337)\nPallas = EllipticCurve(Fp, [0, 5])\nPallas.count_points()","breadcrumbs":"Pasta Curves » Pallas","id":"240","title":"Pallas"},"241":{"body":"curve equation: y2=x3+5 base field: 28948022309329048855892746252171976963363056481941647379679742748393362948097 scalar field: 28948022309329048855892746252171976963363056481941560715954676764349967630337 mina generator: (1,11426906929455361843568202299992114520848200991084027513389447476559454104162) arkworks generator: (−1,2) endo: endo_q = 06819A58283E528E511DB4D81CF70F5A0FED467D47C033AF2AA9D2E050AA0E4F\nendo_r = 12CCCA834ACDBA712CAAD5DC57AAB1B01D1F8BD237AD31491DAD5EBDFDFE4AB9 You can use sage to test this: Fq = GF(28948022309329048855892746252171976963363056481941647379679742748393362948097)\nVesta = EllipticCurve(Fq, [0, 5])\nVesta.count_points()","breadcrumbs":"Pasta Curves » Vesta","id":"241","title":"Vesta"},"242":{"body":"This document specifies kimchi , a zero-knowledge proof system that's a variant of PLONK. This document does not specify how circuits are created or executed, but only how to convert a circuit and its execution into a proof. Table of content: Overview Tables used to describe a circuit Tables produced during proof creation Dependencies Polynomial Commitments Poseidon hash function Pasta Constraints Linearization Permutation Lookup The Lookup Tables The Lookup Selectors Producing the sorted table as the prover Gates Double Generic Gate Poseidon Elliptic Curve Addition Endo Scalar Endo Scalar Multiplication Scalar Multiplication Range Check Foreign Field Addition Foreign Field Multiplication Rotation Xor Gadgets Not And Setup Common Index Lookup Index Prover Index Verifier Index Proof Construction & Verification Proof Structure Proof Creation Proof Verification Fiat-Shamir argument Partial verification Batch verification of proofs Optimizations Security Considerations","breadcrumbs":"Kimchi » Kimchi","id":"242","title":"Kimchi"},"243":{"body":"There are three main algorithms to kimchi: Setup : takes a circuit and produces a prover index, and a verifier index. Proof creation : takes the prover index, and the execution trace of the circuit to produce a proof. Proof verification : takes the verifier index and a proof to verify. As part of these algorithms, a number of tables are created (and then converted into polynomials) to create a proof.","breadcrumbs":"Kimchi » Overview","id":"243","title":"Overview"},"244":{"body":"The following tables are created to describe the circuit: Gates . A circuit is described by a series of gates, that we list in a table. The columns of the tables list the gates, while the rows are the length of the circuit. For each row, only a single gate can take a value 1 while all other gates take the value 0. row Generic Poseidon CompleteAdd VarBaseMul EndoMul EndoMulScalar 0 1 0 0 0 0 0 1 0 1 0 0 0 0 Coefficients . The coefficient table has 15 columns, and is used to tweak the gates. Currently, only the Generic and the Poseidon gates use it (refer to their own sections to see how). All other gates set their values to 0. row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 / / / / / / / / / / / / / / / Wiring (or Permutation, or sigmas) . For gates to take the outputs of other gates as inputs, we use a wiring table to wire registers together. To learn about registers, see the next section. It is defined at every row, but only for the first 7 registers. Each cell specifies a (row, column) tuple that it should be wired to. Cells that are not connected to another cell are wired to themselves. Note that if three or more registers are wired together, they must form a cycle. For example, if register (0, 4) is wired to both registers (80, 6) and (90, 0) then you would have the following table: row 0 1 2 3 4 5 6 0 0,0 0,1 0,2 0,3 80,6 0,5 0,6 ... 80 80,0 80,1 80,2 80,3 80,4 80,5 90,0 ... 90 0,4 90,1 90,2 90,3 90,4 90,5 90,6 The lookup feature is currently optional, as it can add some overhead to the protocol. In the case where you would want to use lookups, the following tables would be needed: Lookup Tables . The different lookup tables that are used in the circuit. For example, the XOR lookup table: l r o 1 0 1 0 1 1 1 1 0 0 0 0 Lookup selectors . A lookup selector is used to perform a number of queries in different lookup tables. Any gate can advertise its use of a lookup selector (so a lookup selector can be associated to several gates), and on which rows they want to use them (current and/or next). In cases where a gate need to use lookups in its current row only, and is the only one performing a specific combination of queries, then its gate selector can be used in place of a lookup selector. As with gates, lookup selectors (including gates used as lookup selectors) are mutually exclusives (only one can be used on a given row). For example, suppose we have two lookup selectors: row ChaChaQuery ChaChaFinalQuery 0 0 0 1 1 0 Where each applies 4 queries. A query is a table describing which lookup table it queries, and the linear combination of the witness to use in the query. For example, the following table describes a query into the XOR table made out of linear combinations of registers (checking that r0​⊕r2​=2⋅r1​): table_id l r o XOR 1, r0 1, r2 2, r1","breadcrumbs":"Kimchi » Tables used to describe a circuit","id":"244","title":"Tables used to describe a circuit"},"245":{"body":"The following tables are created by the prover at runtime: Registers (or Witness) . Registers are also defined at every row, and are split into two types: the IO registers from 0 to 6 usually contain input or output of the gates (note that a gate can output a value on the next row as well). I/O registers can be wired to each other (they'll be forced to have the same value), no matter what row they're on (for example, the register at row:0, col:4 can be wired to the register at row:80, col:6). The rest of the registers, 7 through 14, are called advice registers as they can store values that useful only for the row's active gate. Think of them as intermediary or temporary values needed in the computation when the prover executes a circuit. row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 / / / / / / / / / / / / / / / Wiring (Permutation) trace . You can think of the permutation trace as an extra register that is used to enforce the wiring specified in the wiring table. It is a single column that applies on all the rows as well, which the prover computes as part of a proof. row pt 0 / Queries trace . These are the actual values made by queries, calculated by the prover at runtime, and used to construct the proof. Table trace . Represents the concatenation of all the lookup tables, combined into a single column at runtime by both the prover and the verifier. Sorted trace . Represents the processed (see the lookup section) concatenation of the queries trace and the table trace. It is produced at runtime by the prover. The sorted trace is long enough that it is split in several columns. Lookup (aggregation, or permutation) trace . This is a one column table that is similar to the wiring (permutation) trace we talked above. It is produced at runtime by the prover.","breadcrumbs":"Kimchi » Tables produced during proof creation","id":"245","title":"Tables produced during proof creation"},"246":{"body":"To specify kimchi, we rely on a number of primitives that are specified outside of this specification. In this section we list these specifications, as well as the interfaces we make use of in this specification.","breadcrumbs":"Kimchi » Dependencies","id":"246","title":"Dependencies"},"247":{"body":"Refer to the specification on polynomial commitments . We make use of the following functions from that specification: PolyCom.non_hiding_commit(poly) -> PolyCom::NonHidingCommitment PolyCom.commit(poly) -> PolyCom::HidingCommitment PolyCom.evaluation_proof(poly, commitment, point) -> EvaluationProof PolyCom.verify(commitment, point, evaluation, evaluation_proof) -> bool","breadcrumbs":"Kimchi » Polynomial Commitments","id":"247","title":"Polynomial Commitments"},"248":{"body":"Refer to the specification on Poseidon . We make use of the following functions from that specification: Poseidon.init(params) -> FqSponge Poseidon.update(field_elem) Poseidon.finalize() -> FieldElem specify the following functions on top: Poseidon.produce_challenge() (TODO: uses the endomorphism) Poseidon.to_fr_sponge() -> state_of_fq_sponge_before_eval, FrSponge With the current parameters: S-Box alpha: 7 Width: 3 Rate: 2 Full rounds: 55 Round constants: fp_kimchi , fq_kimchi MDS matrix: fp_kimchi , fq_kimchi","breadcrumbs":"Kimchi » Poseidon hash function","id":"248","title":"Poseidon hash function"},"249":{"body":"Kimchi is made to work on cycles of curves, so the protocol switch between two fields Fq and Fr, where Fq represents the base field and Fr represents the scalar field. See the Pasta curves specification .","breadcrumbs":"Kimchi » Pasta","id":"249","title":"Pasta"},"25":{"body":"Instead of a cryptographic hash function, we can use elliptic curve scalar multiplication to construct a commitment scheme. Here scalar multiplication is used like a one-way function based on the hardness assumption of the elliptic curve discrete logarithm problem (ECDLP). Suppose we have Fp​ a prime order field, with p being large (e.g. something like 2256). Publicly agreed generator point G over an elliptic curve E(Fp​) Another publicly agreed curve point H for which no one knows the discrete logarithm commit(x,r)=xG+rHopen(c)=(x,r) where x∈Fp​ is the value being committed to, r∈Fp​ is a random blinding factor and the commitment c=commit(x,r) is a curve point. These commitments are algebraic (i.e. they do not use a boolean-based cryptographic hash function) and have homomorphic properties: you can add commitments together to form another commitment of the added committed values. For example, if you have commitments A and B, you can perform: A+B​=commit(xa​,ra​)+commit(xb​,rb​)=xa​G+ra​H+xb​G+rb​H=(xa​+xb​)G+(ra​+rb​)H=commit(xa​+xb​,ra​+rb​)​ In other words, the sum of commitments A and B is equal to the commitment of the sum of the two committed values xa​ and xb​ and blinders ra​ and rb​. This is possible because in such a scheme scaling is commutative with adding scalars. As a cryptographic primitive, the ability to find a public curve point H for which no one knows the discrete logarithm may, at first, seem rather mind-blowing and powerful. Actually, it's as easy as it is awesome to find such a point--- simply perform rejection sampling by cryptographically hashing G (or, respectively, the hash output), using the output as the x-coordinate of a candidate point on E and checking whether it's valid. The first valid curve point obtained is H and by the hardness assumption of the ECDLP, no one knows it. Since approximately half of the hash outputs will be valid curve points on E, sampling will terminate very quickly. Indeed, as we will see later, this process can be used to sample many public curve points G1​,…,Gn​ for which the discrete logarithms are unknown; the so-called hash to curve algorithm.","breadcrumbs":"Commitments » Algebraic and homomorphic commitments","id":"25","title":"Algebraic and homomorphic commitments"},"250":{"body":"Kimchi enforces the correct execution of a circuit by creating a number of constraints and combining them together. In this section, we describe all the constraints that make up the main polynomial f once combined. We define the following functions: combine_constraints(range_alpha, constraints), which takes a range of contiguous powers of alpha and a number of constraints. It returns the sum of all the constraints, where each constraint has been multiplied by a power of alpha. In other words it returns: i∑​αi⋅constrainti​ The different ranges of alpha are described as follows: gates . Offset starts at 0 and 21 powers of α are used Permutation . Offset starts at 21 and 3 powers of α are used Note As gates are mutually exclusive (a single gate is used on each row), we can reuse the same range of powers of alpha across all the gates.","breadcrumbs":"Kimchi » Constraints","id":"250","title":"Constraints"},"251":{"body":"TODO","breadcrumbs":"Kimchi » Linearization","id":"251","title":"Linearization"},"252":{"body":"The permutation constraints are the following 4 constraints: The two sides of the coin (with shift0​=1): ​z(x)⋅zkpm(x)⋅αPERM0⋅(w0​(x)+β⋅shift0​x+γ)⋅(w1​(x)+β⋅shift1​x+γ)⋅(w2​(x)+β⋅shift2​x+γ)⋅(w3​(x)+β⋅shift3​x+γ)⋅(w4​(x)+β⋅shift4​x+γ)⋅(w5​(x)+β⋅shift5​x+γ)⋅(w6​(x)+β⋅shift6​x+γ)​​ and ​−1⋅z(xω)⋅zkpm(x)⋅αPERM0⋅(w0​(x)+β⋅σ0​(x)+γ)⋅(w1​(x)+β⋅σ1​(x)+γ)⋅(w2​(x)+β⋅σ2​(x)+γ)⋅(w3​(x)+β⋅σ3​(x)+γ)⋅(w4​(x)+β⋅σ4​(x)+γ)⋅(w5​(x)+β⋅σ5​(x)+γ)⋅(w6​(x)+β⋅σ6​(x)+γ)⋅​​ the initialization of the accumulator: (z(x)−1)L1​(x)αPERM1 and the accumulator's final value: (z(x)−1)Ln−k​(x)αPERM2 You can read more about why it looks like that in this post . The quotient contribution of the permutation is split into two parts perm and bnd. They will be used by the prover. perm(x)=​aPERM0⋅zkpl(x)⋅[z(x)⋅(w0​(x)+γ+x⋅β⋅shift0​)⋅(w1​(x)+γ+x⋅β⋅shift1​)⋅(w2​(x)+γ+x⋅β⋅shift2​)⋅(w3​(x)+γ+x⋅β⋅shift3​)⋅(w4​(x)+γ+x⋅β⋅shift4​)⋅(w5​(x)+γ+x⋅β⋅shift5​)⋅(w6​(x)+γ+x⋅β⋅shift6​)⋅−z(x⋅w)⋅(w0​(x)+γ+σ0​⋅β)⋅(w1​(x)+γ+σ1​⋅β)⋅(w2​(x)+γ+σ2​⋅β)⋅(w3​(x)+γ+σ3​⋅β)⋅(w4​(x)+γ+σ4​⋅β)⋅(w5​(x)+γ+σ5​⋅β)⋅(w6​(x)+γ+σ6​⋅β)⋅]​​ and bnd: bnd(x)=aPERM1⋅x−1z(x)−1​+aPERM2⋅x−sid[n−k]z(x)−1​ The linearization: scalar⋅σ6​(x) where scalar is computed as: z(ζω)βαPERM0zkpl(ζ)⋅(γ+βσ0​(ζ)+w0​(ζ))⋅(γ+βσ1​(ζ)+w1​(ζ))⋅(γ+βσ2​(ζ)+w2​(ζ))⋅(γ+βσ3​(ζ)+w3​(ζ))⋅(γ+βσ4​(ζ)+w4​(ζ))⋅(γ+βσ5​(ζ)+w5​(ζ))⋅​​ To compute the permutation aggregation polynomial, the prover interpolates the polynomial that has the following evaluations. The first evaluation represents the initial value of the accumulator: z(g0)=1 For i=0,⋅,n−4, where n is the size of the domain, evaluations are computed as: z(gi+1)=z1​/z2​ with z1​=​ (w0​(gi+sid(gi)⋅beta⋅shift0​+γ)⋅ (w1​(gi)+sid(gi)⋅beta⋅shift1​+γ)⋅ (w2​(gi)+sid(gi)⋅beta⋅shift2​+γ)⋅ (w3​(gi)+sid(gi)⋅beta⋅shift3​+γ)⋅ (w4​(gi)+sid(gi)⋅beta⋅shift4​+γ)⋅ (w5​(gi)+sid(gi)⋅beta⋅shift5​+γ)⋅ (w6​(gi)+sid(gi)⋅beta⋅shift6​+γ)​​ and z2​=​ (w0​(gi)+σ0​⋅beta+γ)⋅ (w1​(gi)+σ1​⋅beta+γ)⋅ (w2​(gi)+σ2​⋅beta+γ)⋅ (w3​(gi)+σ3​⋅beta+γ)⋅ (w4​(gi)+σ4​⋅beta+γ)⋅ (w5​(gi)+σ5​⋅beta+γ)⋅ (w6​(gi)+σ6​⋅beta+γ)​​ We randomize the evaluations at n - zk_rows + 1 and n - zk_rows + 2 in order to add zero-knowledge to the protocol. For a valid witness, we then have have z(gn−zkr​ows)=1.","breadcrumbs":"Kimchi » Permutation","id":"252","title":"Permutation"},"253":{"body":"Lookups in kimchi allows you to check if a single value, or a series of values, are part of a table. The first case is useful to check for checking if a value belongs to a range (from 0 to 1,000, for example), whereas the second case is useful to check truth tables (for example, checking that three values can be found in the rows of an XOR table) or write and read from a memory vector (where one column is an index, and the other is the value stored at that index). Note Similarly to the generic gate, each values taking part in a lookup can be scaled with a fixed field element. The lookup functionality is an opt-in feature of kimchi that can be used by custom gates. From the user's perspective, not using any gates that make use of lookups means that the feature will be disabled and there will be no overhead to the protocol. Please refer to the lookup RFC for an overview of the lookup feature. In this section, we describe the tables kimchi supports, as well as the different lookup selectors (and their associated queries) The Lookup Tables Kimchi currently supports two lookup tables: /// The table ID associated with the XOR lookup table.\npub const XOR_TABLE_ID: i32 = 0; /// The range check table ID.\npub const RANGE_CHECK_TABLE_ID: i32 = 1; XOR The lookup table for 4-bit xor. Note that it is constructed so that (0, 0, 0) is the last position in the table. This is because tables are extended to the full size of a column (essentially) by padding them with their final value. And, having the value (0, 0, 0) here means that when we commit to this table and use the dummy value in the lookup_sorted columns, those entries that have the dummy value of 0=0+j∗0+j2∗0 will translate into a scalar multiplication by 0, which is free. 12-bit Check The range check table is a single-column table containing the numbers from 0 to 2^12 (excluded). This is used to check that the value fits in 12 bits. Runtime tables Another type of lookup tables has been suggested in the Extended Lookup Tables . The Lookup Selectors XorSelector . Performs 4 queries to the XOR lookup table. l r o - l r o - l r o - l r o 1, r3 1, r7 1, r11 - 1, r4 1, r8 1, r12 - 1, r5 1, r9 1, r13 - 1, r6 1, r10 1, r14 Producing the sorted table as the prover Because of our ZK-rows, we can't do the trick in the plookup paper of wrapping around to enforce consistency between the sorted lookup columns. Instead, we arrange the LookupSorted table into columns in a snake-shape. Like so, _ _\n| | | | |\n| | | | |\n|_| |_| | or, imagining the full sorted array is [ s0, ..., s8 ], like s0 s4 s4 s8\ns1 s3 s5 s7\ns2 s2 s6 s6 So the direction (\"increasing\" or \"decreasing\" (relative to LookupTable) is if i % 2 = 0 { Increasing } else { Decreasing } Then, for each i < max_lookups_per_row, if i % 2 = 0, we enforce that the last element of LookupSorted(i) = last element of LookupSorted(i + 1), and if i % 2 = 1, we enforce that the first element of LookupSorted(i) = first element of LookupSorted(i + 1).","breadcrumbs":"Kimchi » Lookup","id":"253","title":"Lookup"},"254":{"body":"A circuit is described as a series of gates. In this section we describe the different gates currently supported by kimchi, the constraints associated to them, and the way the register table, coefficient table, and permutation can be used in conjunction. TODO: for each gate describe how to create it? Double Generic Gate The double generic gate contains two generic gates. A generic gate is simply the 2-fan in gate specified in the vanilla PLONK protocol that allows us to do operations like: addition of two registers (into an output register) or multiplication of two registers equality of a register with a constant More generally, the generic gate controls the coefficients ci​ in the equation: c0​⋅l+c1​⋅r+c2​⋅o+c3​⋅(l×r)+c4​ The layout of the gate is the following: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 l1 r1 o1 l2 r2 o2 where l1, r1, and o1 (resp. l2, r2, o2) are the left, right, and output registers of the first (resp. second) generic gate. The selectors are stored in the coefficient table as: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 l1 r1 o1 m1 c1 l2 r2 o2 m2 c2 with m1 (resp. m2) the mul selector for the first (resp. second) gate, and c1 (resp. c2) the constant selector for the first (resp. second) gate. The constraints: w0​⋅c0​+w1​⋅c1​+w2​⋅c2​+w0​⋅w1​⋅c3​+c4​ w3​⋅c5​+w4​⋅c6​+w5​⋅c7​+w3​w4​c8​+c9​ where the ci​ are the coefficients. Poseidon The poseidon gate encodes 5 rounds of the poseidon permutation. A state is represents by 3 field elements. For example, the first state is represented by (s0, s0, s0), and the next state, after permutation, is represented by (s1, s1, s1). Below is how we store each state in the register table: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 s0 s0 s0 s4 s4 s4 s1 s1 s1 s2 s2 s2 s3 s3 s3 s5 s5 s5 The last state is stored on the next row. This last state is either used: with another Poseidon gate on that next row, representing the next 5 rounds. or with a Zero gate, and a permutation to use the output elsewhere in the circuit. or with another gate expecting an input of 3 field elements in its first registers. Note As some of the poseidon hash variants might not use 5k rounds (for some k), the result of the 4-th round is stored directly after the initial state. This makes that state accessible to the permutation. We define Mr,c​ as the MDS matrix at row r and column c. We define the S-box operation as wS for S the SPONGE_BOX constant. We store the 15 round constants ri​ required for the 5 rounds (3 per round) in the coefficient table: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 The initial state, stored in the first three registers, are not constrained. The following 4 states (of 3 field elements), including 1 in the next row, are constrained to represent the 5 rounds of permutation. Each of the associated 15 registers is associated to a constraint, calculated as: first round: w6​−(r0​+(M0,0​w0S​+M0,1​w1S​+M0,2​w2S​)) w7​−(r1​+(M1,0​w0S​+M1,1​w1S​+M1,2​w2S​)) w8​−(r2​+(M2,0​w0S​+M2,1​w1S​+M2,2​w2S​)) second round: w9​−(r3​+(M0,0​w6S​+M0,1​w7S​+M0,2​w8S​)) w10​−(r4​+(M1,0​w6S​+M1,1​w7S​+M1,2​w8S​)) w11​−(r5​+(M2,0​w6S​+M2,1​w7S​+M2,2​w8S​)) third round: w12​−(r6​+(M0,0​w9S​+M0,1​w10S​+M0,2​w11S​)) w13​−(r7​+(M1,0​w9S​+M1,1​w10S​+M1,2​w11S​)) w14​−(r8​+(M2,0​w9S​+M2,1​w10S​+M2,2​w11S​)) fourth round: w3​−(r9​+(M0,0​w12S​+M0,1​w13S​+M0,2​w14S​)) w4​−(r10​+(M1,0​w12S​+M1,1​w13S​+M1,2​w14S​)) w5​−(r11​+(M2,0​w12S​+M2,1​w13S​+M2,2​w14S​)) fifth round: w0,next​−(r12​+(M0,0​w3S​+M0,1​w4S​+M0,2​w5S​)) w1,next​−(r13​+(M1,0​w3S​+M1,1​w4S​+M1,2​w5S​)) w2,next​−(r14​+(M2,0​w3S​+M2,1​w4S​+M2,2​w5S​)) where wi,next​ is the polynomial wi​(ωx) which points to the next row. Elliptic Curve Addition The layout is 0 1 2 3 4 5 6 7 8 9 10 x1 y1 x2 y2 x3 y3 inf same_x s inf_z x21_inv where (x1, y1), (x2, y2) are the inputs and (x3, y3) the output. inf is a boolean that is true iff the result (x3, y3) is the point at infinity. The rest of the values are inaccessible from the permutation argument, but same_x is a boolean that is true iff x1 == x2. The following constraints are generated: constraint 1: x0​=w2​−w0​ (w10​⋅x0​−F(1)−w7​) constraint 2: x0​=w2​−w0​ w7​⋅x0​ constraint 3: x0​=w2​−w0​ x1​=w3​−w1​ x2​=w0​⋅w0​ w7​⋅(2⋅w8​⋅w1​−2⋅x2​−x2​)+(F(1)−w7​)⋅(x0​⋅w8​−x1​) constraint 4: w0​+w2​+w4​−w8​⋅w8​ constraint 5: w8​⋅(w0​−w4​)−w1​−w5​ constraint 6: x1​=w3​−w1​ x1​⋅(w7​−w6​) constraint 7: x1​=w3​−w1​ x1​⋅w9​−w6​ Endo Scalar We give constraints for the endomul scalar computation. Each row corresponds to 8 iterations of the inner loop in \"Algorithm 2\" on page 29 of the Halo paper . The state of the algorithm that's updated across iterations of the loop is (a, b). It's clear from that description of the algorithm that an iteration of the loop can be written as (a, b, i) -> ( 2 * a + c_func(r_{2 * i}, r_{2 * i + 1}), 2 * b + d_func(r_{2 * i}, r_{2 * i + 1}) ) for some functions c_func and d_func. If one works out what these functions are on every input (thinking of a two-bit input as a number in {0,1,2,3}), one finds they are given by c_func(x), defined by c_func(0) = 0 c_func(1) = 0 c_func(2) = -1 c_func(3) = 1 d_func(x), defined by d_func(0) = -1 d_func(1) = 1 d_func(2) = 0 d_func(3) = 0 One can then interpolate to find polynomials that implement these functions on {0,1,2,3}. You can use sage , as R = PolynomialRing(QQ, 'x')\nc_func = R.lagrange_polynomial([(0, 0), (1, 0), (2, -1), (3, 1)])\nd_func = R.lagrange_polynomial([(0, -1), (1, 1), (2, 0), (3, 0)]) Then, c_func is given by 2/3 * x^3 - 5/2 * x^2 + 11/6 * x and d_func is given by 2/3 * x^3 - 7/2 * x^2 + 29/6 * x - 1 <=> c_func + (-x^2 + 3x - 1) We lay it out the witness as 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Type n0 n8 a0 b0 a8 b8 x0 x1 x2 x3 x4 x5 x6 x7 ENDO where each xi is a two-bit \"crumb\". We also use a polynomial to check that each xi is indeed in {0,1,2,3}, which can be done by checking that each xi​ is a root of the polyunomial below: crumb(x)\n= x (x - 1) (x - 2) (x - 3)\n= x^4 - 6*x^3 + 11*x^2 - 6*x\n= x *(x^3 - 6*x^2 + 11*x - 6) Each iteration performs the following computations Update n: ni+1​=2⋅ni​+xi​ Update a: ai+1​=2⋅ai​+ci​ Update b: bi+1​=2⋅bi​+di​ Then, after the 8 iterations, we compute expected values of the above operations as: expected_n8 := 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * (2 * n0 + x0) + x1 ) + x2 ) + x3 ) + x4 ) + x5 ) + x6 ) + x7 expected_a8 := 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * (2 * a0 + c0) + c1 ) + c2 ) + c3 ) + c4 ) + c5 ) + c6 ) + c7 expected_b8 := 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * (2 * b0 + d0) + d1 ) + d2 ) + d3 ) + d4 ) + d5 ) + d6 ) + d7 Putting together all of the above, these are the 11 constraints for this gate Checking values after the 8 iterations: Constrain n: 0 = expected_n8 - n8 Constrain a: 0 = expected_a8 - a8 Constrain b: 0 = expected_b8 - b8 Checking the crumbs, meaning each x is indeed in the range {0,1,2,3}: Constrain x0​: 0 = x0 * ( x0^3 - 6 * x0^2 + 11 * x0 - 6 ) Constrain x1​: 0 = x1 * ( x1^3 - 6 * x1^2 + 11 * x1 - 6 ) Constrain x2​: 0 = x2 * ( x2^3 - 6 * x2^2 + 11 * x2 - 6 ) Constrain x3​: 0 = x3 * ( x3^3 - 6 * x3^2 + 11 * x3 - 6 ) Constrain x4​: 0 = x4 * ( x4^3 - 6 * x4^2 + 11 * x4 - 6 ) Constrain x5​: 0 = x5 * ( x5^3 - 6 * x5^2 + 11 * x5 - 6 ) Constrain x6​: 0 = x6 * ( x6^3 - 6 * x6^2 + 11 * x6 - 6 ) Constrain x7​: 0 = x7 * ( x7^3 - 6 * x7^2 + 11 * x7 - 6 ) Endo Scalar Multiplication We implement custom gate constraints for short Weierstrass curve endomorphism optimised variable base scalar multiplication. Given a finite field Fq​ of order q, if the order is not a multiple of 2 nor 3, then an elliptic curve over Fq​ in short Weierstrass form is represented by the set of points (x,y) that satisfy the following equation with a,b∈Fq​ and 4a3+27b2=Fq​​0: E(Fq​):y2=x3+ax+b If P=(xp​,yp​) and T=(xt​,yt​) are two points in the curve E(Fq​), the goal of this operation is to perform the operation 2P±T efficiently as (P±T)+P. S = (P + (b ? T : −T)) + P The same algorithm can be used to perform other scalar multiplications, meaning it is not restricted to the case 2⋅P, but it can be used for any arbitrary k⋅P. This is done by decomposing the scalar k into its binary representation. Moreover, for every step, there will be a one-bit constraint meant to differentiate between addition and subtraction for the operation (P±T)+P: In particular, the constraints of this gate take care of 4 bits of the scalar within a single EVBSM row. When the scalar is longer (which will usually be the case), multiple EVBSM rows will be concatenated. Row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Type i xT yT Ø Ø xP yP n xR yR s1 s3 b1 b2 b3 b4 EVBSM i+1 = = xS yS n' xR' yR' s1' s3' b1' b2' b3' b4' EVBSM The layout of this gate (and the next row) allows for this chained behavior where the output point of the current row S gets accumulated as one of the inputs of the following row, becoming P in the next constraints. Similarly, the scalar is decomposed into binary form and n (n′ respectively) will store the current accumulated value and the next one for the check. For readability, we define the following variables for the constraints: endo := EndoCoefficient xq1 :=(1+(endo−1)⋅b1​)⋅xt​ xq2 :=(1+(endo−1)⋅b3​)⋅xt​ yq1 :=(2⋅b2​−1)⋅yt​ yq2 :=(2⋅b4​−1)⋅yt​ These are the 11 constraints that correspond to each EVBSM gate, which take care of 4 bits of the scalar within a single EVBSM row: First block: (xq1 - xp) * s1 = yq1 - yp (2 * xp – s1^2 + xq1) * ((xp – xr) * s1 + yr + yp) = (xp – xr) * 2 * yp (yr + yp)^2 = (xp – xr)^2 * (s1^2 – xq1 + xr) Second block: (xq2 - xr) * s3 = yq2 - yr (2*xr – s3^2 + xq2) * ((xr – xs) * s3 + ys + yr) = (xr – xs) * 2 * yr (ys + yr)^2 = (xr – xs)^2 * (s3^2 – xq2 + xs) Booleanity checks: Bit flag b1​: 0 = b1 * (b1 - 1) Bit flag b2​: 0 = b2 * (b2 - 1) Bit flag b3​: 0 = b3 * (b3 - 1) Bit flag b4​: 0 = b4 * (b4 - 1) Binary decomposition: Accumulated scalar: n_next = 16 * n + 8 * b1 + 4 * b2 + 2 * b3 + b4 The constraints above are derived from the following EC Affine arithmetic equations: (1) => (xq1​​−xp​)⋅s1​=yq1​​−yp​ (2&3) => (xp​–xr​)⋅s2​=yr​+yp​ (2) => (2⋅xp​+xq1​​–s12​)⋅(s1​+s2​)=2⋅yp​ <=> (2⋅xp​–s12​+xq1​​)⋅((xp​–xr​)⋅s1​+yr​+yp​)=(xp​–xr​)⋅2⋅yp​ (3) => s12​−s22​=xq1​​−xr​ <=> (yr​+yp​)2=(xp​–xr​)2⋅(s12​–xq1​​+xr​) (4) => (xq2​​−xr​)⋅s3​=yq2​​−yr​ (5&6) => (xr​–xs​)⋅s4​=ys​+yr​ (5) => (2⋅xr​+xq2​​–s32​)⋅(s3​+s4​)=2⋅yr​ <=> (2⋅xr​–s32​+xq2​​)⋅((xr​–xs​)⋅s3​+ys​+yr​)=(xr​–xs​)⋅2⋅yr​ (6) => s32​–s42​=xq2​​−xs​ <=> (ys​+yr​)2=(xr​–xs​)2⋅(s32​–xq2​​+xs​) Defining s2​ and s4​ as s2​:=2∗xP​+xT​−s12​2⋅yP​​−s1​ s4​:=2∗xR​+xT​−s32​2⋅yR​​−s3​ Gives the following equations when substituting the values of s2​ and s4​: (xq1 - xp) * s1 = (2 * b1 - 1) * yt - yp (2 * xp – s1^2 + xq1) * ((xp – xr) * s1 + yr + yp) = (xp – xr) * 2 * yp (yr + yp)^2 = (xp – xr)^2 * (s1^2 – xq1 + xr) (xq2 - xr) * s3 = (2 * b2 - 1) * yt - yr (2 * xr – s3^2 + xq2) * ((xr – xs) * s3 + ys + yr) = (xr – xs) * 2 * yr (ys + yr)^2 = (xr – xs)^2 * (s3^2 – xq2 + xs) Scalar Multiplication We implement custom Plonk constraints for short Weierstrass curve variable base scalar multiplication. Given a finite field Fq​ of order q, if the order is not a multiple of 2 nor 3, then an elliptic curve over Fq​ in short Weierstrass form is represented by the set of points (x,y) that satisfy the following equation with a,b∈Fq​ and 4a3+27b2=Fq​​0: E(Fq​):y2=x3+ax+b If P=(xp​,yp​) and Q=(xq​,yq​) are two points in the curve E(Fq​), the algorithm we represent here computes the operation 2P+Q (point doubling and point addition) as (P+Q)+Q. Info Point Q=(xq​,yq​) has nothing to do with the order q of the field Fq​. The original algorithm that is being used can be found in the Section 3.1 of https://arxiv.org/pdf/math/0208038.pdf , which can perform the above operation using 1 multiplication, 2 squarings and 2 divisions (one more squaring) if P=Q), thanks to the fact that computing the Y-coordinate of the intermediate addition is not required. This is more efficient to the standard algorithm that requires 1 more multiplication, 3 squarings in total and 2 divisions. Moreover, this algorithm can be applied not only to the operation 2P+Q, but any other scalar multiplication kP. This can be done by expressing the scalar k in biwise form and performing a double-and-add approach. Nonetheless, this requires conditionals to differentiate 2P from 2P+Q. For that reason, we will implement the following pseudocode from https://github.com/zcash/zcash/issues/3924 (where instead, they give a variant of the above efficient algorithm for Montgomery curves b⋅y2=x3+a⋅x2+x). Acc := [2]T\nfor i = n-1 ... 0: Q := (k_{i + 1} == 1) ? T : -T Acc := Acc + (Q + Acc)\nreturn (k_0 == 0) ? Acc - P : Acc The layout of the witness requires 2 rows. The i-th row will be a VBSM gate whereas the next row will be a ZERO gate. Row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Type i xT yT x0 y0 n n' x1 y1 x2 y2 x3 y3 x4 y4 VBSM i+1 x5 y5 b0 b1 b2 b3 b4 s0 s1 s2 s3 s4 ZERO The gate constraints take care of 5 bits of the scalar multiplication. Each single bit consists of 4 constraints. There is one additional constraint imposed on the final number. Thus, the VarBaseMul gate argument requires 21 constraints. For every bit, there will be one constraint meant to differentiate between addition and subtraction for the operation (P±T)+P: S = (P + (b ? T : −T)) + P We follow these criteria: If the bit is positive, the sign should be a subtraction If the bit is negative, the sign should be an addition Then, paraphrasing the above, we will represent this behavior as: S = (P - (2 * b - 1) * T ) + P Let us call Input the point with coordinates (xI, yI) and Target is the point being added with coordinates (xT, yT). Then Output will be the point with coordinates (xO, yO) resulting from O = ( I ± T ) + I Info Do not confuse our Output point (xO, yO) with the point at infinity that is normally represented as O. In each step of the algorithm, we consider the following elliptic curves affine arithmetic equations: s1​:=xi​−xt​yi​−(2⋅b−1)⋅yt​​ s2​:=2∗xi​+xt​−s12​2⋅yi​​−s1​ xo​:=xt​+s22​−s12​ yo​:=s2​⋅(xi​−xo​)−yi​ For readability, we define the following 3 variables in such a way that s2​ can be expressed as u / t: rx :=s12​−xi​−xt​ t :=xi​− rx ⟺2⋅xi​−s12​+xt​ u :=2⋅yi​− t ⋅s1​⟺2⋅yi​−s1​⋅(2⋅xi​−s12​+xt​) Next, for each bit in the algorithm, we create the following 4 constraints that derive from the above: Booleanity check on the bit b: 0 = b * b - b Constrain s1​: (xI - xT) * s1 = yI – (2b - 1) * yT Constrain Output X-coordinate xo​ and s2​: 0 = u^2 - t^2 * (xO - xT + s1^2) Constrain Output Y-coordinate yo​ and s2​: 0 = (yO + yI) * t - (xI - xO) * u When applied to the 5 bits, the value of the Target point (xT, yT) is maintained, whereas the values for the Input and Output points form the chain: [(x0, y0) -> (x1, y1) -> (x2, y2) -> (x3, y3) -> (x4, y4) -> (x5, y5)] Similarly, 5 different s0..s4 are required, just like the 5 bits b0..b4. Finally, the additional constraint makes sure that the scalar is being correctly expressed into its binary form (using the double-and-add decomposition) as: n′=25⋅n+24⋅b0​+23⋅b1​+22⋅b2​+21⋅b3​+b4​ This equation is translated as the constraint: Binary decomposition: 0 = n' - (b4 + 2 * (b3 + 2 * (b2 + 2 * (b1 + 2 * (b0 + 2*n))))) Range Check The multi range check gadget is comprised of three circuit gates (RangeCheck0, RangeCheck1 and Zero) and can perform range checks on three values (v0​,v1​ and v2​) of up to 88 bits each. Values can be copied as inputs to the multi range check gadget in two ways: (Standard mode) With 3 copies, by copying v0​,v1​ and v2​ to the first cells of the first 3 rows of the gadget. In this mode the first gate coefficient is set to 0. (Compact mode) With 2 copies, by copying v2​ to the first cell of the first row and copying v10​=v0​+2ℓ⋅v1​ to the 2nd cell of row 2. In this mode the first gate coefficient is set to 1. The RangeCheck0 gate can also be used on its own to perform 64-bit range checks by constraining witness cells 1-2 to zero. Byte-order: Each cell value is in little-endian byte order Limbs are mapped to columns in big-endian order (i.e. the lowest columns contain the highest bits) We also have the highest bits covered by copy constraints and plookups, so that we can copy the highest two constraints to zero and get a 64-bit lookup, which are envisioned to be a common case The values are decomposed into limbs as follows: L is a 12-bit lookup (or copy) limb, C is a 2-bit \"crumb\" limb (we call half a nybble a crumb). <----6----> <------8------> v0 = L L L L L L C C C C C C C C v1 = L L L L L L C C C C C C C C <2> <--4--> <---------------18----------------> v2 = C C L L L L C C C C C C C C C C C C C C C C C C Witness structure: Row Contents 0 v0​ 1 v1​ 2 v2​ 3 v0​,v1​,v2​ The first 2 rows contain v0​ and v1​ and their respective decompositions into 12-bit and 2-bit limbs The 3rd row contains v2​ and part of its decomposition: four 12-bit limbs and the 1st 10 crumbs The final row contains v0​'s and v1​'s 5th and 6th 12-bit limbs as well as the remaining 10 crumbs of v2​ Note Because we are constrained to 4 lookups per row, we are forced to postpone some lookups of v0 and v1 to the final row. Constraints: For efficiency, the limbs are constrained differently according to their type: 12-bit limbs are constrained with plookups 2-bit crumbs are constrained with degree-4 constraints x(x−1)(x−2)(x−3) Layout: This is how the three 88-bit inputs v0​,v1​ and v2​ are laid out and constrained. vipj is the jth 12-bit limb of value vi​ vicj is the jth 2-bit crumb limb of value vi​ Gates RangeCheck0 RangeCheck0 RangeCheck1 Zero Rows 0 1 2 3 Cols 0 v0 v1 v2 crumb v2c9 MS:1 copy v0p0 copy v1p0 optional v12 crumb v2c10 2 copy v0p1 copy v1p1 crumb v2c0 crumb v2c11 3 plookup v0p2 plookup v1p2 plookup v2p0 plookup v0p0 4 plookup v0p3 plookup v1p3 plookup v2p1 plookup v0p1 5 plookup v0p4 plookup v1p4 plookup v2p2 plookup v1p0 6 plookup v0p5 plookup v1p5 plookup v2p3 plookup v1p1 7 crumb v0c0 crumb v1c0 crumb v2c1 crumb v2c12 8 crumb v0c1 crumb v1c1 crumb v2c2 crumb v2c13 9 crumb v0c2 crumb v1c2 crumb v2c3 crumb v2c14 10 crumb v0c3 crumb v1c3 crumb v2c4 crumb v2c15 11 crumb v0c4 crumb v1c4 crumb v2c5 crumb v2c16 12 crumb v0c5 crumb v1c5 crumb v2c6 crumb v2c17 13 crumb v0c6 crumb v1c6 crumb v2c7 crumb v2c18 LS:14 crumb v0c7 crumb v1c7 crumb v2c8 crumb v2c19 The 12-bit chunks are constrained with plookups and the 2-bit crumbs are constrained with degree-4 constraints of the form x(x−1)(x−2)(x−3). Note that copy denotes a plookup that is deferred to the 4th gate (i.e. Zero). This is because of the limitation that we have at most 4 lookups per row. The copies are constrained using the permutation argument. Gate types: Different rows are constrained using different CircuitGate types Row CircuitGate Purpose 0 RangeCheck0 Partially constrain v0​ 1 RangeCheck0 Partially constrain v1​ 2 RangeCheck1 Fully constrain v2​ (and trigger plookups constraints on row 3) 3 Zero Complete the constraining of v0​ and v1​ using lookups Note Each CircuitGate type corresponds to a unique polynomial and thus is assigned its own unique powers of alpha RangeCheck0 - Range check constraints This circuit gate is used to partially constrain values v0​ and v1​ Optionally, it can be used on its own as a single 64-bit range check by constraining columns 1 and 2 to zero The rest of v0​ and v1​ are constrained by the lookups in the Zero gate row This gate operates on the Curr row It uses three different types of constraints: copy - copy to another cell (12-bits) plookup - plookup (12-bits) crumb - degree-4 constraint (2-bits) Given value v the layout looks like this Column Curr 0 v 1 copy vp0 2 copy vp1 3 plookup vp2 4 plookup vp3 5 plookup vp4 6 plookup vp5 7 crumb vc0 8 crumb vc1 9 crumb vc2 10 crumb vc3 11 crumb vc4 12 crumb vc5 13 crumb vc6 14 crumb vc7 where the notation vpi and vci defined in the \"Layout\" section above. RangeCheck1 - Range check constraints This circuit gate is used to fully constrain v2​ It operates on the Curr and Next rows It uses two different types of constraints: plookup - plookup (12-bits) crumb - degree-4 constraint (2-bits) Given value v2 the layout looks like this Column Curr Next 0 v2 crumb v2c9 1 optional v12 crumb v2c10 2 crumb v2c0 crumb v2c11 3 plookup v2p0 (ignored) 4 plookup v2p1 (ignored) 5 plookup v2p2 (ignored) 6 plookup v2p3 (ignored) 7 crumb v2c1 crumb v2c12 8 crumb v2c2 crumb v2c13 9 crumb v2c3 crumb v2c14 10 crumb v2c4 crumb v2c15 11 crumb v2c5 crumb v2c16 12 crumb v2c6 crumb v2c17 13 crumb v2c7 crumb v2c18 14 crumb v2c8 crumb v2c19 where the notation v2ci and v2pi defined in the \"Layout\" section above. Foreign Field Addition These circuit gates are used to constrain that left_input +/- right_input = field_overflow * foreign_modulus + result Documentation For more details please see the Foreign Field Addition chapter. Mapping To make things clearer, the following mapping between the variable names used in the code and those of the RFC document can be helpful. left_input_lo -> a0 right_input_lo -> b0 result_lo -> r0 bound_lo -> u0 left_input_mi -> a1 right_input_mi -> b1 result_mi -> r1 bound_mi -> u1 left_input_hi -> a2 right_input_hi -> b2 result_hi -> r2 bound_hi -> u2 field_overflow -> q sign -> s carry_lo -> c0 carry_mi -> c1 bound_carry_lo -> k0 bound_carry_mi -> k1 Note: Our limbs are 88-bit long. We denote with: lo the least significant limb (in little-endian, this is from 0 to 87) mi the middle limb (in little-endian, this is from 88 to 175) hi the most significant limb (in little-endian, this is from 176 to 263) Let left_input_lo, left_input_mi, left_input_hi be 88-bit limbs of the left element Let right_input_lo, right_input_mi, right_input_hi be 88-bit limbs of the right element Let foreign_modulus_lo, foreign_modulus_mi, foreign_modulus_hi be 88-bit limbs of the foreign modulus Then the limbs of the result are result_lo = left_input_lo +/- right_input_lo - field_overflow * foreign_modulus_lo - 2^{88} * result_carry_lo result_mi = left_input_mi +/- right_input_mi - field_overflow * foreign_modulus_mi - 2^{88} * result_carry_mi + result_carry_lo result_hi = left_input_hi +/- right_input_hi - field_overflow * foreign_modulus_hi + result_carry_mi field_overflow =0 or 1 or −1 handles overflows in the field result_carry_i =−1,0,1 are auxiliary variables that handle carries between limbs Apart from the range checks of the chained inputs, we need to do an additional range check for a final bound to make sure that the result is less than the modulus, by adding 2^{3*88} - foreign_modulus to it. (This can be computed easily from the limbs of the modulus) Note that 2^{264} as limbs represents: (0, 0, 0, 1) then: The upper-bound check can be calculated as: bound_lo = result_lo - foreign_modulus_lo - bound_carry_lo * 2^{88} bound_mi = result_mi - foreign_modulus_mi - bound_carry_mi * 2^{88} + bound_carry_lo bound_hi = result_hi - foreign_modulus_hi + 2^{88} + bound_carry_mi Which is equivalent to another foreign field addition with right input 2^{264}, q = 1 and s = 1 bound_lo = result_lo + s * 0 - q * foreign_modulus_lo - bound_carry_lo * 2^{88} bound_mi = result_mi + s * 0 - q * foreign_modulus_mi - bound_carry_mi * 2^{88} + bound_carry_lo bound_hi = result_hi + s * 2^{88} - q * foreign_modulus_hi + bound_carry_mi bound_carry_i =0 or 1 or −1 are auxiliary variables that handle carries between limbs The range check of bound can be skipped until the end of the operations and result is an intermediate value that is unused elsewhere (since the final result must have had the right amount of moduli subtracted along the way, meaning a multiple of the modulus). In other words, intermediate results could potentially give a valid witness that satisfies the constraints but where the result is larger than the modulus (yet smaller than 2^{264}). The reason that we have a final bound check is to make sure that the final result (min_result) is indeed the minimum one (meaning less than the modulus). A more optimized version of these constraints is able to reduce by 2 the number of constraints and by 1 the number of witness cells needed. The idea is to condense the low and middle limbs in one longer limb of 176 bits (which fits inside our native field) and getting rid of the low carry flag. With this idea in mind, the sole carry flag we need is the one located between the middle and the high limbs. Layout The sign of the operation (whether it is an addition or a subtraction) is stored in the fourth coefficient as a value +1 (for addition) or -1 (for subtraction). The first 3 coefficients are the 3 limbs of the foreign modulus. One could lay this out as a double-width gate for chained foreign additions and a final row, e.g.: col ForeignFieldAdd chain ForeignFieldAdd final ForeignFieldAdd final Zero 0 left_input_lo (copy) result_lo (copy) min_result_lo (copy) bound_lo (copy) 1 left_input_mi (copy) result_mi (copy) min_result_mi (copy) bound_mi (copy) 2 left_input_hi (copy) result_hi (copy) min_result_hi (copy) bound_hi (copy) 3 right_input_lo (copy) 0 (check) 4 right_input_mi (copy) 0 (check) 5 right_input_hi (copy) 2^88 (check) 6 field_overflow (copy?) 1 (check) 7 carry bound_carry 8 9 10 11 12 13 14 We reuse the foreign field addition gate for the final bound check since this is an addition with a specific parameter structure. Checking that the correct right input, overflow, and overflow are used shall be done by copy constraining these values with a public input value. One could have a specific gate for just this check requiring less constrains, but the cost of adding one more selector gate outweights the savings of one row and a few constraints of difference. Integration Copy final overflow bit from public input containing value 1 * Range check the final bound Foreign Field Multiplication This gadget is used to constrain that left_input * right_input = quotient * foreign_field_modulus + remainder Documentation For more details please see the Foreign Field Multiplication chapter or the original Foreign Field Multiplication RFC Notations For clarity, we use more descriptive variable names in the code than in the RFC, which uses mathematical notations. In order to relate the two documents, the following mapping between the variable names used in the code and those of the RFC can be helpful. left_input0 => a0 right_input0 => b0 quotient0 => q0 remainder01 => r01\nleft_input1 => a1 right_input1 => b1 quotient1 => q1\nleft_input2 => a2 right_input2 => b2 quotient2 => q2 remainder2 => r2 product1_lo => p10 product1_hi_0 => p110 product1_hi_1 => p111 carry0 => v0 carry1_lo => v10 carry1_hi => v11 quotient_hi_bound => q'2 Suffixes The variable names in this code uses descriptive suffixes to convey information about the positions of the bits referred to. When a word is split into up to n parts we use: 0, 1 ... n (where n is the most significant). For example, if we split word x into three limbs, we'd name them x0, x1 and x2 or x[0], x[1] and x[2]. Continuing in this fashion, when one of those words is subsequently split in half, then we add the suffixes _lo and _hi, where hi corresponds to the most significant bits. For our running example, x1 would become x1_lo and x1_hi. If we are splitting into more than two things, then we pick meaningful names for each. So far we've explained our conventions for a splitting depth of up to 2. For splitting deeper than two, we simply cycle back to our depth 1 suffixes again. So for example, x1_lo would be split into x1_lo_0 and x1_lo_1. Parameters hi_foreign_field_modulus := high limb of foreign field modulus $f$ (stored in gate coefficient 0) neg_foreign_field_modulus := negated foreign field modulus $f'$ (stored in gate coefficients 1-3) n := the native field modulus is obtainable from F, the native field's trait bound Witness left_input := left foreign field element multiplicand $ ~\\in F_f$ right_input := right foreign field element multiplicand $ ~\\in F_f$ quotient := foreign field quotient $ ~\\in F_f$ remainder := foreign field remainder $ ~\\in F_f$ carry0 := 2 bit carry carry1_lo := low 88 bits of carry1 carry1_hi := high 3 bits of carry1 product1_lo := lowest 88 bits of middle intermediate product product1_hi_0 := lowest 88 bits of middle intermediate product's highest 88 + 2 bits product1_hi_1 := highest 2 bits of middle intermediate product quotient_hi_bound := quotient high bound for checking q2 ≤ f2 Layout The foreign field multiplication gate's rows are laid out like this col ForeignFieldMul Zero 0 left_input0 (copy) remainder01 (copy) 1 left_input1 (copy) remainder2 (copy) 2 left_input2 (copy) quotient0 (copy) 3 right_input0 (copy) quotient1 (copy) 4 right_input1 (copy) quotient2 (copy) 5 right_input2 (copy) quotient_hi_bound (copy) 6 product1_lo (copy) product1_hi_0 (copy) 7 carry1_0 (plookup) product1_hi_1 (dummy) 8 `carry1_12 (plookup) carry1_48 (plookup) 9 carry1_24 (plookup) carry1_60 (plookup) 10 carry1_36 (plookup) carry1_72 (plookup) 11 carry1_84 carry0 12 carry1_86 13 carry1_88 14 carry1_90 Rotation Rotation of a 64-bit word by a known offset Rot64 onstrains known-length rotation of 64-bit words: This circuit gate is used to constrain that a 64-bit word is rotated by $r < 64$ bits to the \"left\". The rotation is performed towards the most significant side (thus, the new LSB is fed with the old MSB). This gate operates on the Curr and Next rows. The idea is to split the rotation operation into two parts: Shift to the left Add the excess bits to the right We represent shifting with multiplication modulo $2^{64}$. That is, for each word to be rotated, we provide in the witness a quotient and a remainder, similarly to ForeignFieldMul such that the following operation holds: $$word \\cdot 2^{rot} = quotient \\cdot 2^{64} + remainder$$ Then, the remainder corresponds to the shifted word, and the quotient corresponds to the excess bits. $$word \\cdot 2^{rot} = excess \\cdot 2^{64} + shifted$$ Thus, in order to obtain the rotated word, we need to add the quotient and the remainder as follows: $$rotated = shifted + excess$$ The input word is known to be of length 64 bits. All we need for soundness is check that the shifted and excess parts of the word have the correct size as well. That means, we need to range check that: $$ \\begin{aligned} excess &< 2^{rot}\\ shifted &< 2^{64} \\end{aligned} $$ The latter can be obtained with a RangeCheck0 gate setting the two most significant limbs to zero. The former is equivalent to the following check: $$bound = excess - 2^{rot} + 2^{64} < 2^{64}$$ which is doable with the constraints in a RangeCheck0 gate. Since our current row within the Rot64 gate is almost empty, we can use it to perform the range check within the same gate. Then, using the following layout and assuming that the gate has a coefficient storing the value $2^{rot}$, which is publicly known Gate Rot64 RangeCheck0 gadgets (designer's duty) Column Curr Next ------ ------------------- ---------------- 0 copy word shifted 1 copy rotated 0 2 excess 0 3 bound_limb0 shifted_limb0 4 bound_limb1 shifted_limb1 5 bound_limb2 shifted_limb2 6 bound_limb3 shifted_limb3 7 bound_crumb0 shifted_crumb0 8 bound_crumb1 shifted_crumb1 9 bound_crumb2 shifted_crumb2 10 bound_crumb3 shifted_crumb3 11 bound_crumb4 shifted_crumb4 12 bound_crumb5 shifted_crumb5 13 bound_crumb6 shifted_crumb6 14 bound_crumb7 shifted_crumb7 In Keccak, rotations are performed over a 5x5 matrix state of w-bit words each cell. The values used to perform the rotation are fixed, public, and known in advance, according to the following table, depending on the coordinate of each cell within the 5x5 matrix state: y \\ x 0 1 2 3 4 0 0 36 3 105 210 1 1 300 10 45 66 2 190 6 171 15 253 3 28 55 153 21 120 4 91 276 231 136 78 But since we will always be using 64-bit words in our Keccak usecase ($w = 64$), we can have an equivalent table with these values modulo 64 to avoid needing multiple passes of the rotation gate (a single step would cause overflows otherwise): y \\ x 0 1 2 3 4 0 0 36 3 41 18 1 1 44 10 45 2 2 62 6 43 15 61 3 28 55 25 21 56 4 27 20 39 8 14 Since there is one value of the coordinates (x, y) where the rotation is 0 bits, we can skip that step in the gadget. This will save us one gate, and thus the whole 25-1=24 rotations will be performed in just 48 rows. Xor Xor16 - Chainable XOR constraints for words of multiples of 16 bits. This circuit gate is used to constrain that in1 xored with in2 equals out The length of in1, in2 and out must be the same and a multiple of 16bits. This gate operates on the Curr and Next rows. It uses three different types of constraints: copy - copy to another cell (32-bits) plookup - xor-table plookup (4-bits) decomposition - the constraints inside the gate The 4-bit nybbles are assumed to be laid out with 0 column being the least significant nybble. Given values in1, in2 and out, the layout looks like this: Column Curr Next 0 copy in1 copy in1' 1 copy in2 copy in2' 2 copy out copy out' 3 plookup0 in1_0 4 plookup1 in1_1 5 plookup2 in1_2 6 plookup3 in1_3 7 plookup0 in2_0 8 plookup1 in2_1 9 plookup2 in2_2 10 plookup3 in2_3 11 plookup0 out_0 12 plookup1 out_1 13 plookup2 out_2 14 plookup3 out_3 One single gate with next values of in1', in2' and out' being zero can be used to check that the original in1, in2 and out had 16-bits. We can chain this gate 4 times as follows to obtain a gadget for 64-bit words XOR: Row CircuitGate Purpose 0 Xor16 Xor 2 least significant bytes of the words 1 Xor16 Xor next 2 bytes of the words 2 Xor16 Xor next 2 bytes of the words 3 Xor16 Xor 2 most significant bytes of the words 4 Generic Zero values, can be reused as generic gate Info We could halve the number of rows of the 64-bit XOR gadget by having lookups for 8 bits at a time, but for now we will use the 4-bit XOR table that we have. Rough computations show that if we run 8 or more Keccaks in one circuit we should use the 8-bit XOR table.","breadcrumbs":"Kimchi » Gates","id":"254","title":"Gates"},"255":{"body":"Here we describe basic gadgets that we build using a combination of the gates described above. Not We implement NOT, i.e. bitwise negation, as a gadget in two different ways, needing no new gate type for it. Instead, it reuses the XOR gadget and the Generic gate. The first version of the NOT gadget reuses Xor16 by making the following observation: the bitwise NOT operation is equivalent to the bitwise XOR operation with the all one words of a certain length . In other words, $$\\neg x = x \\oplus 1^ $$ where $1^ $ denotes a bitstring of all ones of length $|x|$. Let $x_i$ be the $i$-th bit of $x$, the intuition is that if $x_i = 0$ then XOR with $1$ outputs $1$, thus negating $x_i$. Similarly, if $x_i = 1$ then XOR with 1 outputs 0, again negating $x_i$. Thus, bitwise XOR with $1^*$ is equivalent to bitwise negation (i.e. NOT). Then, if we take the XOR gadget with a second input to be the all one word of the same length, that gives us the NOT gadget. The correct length can be imposed by having a public input containing the 2^bits - 1 value and wiring it to the second input of the XOR gate. This approach needs as many rows as an XOR would need, for a single negation, but it comes with the advantage of making sure the input is of a certain length. The other approach can be more efficient if we already know the length of the inputs. For example, the input may be the input of a range check gate, or the output of a previous XOR gadget (which will be the case in our Keccak usecase). In this case, we simply perform the negation as a subtraction of the input word from the all one word (which again can be copied from a public input). This comes with the advantage of holding up to 2 word negations per row (an eight-times improvement over the XOR approach), but it requires the user to know the length of the input. NOT Layout using XOR Here we show the layout of the NOT gadget using the XOR approach. The gadget needs a row with a public input containing the all-one word of the given length. Then, a number of XORs follow, and a final Zero row is needed. In this case, the NOT gadget needs $\\ceil(\\frac{|x|}{16})$ Xor16 gates, that means one XOR row for every 16 bits of the input word. Row CircuitGate Purpose pub Generic Leading row with the public $1^*$ value i...i+n-1 Xor16 Negate every 4 nybbles of the word, from least to most significant i+n Generic Constrain that the final row is all zeros for correctness of Xor gate NOT Layout using Generic gates Here we show the layout of the NOT gadget using the Generic approach. The gadget needs a row with a public input containing the all-one word of the given length, exactly as above. Then, one Generic gate reusing the all-one word as left inputs can be used to negate up to two words per row. This approach requires that the input word is known (or constrained) to have a given length. Row CircuitGate Purpose pub Generic Leading row with the public $1^*$ value i Generic Negate one or two words of the length given by the length of the all-one word And We implement the AND gadget making use of the XOR gadget and the Generic gate. A new gate type is not needed, but we could potentially add an And16 gate type reusing the same ideas of Xor16 so as to save one final generic gate, at the cost of one additional AND lookup table that would have the same size as that of the Xor. For now, we are willing to pay this small overhead and produce AND gadget as follows: We observe that we can express bitwise addition as follows: $$A + B = (A \\oplus B) + 2 \\cdot (A \\wedge B)$$ where $\\oplus$ is the bitwise XOR operation, $\\wedge$ is the bitwise AND operation, and $+$ is the addition operation. In other words, the value of the addition is nothing but the XOR of its operands, plus the carry bit if both operands are 1. Thus, we can rewrite the above equation to obtain a definition of the AND operation as follows: $$A \\wedge B = \\frac{A + B - (A \\oplus B)}{2}$$ Let us define the following operations for better readability: a + b = sum\na x b = xor\na ^ b = and Then, we can rewrite the above equation as follows: $$ 2 \\cdot and = sum - xor $$ which can be expressed as a double generic gate. Then, our AND gadget for $n$ bytes looks as follows: $n/8$ Xor16 gates 1 (single) Generic gate to check that the final row of the XOR chain is all zeros. 1 (double) Generic gate to check sum $a + b = sum$ and the conjunction equation $2\\cdot and = sum - xor$. Finally, we connect the wires in the following positions (apart from the ones already connected for the XOR gates): Column 2 of the first Xor16 row (the output of the XOR operation) is connected to the right input of the second generic operation of the last row. Column 2 of the first generic operation of the last row is connected to the left input of the second generic operation of the last row. Meaning, the xor in a x b = xor is connected to the xor in 2 \\cdot and = sum - xor the sum in a + b = sum is connected to the sum in 2 \\cdot and = sum - xor","breadcrumbs":"Kimchi » Gadgets","id":"255","title":"Gadgets"},"256":{"body":"In this section we specify the setup that goes into creating two indexes from a circuit: A prover index , necessary for the prover to create proofs. A verifier index , necessary for the verifier to verify proofs. Note The circuit creation part is not specified in this document. It might be specified in a separate document, or we might want to specify how to create the circuit description tables. As such, the transformation of a circuit into these two indexes can be seen as a compilation step. Note that the prover still needs access to the original circuit to create proofs, as they need to execute it to create the witness (register table).","breadcrumbs":"Kimchi » Setup","id":"256","title":"Setup"},"257":{"body":"In this section we describe data that both the prover and the verifier index share. URS (Uniform Reference String) The URS is a set of parameters that is generated once, and shared between the prover and the verifier. It is used for polynomial commitments, so refer to the poly-commitment specification for more details. Note Kimchi currently generates the URS based on the circuit, and attach it to the index. So each circuit can potentially be accompanied with a different URS. On the other hand, Mina reuses the same URS for multiple circuits ( see zkapps for more details ). Domain . A domain large enough to contain the circuit and the zero-knowledge rows (used to provide zero-knowledge to the protocol). Specifically, the smallest subgroup in our field that has order greater or equal to n + ZK_ROWS, with n is the number of gates in the circuit. TODO: what if the domain is larger than the URS? Ordering of elements in the domain Note that in this specification we always assume that the first element of a domain is $1$. Shifts . As part of the permutation, we need to create PERMUTS shifts. To do that, the following logic is followed (in pseudo code): (TODO: move shift creation within the permutation section?) shifts[0] = 1 # first shift is identity for i in 0..7: # generate 7 shifts i = 7 shift, i = sample(domain, i) while shifts.contains(shift) do: shift, i = sample(domain, i) shift[i] = shift def sample(domain, i): i += 1 shift = Field(Blake2b512(to_be_bytes(i))) while is_not_quadratic_non_residue(shift) || domain.contains(shift): i += 1 shift = Field(Blake2b512(to_be_bytes(i))) return shift, i Public . This variable simply contains the number of public inputs. (TODO: actually, it's not contained in the verifier index) The compilation steps to create the common index are as follow: If the circuit is less than 2 gates, abort. Compute the number of zero-knowledge rows (zk_rows) that will be required to achieve zero-knowledge. The following constraints apply to zk_rows: The number of chunks c results in an evaluation at zeta and zeta * omega in each column for 2*c evaluations per column, so zk_rows >= 2*c + 1. The permutation argument interacts with the c chunks in parallel, so it is possible to cross-correlate between them to compromise zero knowledge. We know that there is some c >= 1 such that zk_rows = 2*c + k from the above. Thus, attempting to find the evaluation at a new point, we find that: the evaluation of every witness column in the permutation contains k unknowns; the evaluations of the permutation argument aggregation has k-1 unknowns; the permutation argument applies on all but zk_rows - 3 rows; and thus we form the equation zk_rows - 3 < 7 * k + (k - 1) to ensure that we can construct fewer equations than we have unknowns. This simplifies to k > (2 * c - 2) / 7, giving zk_rows > (16 * c - 2) / 7. We can derive c from the max_poly_size supported by the URS, and thus we find zk_rows and domain_size satisfying the fixpoint zk_rows = (16 * (domain_size / max_poly_size) + 5) / 7\ndomain_size = circuit_size + zk_rows Create a domain for the circuit. That is, compute the smallest subgroup of the field that has order greater or equal to n + zk_rows elements. Pad the circuit: add zero gates to reach the domain size. sample the PERMUTS shifts.","breadcrumbs":"Kimchi » Common Index","id":"257","title":"Common Index"},"258":{"body":"If lookup is used, the following values are added to the common index: LookupSelectors . The list of lookup selectors used. In practice, this tells you which lookup tables are used. TableIds . This is a list of table ids used by the Lookup gate. MaxJointSize . This is the maximum number of columns appearing in the lookup tables used by the lookup selectors. For example, the XOR lookup has 3 columns. To create the index, follow these steps: If no lookup is used in the circuit, do not create a lookup index Get the lookup selectors and lookup tables that are specified implicitly Concatenate explicit runtime lookup tables with the ones (implicitly) used by gates. Get the highest number of columns max_table_width that a lookup table can have. Create the concatenated table of all the fixed lookup tables. It will be of height the size of the domain, and of width the maximum width of any of the lookup tables. In addition, create an additional column to store all the tables' table IDs. For example, if you have a table with ID 0 1 2 3 5 6 7 0 0 0 and another table with ID 1 8 9 the concatenated table in a domain of size 5 looks like this: 1 2 3 5 6 7 0 0 0 8 9 0 0 0 0 with the table id vector: table id 0 0 0 1 0 To do this, for each table: Update the corresponding entries in a table id vector (of size the domain as well) with the table ID of the table. Copy the entries from the table to new rows in the corresponding columns of the concatenated table. Fill in any unused columns with 0 (to match the dummy value) Pad the end of the concatened table with the dummy value. Pad the end of the table id vector with 0s. pre-compute polynomial and evaluation form for the look up tables pre-compute polynomial and evaluation form for the table IDs, only if a table with an ID different from zero was used.","breadcrumbs":"Kimchi » Lookup Index","id":"258","title":"Lookup Index"},"259":{"body":"Both the prover and the verifier index, besides the common parts described above, are made out of pre-computations which can be used to speed up the protocol. These pre-computations are optimizations, in the context of normal proofs, but they are necessary for recursion. pub struct ProverIndex> { /// constraints system polynomials #[serde(bound = \"ConstraintSystem: Serialize + DeserializeOwned\")] pub cs: ConstraintSystem, /// The symbolic linearization of our circuit, which can compile to concrete types once certain values are learned in the protocol. #[serde(skip)] pub linearization: Linearization>, Column>, /// The mapping between powers of alpha and constraints #[serde(skip)] pub powers_of_alpha: Alphas, /// polynomial commitment keys #[serde(skip)] #[serde(bound(deserialize = \"OpeningProof::SRS: Default\"))] pub srs: Arc, /// maximal size of polynomial section pub max_poly_size: usize, #[serde(bound = \"ColumnEvaluations: Serialize + DeserializeOwned\")] pub column_evaluations: ColumnEvaluations, /// The verifier index corresponding to this prover index #[serde(skip)] pub verifier_index: Option>, /// The verifier index digest corresponding to this prover index #[serde_as(as = \"Option\")] pub verifier_index_digest: Option,\n}","breadcrumbs":"Kimchi » Prover Index","id":"259","title":"Prover Index"},"26":{"body":"The homomorphic commitment commit(x,r)=xG+rH described above is known as a Pedersen commitment. If you remove the rH term you get a non-hiding commitment, called a Pedersen hash . Both rely on the ECDLP hardness assumption. This means that, at least theoretically, you might be lucky (or have a quantum computer) and figure out that H=hG, which would allow you to find different values x′ and h′ to open the commitment. We say that pedersen commitments are computationally binding and not unconditionally binding. For example, you could express c=xG+rH alternatively as c=xG+rhG=(x+rh)G and compute a satisfying opening pair x′=rh and r′=hx​. On the other hand, Pedersen commitments are unconditionally hiding , as there is no way (even with a magic computer) to reveal what x is without knowing r. Lack of perfect binding is the reason why most of the \"proofs\" we will see later in this book are not referred to as proofs, but instead are referred to as arguments of knowledge (although we may care little about this distinction). Just remember that you need perfect binding to be called a proof. Interestingly, it is impossible to have a commitment scheme that has both perfect hiding and perfect binding. To recap, in cryptography the following distinctions are important Perfect. The property that an algorithm is statistically sound without hardness assumptions, also known as unconditional or statistical soundness. Computational. The algorithm relies on a hardness assumption or computational limitation for soundness. Thus, said another way, Pedersen commitments provide perfect hiding and computational binding.","breadcrumbs":"Commitments » Pedersen commitments","id":"26","title":"Pedersen commitments"},"260":{"body":"Same as the prover index, we have a number of pre-computations as part of the verifier index. #[serde_as]\n#[derive(Serialize, Deserialize, Debug, Clone)]\npub struct LookupVerifierIndex { pub joint_lookup_used: bool, #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub lookup_table: Vec>, #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub lookup_selectors: LookupSelectors>, /// Table IDs for the lookup values. /// This may be `None` if all lookups originate from table 0. #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub table_ids: Option>, /// Information about the specific lookups used pub lookup_info: LookupInfo, /// An optional selector polynomial for runtime tables #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub runtime_tables_selector: Option>,\n} #[serde_as]\n#[derive(Serialize, Deserialize, Debug, Clone)]\npub struct VerifierIndex> { /// evaluation domain #[serde_as(as = \"o1_utils::serialization::SerdeAs\")] pub domain: D, /// maximal size of polynomial section pub max_poly_size: usize, /// the number of randomized rows to achieve zero knowledge pub zk_rows: u64, /// polynomial commitment keys #[serde(skip)] #[serde(bound(deserialize = \"OpeningProof::SRS: Default\"))] pub srs: Arc, /// number of public inputs pub public: usize, /// number of previous evaluation challenges, for recursive proving pub prev_challenges: usize, // index polynomial commitments /// permutation commitment array #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub sigma_comm: [PolyComm; PERMUTS], /// coefficient commitment array #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub coefficients_comm: [PolyComm; COLUMNS], /// coefficient commitment array #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub generic_comm: PolyComm, // poseidon polynomial commitments /// poseidon constraint selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub psm_comm: PolyComm, // ECC arithmetic polynomial commitments /// EC addition selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub complete_add_comm: PolyComm, /// EC variable base scalar multiplication selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub mul_comm: PolyComm, /// endoscalar multiplication selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub emul_comm: PolyComm, /// endoscalar multiplication scalar computation selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub endomul_scalar_comm: PolyComm, /// RangeCheck0 polynomial commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub range_check0_comm: Option>, /// RangeCheck1 polynomial commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub range_check1_comm: Option>, /// Foreign field addition gates polynomial commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub foreign_field_add_comm: Option>, /// Foreign field multiplication gates polynomial commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub foreign_field_mul_comm: Option>, /// Xor commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub xor_comm: Option>, /// Rot commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub rot_comm: Option>, /// wire coordinate shifts #[serde_as(as = \"[o1_utils::serialization::SerdeAs; PERMUTS]\")] pub shift: [G::ScalarField; PERMUTS], /// zero-knowledge polynomial #[serde(skip)] pub permutation_vanishing_polynomial_m: OnceCell>, // TODO(mimoo): isn't this redundant with domain.d1.group_gen ? /// domain offset for zero-knowledge #[serde(skip)] pub w: OnceCell, /// endoscalar coefficient #[serde(skip)] pub endo: G::ScalarField, #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub lookup_index: Option>, #[serde(skip)] pub linearization: Linearization>, Column>, /// The mapping between powers of alpha and constraints #[serde(skip)] pub powers_of_alpha: Alphas,\n}","breadcrumbs":"Kimchi » Verifier Index","id":"260","title":"Verifier Index"},"261":{"body":"Originally, kimchi is based on an interactive protocol that was transformed into a non-interactive one using the Fiat-Shamir transform. For this reason, it can be useful to visualize the high-level interactive protocol before the transformation: sequenceDiagram participant Prover participant Verifier Note over Prover,Verifier: Prover produces commitments to secret polynomials Prover->>Verifier: public input & witness commitment Verifier->>Prover: beta & gamma Prover->>Verifier: permutation commitment opt lookup Prover->>Verifier: sorted Prover->>Verifier: aggreg end Note over Prover,Verifier: Prover produces commitment to quotient polynomial Verifier->>Prover: alpha Prover->>Verifier: quotient commitment Note over Prover,Verifier: Verifier produces an evaluation point Verifier->>Prover: zeta Note over Verifier: change of verifier (change of sponge) opt recursion Prover->>Verifier: recursion challenges end Note over Prover,Verifier: Prover provides needed evaluations for the linearization Prover->>Verifier: ft(zeta * omega) Prover->>Verifier: negated public input p(zeta) & p(zeta * omega) Prover->>Verifier: the 15 registers w_i(zeta) & w_i(zeta * omega) Prover->>Verifier: permutation poly z(zeta) & z(zeta * omega) Prover->>Verifier: the 6 sigmas s_i(zeta) & s_i(zeta * omega) Prover->>Verifier: the 15 coefficients c_i(zeta) & c_i(zeta * omega) opt lookup Prover->>Verifier: sorted(zeta) & sorted(zeta * omega) Prover->>Verifier: aggreg(zeta) & aggreg(zeta * omega) Prover->>Verifier: table(zeta) & table(zeta * omega) end Prover->>Verifier: the generic selector gen(zeta) & gen(zeta * omega) Prover->>Verifier: the poseidon selector pos(zeta) & pos(zeta * omega) Note over Prover,Verifier: Batch verification of evaluation proofs Verifier->>Prover: v, u Note over Verifier: change of verifier (change of sponge) Prover->>Verifier: aggregated evaluation proof (involves more interaction) The Fiat-Shamir transform simulates the verifier messages via a hash function that hashes the transcript of the protocol so far before outputing verifier messages. You can find these operations under the proof creation and proof verification algorithms as absorption and squeezing of values with the sponge.","breadcrumbs":"Kimchi » Proof Construction & Verification","id":"261","title":"Proof Construction & Verification"},"262":{"body":"A proof consists of the following data structures: /// Evaluations of a polynomial at 2 points\n#[serde_as]\n#[derive(Copy, Clone, Serialize, Deserialize, Default, Debug)]\n#[cfg_attr( feature = \"ocaml_types\", derive(ocaml::IntoValue, ocaml::FromValue, ocaml_gen::Struct)\n)]\n#[serde(bound( serialize = \"Vec: serde_with::SerializeAs\", deserialize = \"Vec: serde_with::DeserializeAs<'de, Evals>\"\n))]\npub struct PointEvaluations { /// Evaluation at the challenge point zeta. #[serde_as(as = \"Vec\")] pub zeta: Evals, /// Evaluation at `zeta . omega`, the product of the challenge point and the group generator. #[serde_as(as = \"Vec\")] pub zeta_omega: Evals,\n} // TODO: this should really be vectors here, perhaps create another type for chunked evaluations?\n/// Polynomial evaluations contained in a `ProverProof`.\n/// - **Chunked evaluations** `Field` is instantiated with vectors with a length\n/// that equals the length of the chunk\n/// - **Non chunked evaluations** `Field` is instantiated with a field, so they\n/// are single-sized#[serde_as]\n#[serde_as]\n#[derive(Debug, Clone, Serialize, Deserialize)]\npub struct ProofEvaluations { /// public input polynomials pub public: Option, /// witness polynomials pub w: [Evals; COLUMNS], /// permutation polynomial pub z: Evals, /// permutation polynomials /// (PERMUTS-1 evaluations because the last permutation is only used in /// commitment form) pub s: [Evals; PERMUTS - 1], /// coefficient polynomials pub coefficients: [Evals; COLUMNS], /// evaluation of the generic selector polynomial pub generic_selector: Evals, /// evaluation of the poseidon selector polynomial pub poseidon_selector: Evals, /// evaluation of the elliptic curve addition selector polynomial pub complete_add_selector: Evals, /// evaluation of the elliptic curve variable base scalar multiplication /// selector polynomial pub mul_selector: Evals, /// evaluation of the endoscalar multiplication selector polynomial pub emul_selector: Evals, /// evaluation of the endoscalar multiplication scalar computation selector /// polynomial pub endomul_scalar_selector: Evals, // Optional gates /// evaluation of the RangeCheck0 selector polynomial pub range_check0_selector: Option, /// evaluation of the RangeCheck1 selector polynomial pub range_check1_selector: Option, /// evaluation of the ForeignFieldAdd selector polynomial pub foreign_field_add_selector: Option, /// evaluation of the ForeignFieldMul selector polynomial pub foreign_field_mul_selector: Option, /// evaluation of the Xor selector polynomial pub xor_selector: Option, /// evaluation of the Rot selector polynomial pub rot_selector: Option, // lookup-related evaluations /// evaluation of lookup aggregation polynomial pub lookup_aggregation: Option, /// evaluation of lookup table polynomial pub lookup_table: Option, /// evaluation of lookup sorted polynomials pub lookup_sorted: [Option; 5], /// evaluation of runtime lookup table polynomial pub runtime_lookup_table: Option, // lookup selectors /// evaluation of the runtime lookup table selector polynomial pub runtime_lookup_table_selector: Option, /// evaluation of the Xor range check pattern selector polynomial pub xor_lookup_selector: Option, /// evaluation of the Lookup range check pattern selector polynomial pub lookup_gate_lookup_selector: Option, /// evaluation of the RangeCheck range check pattern selector polynomial pub range_check_lookup_selector: Option, /// evaluation of the ForeignFieldMul range check pattern selector /// polynomial pub foreign_field_mul_lookup_selector: Option,\n} /// Commitments linked to the lookup feature\n#[serde_as]\n#[derive(Debug, Clone, Serialize, Deserialize)]\n#[serde(bound = \"G: ark_serialize::CanonicalDeserialize + ark_serialize::CanonicalSerialize\")]\npub struct LookupCommitments { /// Commitments to the sorted lookup table polynomial (may have chunks) pub sorted: Vec>, /// Commitment to the lookup aggregation polynomial pub aggreg: PolyComm, /// Optional commitment to concatenated runtime tables pub runtime: Option>,\n} /// All the commitments that the prover creates as part of the proof.\n#[serde_as]\n#[derive(Debug, Clone, Serialize, Deserialize)]\n#[serde(bound = \"G: ark_serialize::CanonicalDeserialize + ark_serialize::CanonicalSerialize\")]\npub struct ProverCommitments { /// The commitments to the witness (execution trace) pub w_comm: [PolyComm; COLUMNS], /// The commitment to the permutation polynomial pub z_comm: PolyComm, /// The commitment to the quotient polynomial pub t_comm: PolyComm, /// Commitments related to the lookup argument pub lookup: Option>,\n} /// The proof that the prover creates from a\n/// [ProverIndex](super::prover_index::ProverIndex) and a `witness`.\n#[serde_as]\n#[derive(Debug, Clone, Serialize, Deserialize)]\n#[serde(bound = \"G: ark_serialize::CanonicalDeserialize + ark_serialize::CanonicalSerialize\")]\npub struct ProverProof { /// All the polynomial commitments required in the proof pub commitments: ProverCommitments, /// batched commitment opening proof #[serde(bound( serialize = \"OpeningProof: Serialize\", deserialize = \"OpeningProof: Deserialize<'de>\" ))] pub proof: OpeningProof, /// Two evaluations over a number of committed polynomials pub evals: ProofEvaluations>>, /// Required evaluation for [Maller's /// optimization](https://o1-labs.github.io/proof-systems/kimchi/maller_15.html#the-evaluation-of-l) #[serde_as(as = \"o1_utils::serialization::SerdeAs\")] pub ft_eval1: G::ScalarField, /// The challenges underlying the optional polynomials folded into the proof pub prev_challenges: Vec>,\n} /// A struct to store the challenges inside a `ProverProof`\n#[serde_as]\n#[derive(Debug, Clone, Deserialize, Serialize)]\n#[serde(bound = \"G: ark_serialize::CanonicalDeserialize + ark_serialize::CanonicalSerialize\")]\npub struct RecursionChallenge\nwhere G: AffineRepr,\n{ /// Vector of scalar field elements #[serde_as(as = \"Vec\")] pub chals: Vec, /// Polynomial commitment pub comm: PolyComm,\n} The following sections specify how a prover creates a proof, and how a verifier validates a number of proofs.","breadcrumbs":"Kimchi » Proof Structure","id":"262","title":"Proof Structure"},"263":{"body":"To create a proof, the prover expects: A prover index, containing a representation of the circuit (and optionaly pre-computed values to be used in the proof creation). The (filled) registers table, representing parts of the execution trace of the circuit. Note The public input is expected to be passed in the first Public rows of the registers table. The following constants are set: EVAL_POINTS = 2. This is the number of points that the prover has to evaluate their polynomials at. ($\\zeta$ and $\\zeta\\omega$ where $\\zeta$ will be deterministically generated.) ZK_ROWS = 3. This is the number of rows that will be randomized to provide zero-knowledgeness. Note that it only needs to be greater or equal to the number of evaluations (2) in the protocol. Yet, it contains one extra row to take into account the last constraint (final value of the permutation accumulator). (TODO: treat the final constraint separately so that ZK_ROWS = 2) The prover then follows the following steps to create the proof: Ensure we have room in the witness for the zero-knowledge rows. We currently expect the witness not to be of the same length as the domain, but instead be of the length of the (smaller) circuit. If we cannot add zk_rows rows to the columns of the witness before reaching the size of the domain, abort. Pad the witness columns with Zero gates to make them the same length as the domain. Then, randomize the last zk_rows of each columns. Setup the Fq-Sponge. Absorb the digest of the VerifierIndex. Absorb the commitments of the previous challenges with the Fq-sponge. Compute the negated public input polynomial as the polynomial that evaluates to $-p_i$ for the first public_input_size values of the domain, and $0$ for the rest. Commit (non-hiding) to the negated public input polynomial. Absorb the commitment to the public polynomial with the Fq-Sponge. Note: unlike the original PLONK protocol, the prover also provides evaluations of the public polynomial to help the verifier circuit. This is why we need to absorb the commitment to the public polynomial at this point. Commit to the witness columns by creating COLUMNS hidding commitments. Note: since the witness is in evaluation form, we can use the commit_evaluation optimization. Absorb the witness commitments with the Fq-Sponge. Compute the witness polynomials by interpolating each COLUMNS of the witness. As mentioned above, we commit using the evaluations form rather than the coefficients form so we can take advantage of the sparsity of the evaluations (i.e., there are many 0 entries and entries that have less-than-full-size field elemnts.) If using lookup: if using runtime table: check that all the provided runtime tables have length and IDs that match the runtime table configuration of the index we expect the given runtime tables to be sorted as configured, this makes it easier afterwards calculate the contribution to the second column of the lookup table (the runtime vector) If queries involve a lookup table with multiple columns then squeeze the Fq-Sponge to obtain the joint combiner challenge $j'$, otherwise set the joint combiner challenge $j'$ to $0$. Derive the scalar joint combiner $j$ from $j'$ using the endomorphism (TODO: specify) If multiple lookup tables are involved, set the table_id_combiner as the $j^i$ with $i$ the maximum width of any used table. Essentially, this is to add a last column of table ids to the concatenated lookup tables. Compute the dummy lookup value as the combination of the last entry of the XOR table (so (0, 0, 0)). Warning: This assumes that we always use the XOR table when using lookups. Compute the lookup table values as the combination of the lookup table entries. Compute the sorted evaluations. Randomize the last EVALS rows in each of the sorted polynomials in order to add zero-knowledge to the protocol. Commit each of the sorted polynomials. Absorb each commitments to the sorted polynomials. Sample $\\beta$ with the Fq-Sponge. Sample $\\gamma$ with the Fq-Sponge. If using lookup: Compute the lookup aggregation polynomial. Commit to the aggregation polynomial. Absorb the commitment to the aggregation polynomial with the Fq-Sponge. Compute the permutation aggregation polynomial $z$. Commit (hidding) to the permutation aggregation polynomial $z$. Absorb the permutation aggregation polynomial $z$ with the Fq-Sponge. Sample $\\alpha'$ with the Fq-Sponge. Derive $\\alpha$ from $\\alpha'$ using the endomorphism (TODO: details) TODO: instantiate alpha? Compute the quotient polynomial (the $t$ in $f = Z_H \\cdot t$). The quotient polynomial is computed by adding all these polynomials together: the combined constraints for all the gates the combined constraints for the permutation TODO: lookup the negated public polynomial and by then dividing the resulting polynomial with the vanishing polynomial $Z_H$. TODO: specify the split of the permutation polynomial into perm and bnd? commit (hiding) to the quotient polynomial $t$ Absorb the commitment of the quotient polynomial with the Fq-Sponge. Sample $\\zeta'$ with the Fq-Sponge. Derive $\\zeta$ from $\\zeta'$ using the endomorphism (TODO: specify) If lookup is used, evaluate the following polynomials at $\\zeta$ and $\\zeta \\omega$: the aggregation polynomial the sorted polynomials the table polynonial Chunk evaluate the following polynomials at both $\\zeta$ and $\\zeta \\omega$: $s_i$ $w_i$ $z$ lookup (TODO, see this issue ) generic selector poseidon selector By \"chunk evaluate\" we mean that the evaluation of each polynomial can potentially be a vector of values. This is because the index's max_poly_size parameter dictates the maximum size of a polynomial in the protocol. If a polynomial $f$ exceeds this size, it must be split into several polynomials like so: $$f(x) = f_0(x) + x^n f_1(x) + x^{2n} f_2(x) + \\cdots$$ And the evaluation of such a polynomial is the following list for $x \\in {\\zeta, \\zeta\\omega}$: $$(f_0(x), f_1(x), f_2(x), \\ldots)$$ TODO: do we want to specify more on that? It seems unnecessary except for the t polynomial (or if for some reason someone sets that to a low value) Evaluate the same polynomials without chunking them (so that each polynomial should correspond to a single value this time). Compute the ft polynomial. This is to implement Maller's optimization . construct the blinding part of the ft polynomial commitment see this section Evaluate the ft polynomial at $\\zeta\\omega$ only. Setup the Fr-Sponge Squeeze the Fq-sponge and absorb the result with the Fr-Sponge. Absorb the previous recursion challenges. Compute evaluations for the previous recursion challenges. Absorb the unique evaluation of ft: $ft(\\zeta\\omega)$. Absorb all the polynomial evaluations in $\\zeta$ and $\\zeta\\omega$: the public polynomial z generic selector poseidon selector the 15 register/witness 6 sigmas evaluations (the last one is not evaluated) Sample $v'$ with the Fr-Sponge Derive $v$ from $v'$ using the endomorphism (TODO: specify) Sample $u'$ with the Fr-Sponge Derive $u$ from $u'$ using the endomorphism (TODO: specify) Create a list of all polynomials that will require evaluations (and evaluation proofs) in the protocol. First, include the previous challenges, in case we are in a recursive prover. Then, include: the negated public polynomial the ft polynomial the permutation aggregation polynomial z polynomial the generic selector the poseidon selector the 15 registers/witness columns the 6 sigmas the optional gates optionally, the runtime table if using lookup: add the lookup sorted polynomials add the lookup aggreg polynomial add the combined table polynomial if present, add the runtime table polynomial the lookup selectors Create an aggregated evaluation proof for all of these polynomials at $\\zeta$ and $\\zeta\\omega$ using $u$ and $v$.","breadcrumbs":"Kimchi » Proof Creation","id":"263","title":"Proof Creation"},"264":{"body":"TODO: we talk about batch verification, but is there an actual batch operation? It seems like we're just verifying an aggregated opening proof We define two helper algorithms below, used in the batch verification of proofs. Fiat-Shamir argument We run the following algorithm: Setup the Fq-Sponge. This sponge mostly absorbs group Absorb the digest of the VerifierIndex. Absorb the commitments of the previous challenges with the Fq-sponge. Absorb the commitment of the public input polynomial with the Fq-Sponge. Absorb the commitments to the registers / witness columns with the Fq-Sponge. If lookup is used: If it involves queries to a multiple-column lookup table, then squeeze the Fq-Sponge to obtain the joint combiner challenge $j'$, otherwise set the joint combiner challenge $j'$ to $0$. Derive the scalar joint combiner challenge $j$ from $j'$ using the endomorphism. (TODO: specify endomorphism) absorb the commitments to the sorted polynomials. Sample the first permutation challenge $\\beta$ with the Fq-Sponge. Sample the second permutation challenge $\\gamma$ with the Fq-Sponge. If using lookup, absorb the commitment to the aggregation lookup polynomial. Absorb the commitment to the permutation trace with the Fq-Sponge. Sample the quotient challenge $\\alpha'$ with the Fq-Sponge. Derive $\\alpha$ from $\\alpha'$ using the endomorphism (TODO: details). Enforce that the length of the $t$ commitment is of size 7. Absorb the commitment to the quotient polynomial $t$ into the argument. Sample $\\zeta'$ with the Fq-Sponge. Derive $\\zeta$ from $\\zeta'$ using the endomorphism (TODO: specify). Setup the Fr-Sponge. This sponge absorbs elements from Squeeze the Fq-sponge and absorb the result with the Fr-Sponge. Absorb the previous recursion challenges. Compute evaluations for the previous recursion challenges. Evaluate the negated public polynomial (if present) at $\\zeta$ and $\\zeta\\omega$. NOTE: this works only in the case when the poly segment size is not smaller than that of the domain. Absorb the unique evaluation of ft: $ft(\\zeta\\omega)$. Absorb all the polynomial evaluations in $\\zeta$ and $\\zeta\\omega$: the public polynomial z generic selector poseidon selector the 15 register/witness 6 sigmas evaluations (the last one is not evaluated) Sample the \"polyscale\" $v'$ with the Fr-Sponge. Derive $v$ from $v'$ using the endomorphism (TODO: specify). Sample the \"evalscale\" $u'$ with the Fr-Sponge. Derive $u$ from $u'$ using the endomorphism (TODO: specify). Create a list of all polynomials that have an evaluation proof. Compute the evaluation of $ft(\\zeta)$. Partial verification For every proof we want to verify, we defer the proof opening to the very end. This allows us to potentially batch verify a number of partially verified proofs. Essentially, this steps verifies that $f(\\zeta) = t(\\zeta) * Z_H(\\zeta)$. Check the length of evaluations inside the proof. Commit to the negated public input polynomial. Run the Fiat-Shamir argument . Combine the chunked polynomials' evaluations (TODO: most likely only the quotient polynomial is chunked) with the right powers of $\\zeta^n$ and $(\\zeta * \\omega)^n$. Compute the commitment to the linearized polynomial $f$. To do this, add the constraints of all of the gates, of the permutation, and optionally of the lookup. (See the separate sections in the constraints section.) Any polynomial should be replaced by its associated commitment, contained in the verifier index or in the proof, unless a polynomial has its evaluation provided by the proof in which case the evaluation should be used in place of the commitment. Compute the (chuncked) commitment of $ft$ (see Maller's optimization ). List the polynomial commitments, and their associated evaluations, that are associated to the aggregated evaluation proof in the proof: recursion public input commitment ft commitment (chunks of it) permutation commitment index commitments that use the coefficients witness commitments coefficient commitments sigma commitments optional gate commitments lookup commitments Batch verification of proofs Below, we define the steps to verify a number of proofs (each associated to a verifier index ). You can, of course, use it to verify a single proof. If there's no proof to verify, the proof validates trivially. Ensure that all the proof's verifier index have a URS of the same length. (TODO: do they have to be the same URS though? should we check for that?) Validate each proof separately following the partial verification steps. Use the PolyCom.verify to verify the partially evaluated proofs.","breadcrumbs":"Kimchi » Proof Verification","id":"264","title":"Proof Verification"},"265":{"body":"commit_evaluation: TODO","breadcrumbs":"Kimchi » Optimizations","id":"265","title":"Optimizations"},"266":{"body":"TODO","breadcrumbs":"Kimchi » Security Considerations","id":"266","title":"Security Considerations"},"267":{"body":"Note Note that the URS is called SRS in our codebase at the moment. SRS is incorrect, as it is used in the presence of a trusted setup (which kimchi does not have). This needs to be fixed. The URS comprises of: Gs: an arbitrary ordered list of curve points that can be used to commit to a polynomial in a non-hiding way. H: a blinding curve point that can be used to add hidding to a polynomial commitment scheme. The URS is generated deterministically, and thus can be rederived if not stored. As different circuits of different size do not share the same domain, and the size of the domain is directly equal to the number of Gs curve points we use to commit to polynomials, we often want to share the same URS of maximum size across many circuits (constraint systems). Circuit associated with smaller domains will simply use a truncation of the shared URS. Note A circuit's domain is generated as the first power of 2 that is equal, or larger, to the number of gates used in the circuit (including zero-knowledge gates).","breadcrumbs":"Universal Reference String (URS) » Universal Reference String (URS)","id":"267","title":"Universal Reference String (URS)"},"268":{"body":"TODO: specify this part","breadcrumbs":"Universal Reference String (URS) » Group map","id":"268","title":"Group map"},"269":{"body":"The following pseudo-code specifies how the Gs curve points, as well as the H curve point, are generated. def create(depth): # generators Gs = [] for i in range(0, depth): digest = Blake2b512.hash(be_encode(i)) Gs.push(point_of_random_bytes(digest)) # blinding point digest = (b\"srs_misc\" || be_encode(0)) H = point_of_random_bytes(digest) # return (Gs, H) def point_of_random_bytes(random_bytes): # packing in bit-representation const N: usize = 31 let mut bits = [false; 8 * N] for i in range(0, N) { for j in range(0, 8) { bits[8 * i + j] = (random_bytes[i] >> j) & 1 == 1; } } let n = BigInt::from_bits_be(&bits); let t = G::BaseField::from_repr(n) return map.to_group(t) TODO: specify map.to_group in the previous section.","breadcrumbs":"Universal Reference String (URS) » Curve points to commit to a polynomial","id":"269","title":"Curve points to commit to a polynomial"},"27":{"body":"We can commit to several values x1​,⋯,xn​ by sending separate Pedersen commitments to all of these values as such: x1​G+r1​H,⋮xn​G+rn​H But we can instead batch/aggregate all of these commitments together as a single commitment: x1​G1​+⋯+xn​Gn​+rH with G1​,⋯,Gn​,H independent bases with unknown discrete logarithms. If you represent xs and the Gs as two vectors x=(x1​,⋯,xn​) and G=(G1​,⋯,Gn​), we can quickly write the previous statement as an inner product xG+rH Vector commitments (sometimes referred to as multi-commitments) are a powerful construction because an arbitrarily large vector can be committed with a single curve point. The naive approach to constructing an opening proof for a length n vector commitment has size O(n). It is simply the tuple (x1​,…,xn​,r). As we will see later, opening proofs for vector commitments is an interesting topic and there is a much more efficient algorithm.","breadcrumbs":"Commitments » Vector commitments","id":"27","title":"Vector commitments"},"270":{"body":"As there is no limit to the number of commitment curve points you can generate, we only provide the first three ones to serve as test vectors. TODO: specify the encoding","breadcrumbs":"Universal Reference String (URS) » URS values in practice","id":"270","title":"URS values in practice"},"271":{"body":"Gs . G0 = (x=121C4426885FD5A9701385AAF8D43E52E7660F1FC5AFC5F6468CC55312FC60F8, y=21B439C01247EA3518C5DDEB324E4CB108AF617780DDF766D96D3FD8AB028B70)\nG1 = (x=26C9349FF7FB4AB230A6F6AEF045F451FBBE9B37C43C3274E2AA4B82D131FD26, y=1996274D67EC0464C51F79CCFA1F511C2AABB666ABE67733EE8185B71B27A504)\nG2 = (x=26985F27306586711466C5B2C28754AA62FE33516D75CEF1F7751F1A169713FD, y=2E8930092FE6A18B331CE0E6E27B413AA18E76394F18A2835DA9FAE10AA3229D) H : H = (x=092060386301C999AAB4F263757836369CA27975E28BC7A8E5B2CE5B26262201, y=314FC4D83AE66A509F9D41BE6165F2606A209A9B5805EE85CE20249C5EBCBE26)","breadcrumbs":"Universal Reference String (URS) » Vesta","id":"271","title":"Vesta"},"272":{"body":"G0 = (x363D83141FD1E0540718FADBA7278ABAEEDB46D7A3F050F2CFF1DF4F300C9C30, y=034C68F4079B4F338A19BE2D7BFA44B395C65B9790DD273F361327446C778764)\nG1 = (x2CC40B77D87665244AE5EB5304E8744004C80061AD08476A0F0656C13134EA45, y=28146EC860159DB55CB5EA5B14F0AA2F8751DEDFE0DDAFD1C313B15575C4B4AC)\nG2 = (x2808BC21BEB90314377BF6130285FABE6CE4B8A4457FB25BC95EBA0083DF27E3, y=1E04E53DD6395FAB8018D7FE98F9C7FAB39C40BFBE48589626A7B8532728B002) H : H = (x221B959DACD2052AAE26193FCA36B53279866A4FBBAB0D5A2F828B5FD7778201, y=058C8F1105CAE57F4891EADC9B85C8954E5067190E155E61D66855ACE69C16C0)","breadcrumbs":"Universal Reference String (URS) » Pallas","id":"272","title":"Pallas"},"273":{"body":"Pickles is Mina’s inductive zk-SNARK composition system. It lets you construct proofs with zk-SNARKs and combine them in flexible ways to deliver incremental verifiable computation. Pickles uses a pair of amicable curves called Pasta in order to deliver incremental verifiable computation efficiently. These curves are referred to as \"tick\" and \"tock\" within the Mina source code. Tick - Vesta (a.k.a. Step), constraint domain size 218 (block and transaction proofs) Tock - Pallas (a.k.a. Wrap), constraint domain size 217 (signatures) The Tock prover does less (only performs recursive verifications and no other logic), so it requires fewer constraints and has a smaller domain size. Internally Pickles refers to Tick and Tock as Step and Wrap, respectively. Tock is used to prove the verification of a Tick proof and outputs a Tick proof. Tick is used to prove the verification of a Tock proof and outputs a Tock proof. In other words, Provetock​(Verify(Tick))=Tickproof​ Provetick​(Verify(Tock))=Tockproof​ Both Tick and Tock can verify at most 2 proofs of the opposite kind, though, theoretically more is possible. Currently, in Mina we have the following situation. Every Tock always wraps 1 Tick proof, such as Tick proofs can verify 2 Tock proofs Blockchain SNARK takes previous blockchain SNARK proof and a transaction proof Verifying two Tock transaction proofs","breadcrumbs":"Pickles » Pickles","id":"273","title":"Pickles"},"274":{"body":"vrf is appendix C of https://eprint.iacr.org/2017/573.pdf","breadcrumbs":"Consensus » Consensus","id":"274","title":"Consensus"},"28":{"body":"To construct SNARKs we use polynomial commitments. A polynomial commitment scheme for a field F (or it could even be a ring) is a way of committing to a polynomial f∈F[x] to get a commitment c, in such a way that for any α∈F, you can provide y=f(α), along with an \"opening proof\" π that proves that the polynomial committed to in c equals y when evaluated at α. In other words, it is a type of commitment C, a type of randomness R, a type of opening proof P along with algorithms commit:F[x]×R→Copen:C×F×(F[x]×R)→F×Pverify:C×(F×P)→Bool such that for any f∈F[x],r∈R,α∈F , we have c:=commit(f,r)verify(c,open(c,α,(f,r)))=true and if b=f(α) then it is not possible to compute π∈P such that verify(c,(b,π))=true In other words, if b=f(α) then every π which is feasible to compute results in verify(c,(b,π))=false. One thing that's pretty cool is that because polynomial commitment schemes let you construct zk-SNARKs, polynomial commitment schemes imply commitment schemes with arbitrary opening functionality. TODO","breadcrumbs":"Commitments » Polynomial commitments","id":"28","title":"Polynomial commitments"},"29":{"body":"All known constructions of polynomial commitment schemes are a bit complicated. The easiest to describe is called the Kate (pronounced \"kah-TAY\") scheme, also known as \"KZG\". It requires a \"prime-order group with a pairing\", which is three groups G1​,G2​,GT​ of prime order p (hence, all isomorphic cyclic groups) together with a function e:G1​×G2​→GT​ such that for any a1​,a2​∈Z, g1​∈G1​, g2​∈G2​, we have e(a1​⋅g1​,a2​⋅g2​)=a1​a2​⋅e(g1​,g2​) e is called a \"pairing\" or a \"bilinear pairing\". What this lets us do is \"multiply in the scalar\" but only once. Fix a degree bound d on the polynomials we would like to be able to commit to. The KZG scheme, will let us commit to polynomials in Fp​[x]| |<------ b | OT Prot. | | |-------> x_b +----------+ Due to a result of Impagliazzo and Rudich in this paper , it is very unlikely that OT is possible without the use of public-key cryptography. However, OT can be efficiently extended. That is, starting with a small number of base OTs, one could create many more OTs with only symmetric primitives. The seminal work of IKNP03 presented a very efficient protocol for extending OTs, requiring only black-box use of symmetric primitives and λ base OTs, where λ is the security parameter. This doc focuses on the family of protocols inspired by IKNP.","breadcrumbs":"Two Party Computation » Oblivious Transfer » Oblivious Transfer","id":"66","title":"Oblivious Transfer"},"67":{"body":"We introduce the CO15 OT as the base OT. Notation . The protocol is described over an additive group (G,G,p,+) of prime order p generated by G. Denote H:(G×G)×G→{0,1}λ as a key-derivation function to extract a λ-bit key from group elements. Note that H here is different from the tweakable hash function in garbled circuit. In the CO15 protocol, the sender holds n 128-bit strings {(Xi0​,Xi1​)}i∈[n]​ and the receiver holds n-bit string b∈{0,1}n. The protocol is as follows. The sender samples y←Fp​, and computes S=y⋅G and T=y⋅S. The sender sends S to the receiver, who aborts if S∈/G. For i∈[n], the receiver samples xi​←Fp​, and computes: Ri​=bi​S+xi​G The receiver sends {Ri​}i∈[n]​ to the sender, who aborts if Ri​∈/G. For i∈[n], the sender computes ki0​=H(S,Ri​,yRi​), ki1​=H(S,Ri​,yRi​−T) and sends {(ci0​=ki0​⊕Xi0​,ci1​=ki1​⊕Xi1​)}i∈[n]​ to the receiver. For i∈[n], the receiver computes ki​=H(S,Ri​,xi​S), and outputs cibi​​⊕ki​ Correctness . The receiver always computes the hash of xi​S=(xi​y)G. The sender sends the hashes of yRi​ and yRi​−T. If bi​=0, ki0​ is the hash of yRi​=(yxi​)G, then the receive will get Xi0​. if bi​=1, ki1​ is the hash of yRi​−T=y(S+xi​G)−T=(yxi​)G, then the receiver will get Xi1​. Security . We refer the security analysis to CO15 .","breadcrumbs":"Two Party Computation » Oblivious Transfer » Base OT » Base OT","id":"67","title":"Base OT"},"68":{"body":"To meet the large-scale demand of OTs, OT extensions can be used. An OT extension protocol works by running a small number of base OTs that are used as a base for obtaining many OTs via the use of cheap symmetric cryptographic operations only. This is conceptually similar to hybrid encryption. In this subsection, we focus on the maliciously secure OT extension presented in KOS15 . To be compatible with the garbled circuit optimizations, Correlated OT (COT) is considered here. COT is a variant of OT, where the sender chooses a secret string Δ and the receiver has a choice bit b. After the protocol, then sender obtains two strings X0​=X and X1​=X⊕Δ and the receiver obtains Xb​, where X is a random string. Sender Receiver +----------+ Delta ------->| |<------ b | COT Prot.| x_0 = x, x_1 = x + Delta <-------| |-------> x_b +----------+ Notation . Let λ be the security parameter, we use λ=128 in zkOracles. Identify F2λ​ with the finite field F2λ​. Use \"⋅\" for multiplication in F2λ​ and \"⋆\" for the component-wise product in F2λ​. Given a matrix A, denote its rows by subindices ai​ and its columns by superindices ak. Use v[i] to denote the i-th entry of v. The KOS15 OT protocol to generate ℓ(≫λ) OTs is described as follows. Initialize The receiver samples λ pairs of λ-bit seeds {(k0i​,k1i​)}i=1λ​. The sender choose a uniformly random λ-bit string Δ. The two parties calls the base OT with inputs Δ and {(k0i​,k1i​)}i=1λ​. Note that the sender acts as the receiver in the base OT, and the receiver acts as the sender in the base OT. The sender obtains kΔi​i​ for i∈[λ]. Extend The receiver takes as input the choice bits x1​,...,xℓ​, defines ℓ′=ℓ+λ+s, where s is the statistic parameter, and we set s=40 in zkOracles. Let x=x1​∥x2​∥...∥xℓ​∥x′∈F2ℓ′​, with x′∈F2ℓ′−ℓ​ uniformly chosen. The receiver defines vectors x1​,...,xℓ′​ as xi​=x[i]⋅(1,...,1)∈F2λ​ for i∈[ℓ′]. The receiver expands k0i​ and k1i​ with a pseudo random generator (PRG), letting t0i​=PRG(k0i​)∈F2ℓ′​ and t1i​=PRG(k1i​)∈F2ℓ′​ for i∈[λ] The sender expands kΔi​i​ with the same PRG and gets tΔi​i​=PRG(kΔi​i​) for i∈[λ] The receiver computes ui=t0i​⊕t1i​⊕x∈F2ℓ′​ for i∈[λ] and sends them to the sender. The sender computes qi=Δi​⋅ui⊕tΔi​i​=t0i​⊕Δi​⋅x∈F2ℓ′​ for i∈[λ] Let qj​ be the j-th row of the ℓ′×λ matrix Q=[q1∣q2∣⋯∣qλ], and similarly let tj​ be the j-th row of T=[t01​∣t02​∣⋯∣t0λ​]. Note that qj​=tj​⊕(xj​⋆Δ)=tj​⊕(x[j]⋅Δ) for j∈[ℓ′] Correlation Check The sender and receiver run a πRAND​(F2λℓ′​) protocol to obtain random elements χ1​,...,χℓ′​. πRAND​(F2λℓ′​) will be described later. The receiver computes x=j=1∑ℓ′​x[j]⋅χj​ and t=j=1∑ℓ′​tj​⋅χj​ and sends them to the sender. The sender computes q=j=1∑ℓ′​qj​⋅χj​ and checks t=q⊕x⋅Δ. If the check fails, abort. Output The sender computes vj​=H(j,qj​)⊕H(j,qj​⊕Δ)⊕Δ for j∈[ℓ] and sends them to the receiver. H is a tweakable hash function as used in garbled circuit. The sender outputs (H(j,qj​), H(j,qj​)⊕Δ) For j∈[ℓ], the receiver outputs the following: If xj​=0, outputs H(j,tj​); If xj​=1, outputs H(j,tj​)⊕vj​ This ends the description of the KOS15 protocol. We note that a very recent work pointed out that there is a flaw in the security proof of the main lemma of KOS15 and provided a fix, which is less efficient. They also pointed out that this the original KOS15 protocol is still secure in practice. Therefore, we choose the original KOS15 protocol in zkOracles in this version. We begin to describe the πRAND​(F2λℓ​) protocol as follows. The sender and receiver locally choose uniformly 128-bit random seeds s0​ and s1​, respectively. The sender and receiver compute commitments of the seeds by choosing 128-bit random strings r0​,r1​ and generate c0​=SHA256(s0​∥r0​) and c1​=SHA256(s1​∥r1​), respectively. The sender sends c0​ to the receiver, and the receiver sends c1​ to the sender. On receiving the commitments, the sender and receiver open the commitments by sending (s0​,r0​) and (s1​,r1​) to each other, respectively. The sender checks c1​ by re-computing it with s1​,r1​, and the receiver checks c0​ by re-computing it with s0​,r0​. If checks fail, abort. The sender and receiver locally compute s=s0​⊕s1​ and generate random elements with a PRG as χ1​,...,χℓ​←PRG(s)","breadcrumbs":"Two Party Computation » Oblivious Transfer » OT Extension » OT Extension","id":"68","title":"OT Extension"},"69":{"body":"With garbled circuit and oblivious transfer, we are ready to describe the well-known Yao's method to construct secure two-party computation protocols for any polynomial-size function. Given any function F(x,y), where x is the private input of Alice, and y is the private input of Bob. Let x=x1​∥x2​∥⋯∥xℓ1​​ and y=y1​∥y2​∥⋯∥yℓ2​​, where xi​ and yi​ are bits. Let Alice be the garbler and Bob be the evaluator, Yao's protocol is described as follows. Alice and Bob run a COT protocol to generate ℓ2​ OTs, where Alice is the sender and Bob is the receiver. Alice obtains (Y1​,Y1​⊕Δ),...,(Yℓ2​​,Yℓ2​​⊕Δ), and Bob obtains (Y1​⊕y1​⋅Δ,...,Yℓ2​​⊕yℓ2​​⋅Δ) Alice chooses uniformly random labels X1​,X2​,...,Xℓ1​​. Alice uses (X1​,...,Xℓ1​​,Y1​,...,Yℓ2​​) and global Δ to generate the garbled circuit of F and sends GC(F) to Bob. Alice also sends the decoding information to Bob. Alice encodes her inputs and sends X1​⊕x1​⋅Δ,...,Xℓ1​​⊕xℓ1​​⋅Δ to Bob. With the encoded X labels and Y labels, Bob evaluates GC(F) to get the output labels, and then decodes these output labels with decoding information and gets the output bits.","breadcrumbs":"Two Party Computation » Full Protocol » Full Protocol","id":"69","title":"Full Protocol"},"7":{"body":"When cryptographers talk about \"groups\", they usually mean a \"computational group\", which is a group equipped with efficient algorithms for computing the group operation and inversion. This is different because a group in the mathematical sense need not have its operations be computable at all.","breadcrumbs":"Groups » More things to know","id":"7","title":"More things to know"},"70":{"body":"Many modern proof systems (and I think all that are in use) are constructed according to the following recipe. You start out with a class of computations. You devise a way to arithmetize those computations. That is, to express your computation as a statement about polynomials. More specifically, you describe what is often called an \"algebraic interactive oracle proof\" (AIOP) that encodes your computation. An AIOP is a protocol describing an interaction between a prover and a verifier, in which the prover sends the verifier some \"polynomial oracles\" (basically a black box function that given a point evaluates a polynomial at that point), the verifier sends the prover random challenges, and at the end, the verifier queries the prover's polynomials at points of its choosing and makes a decision as to whether it has been satisfied by the proof. An AIOP is an imagined interaction between parties. It is an abstract description of the protocol that will be \"compiled\" into a SNARK. There are several \"non-realistic\" aspects about it. One is that the prover sends the verifier black-box polynomials that the verifier can evaluate. These polynomials have degree comparable to the size of the computation being verified. If we implemented these \"polynomial oracles\" by having the prover really send the O(n) size polynomials (say by sending all their coefficients), then we would not have a zk-SNARK at all, since the verifier would have to read this linearly sized polynomial so we would lose succinctness, and the polynomials would not be black-box functions, so we may lose zero-knowledge. Instead, when we concretely instantiate the AIOP, we have the prover send constant-sized, hiding polynomial commitments . Then, in the phase of the AIOP where the verifier queries the polynomials, the prover sends an opening proof for the polynomial commitments which the verifier can check, thus simulating the activity of evaluating the prover's polynomials on your own. So this is the next step of making a SNARK: instantiating the AIOP with a polynomial commitment scheme of one's choosing. There are several choices here and these affect the properties of the SNARK you are constructing, as the SNARK will inherit efficiency and setup properties of the polynomial commitment scheme used. An AIOP describes an interactive protocol between the verifier and the prover. In reality, typically, we also want our proofs to be non-interactive. This is accomplished by what is called the Fiat--Shamir transformation . The basic idea is this: all that the verifier is doing is sampling random values to send to the prover. Instead, to generate a \"random\" value, the prover simulates the verifier by hashing its messages. The resulting hash is used as the \"random\" challenge. At this point we have a fully non-interactive proof. Let's review our steps. Start with a computation. Translate the computation into a statement about polynomials and design a corresponding AIOP. Compile the AIOP into an interactive protocol by having the prover send hiding polynomial commitments instead of polynomial oracles. Get rid of the verifier-interaction by replacing it with a hash function. I.e., apply the Fiat--Shamir transform.","breadcrumbs":"Proof Systems » Proof Systems Design Overview","id":"70","title":"Proof Systems Design Overview"},"71":{"body":"Finally we can discuss zk-SNARKs. In these notes right now I will not start with intuitive motivations for SNARKs because I can't think of them right now, but maybe later. Let f:A×W→Bool be a program (in some yet to be specified programming language). For a:A, zk-SNARKs let you produce proofs for statements of the form I know w:W such that f(a,w)=true in such a way that no information about w is revealed by the proof (this is the zero-knowledge or ZK part of them). In other words, zk-SNARKs allow you to prove execution of programs, where you are also allowed to hide a selected portion (i.e., the W portion) of the data involved in the computation. We could also phrase zk-SNARKs in terms of computing functions f:A×W→B, which seems more general, but actually this would be taken care of by the previous sense where we can only use boolean-valued functions, by proving against the function g:(A×B)×W→Bool defined by g((a,b),w):=b=?​f(a,w). SNARK stands for Succinct Non-interactive ARguments of Knowledge . If it also satisfies zero-knowledge, then it is called a zk-SNARK. Preprocessing SNARKs allow the verifier to precompute some encodings of the relation to run independently of the instance length. Definitions of these schemes owe the succinctness property to the fact that both the verifier and the proof length are (at most) polylogarithmic in the size of the computation (the witness length). Recent schemes such as Plonk, go beyond that definition and provide zkSNARKs with constant proof length. Unlike the verifier in preprocessing SNARKs, the prover can't be faster than linear, as it has to read the circuit at least.","breadcrumbs":"Proof Systems » zk-SNARKs » zk-SNARKs","id":"71","title":"zk-SNARKs"},"72":{"body":"We will define a notion of computation (a programming language basically) for which it is easy to construct zk-SNARKs.","breadcrumbs":"Proof Systems » zk-SNARKs » Expressing computations with \"iterating over trace tables\"","id":"72","title":"Expressing computations with \"iterating over trace tables\""},"73":{"body":"The proof system in Mina is a variant of PlonK . To understand PlonK, you can refer to our series of videos on the scheme. In this section we explain our variant, called kimchi . kimchi is not formally a zk-SNARK, as it is not succinct in the proof size. zk-SNARKs must have a log(n) proof size where n is the number of gates in the circuit. In kimchi, due to the Bootleproof polynomial commitment approach, our proofs are log(d) where d is the maximum degree of the committed polynomials (in the order of the number of rows). In practice, our proofs are in the order of dozens of kilobytes. Note that kimchi is a zk-SNARK in terms of verification complexity (log(n)) and the proving complexity is quasilinear (O(nlogn)) – recall that the prover cannot be faster than linear. Essentially what PlonK allows you to do is to, given a program with inputs and outputs, take a snapshot of its execution. Then, you can remove parts of the inputs, or outputs, or execution trace, and still prove to someone that the execution was performed correctly for the remaining inputs and outputs of the snapshot. There are a lot of steps involved in the transformation of \"I know an input to this program such that when executed with these other inputs it gives this output\" to \"here's a sequence of operations you can execute to convince yourself of that\". At the end of this chapter, you will understand that the protocol boils down to filling a number of tables (illustrated in the diagram below): tables that specify the circuit, and tables that describes an execution trace of the circuit (given some secret and public inputs). kimchi","breadcrumbs":"Overview » Overview","id":"73","title":"Overview"},"74":{"body":"Size: number of rows Columns: number of variables per rows Cell: a pair (row, column) Witness: the values assigned in all the cells Gate: polynomials that act on the variables in a row Selector vector: a vector of values 1 or 0 (not true for constant vector I think) that toggles gates and variables in a row Gadget: a series of contiguous rows with some specific gates set (via selector vectors)","breadcrumbs":"Overview » Glossary » Glossary","id":"74","title":"Glossary"},"75":{"body":"PlonK needs a domain to encode the circuit on (for each point we can encode a row/constraint/gate). We choose a domain H such that it is a multiplicative subgroup of the scalar field of our curve.","breadcrumbs":"Domain » Domain","id":"75","title":"Domain"},"76":{"body":"Furthermore, as FFT (used for interpolation, see the section on FFTs ) works best in domains of size 2k for some k, we choose H to be a subgroup of order 2k. Since the pasta curves both have scalar fields of prime orders p and q, their multiplicative subgroup is of order p−1 and q−1 respectively (without the zero element). The pasta curves were generated specifically so that both p−1 and q−1 are multiples of 232. We say that they have 2-adicity of 32. Looking at the definition of one of the pasta fields in Rust you can see that it is defined specifically for a trait related to FFTs: impl FftParameters for FqParameters { type BigInt = BigInteger; const TWO_ADICITY: u32 = 32; #[rustfmt::skip] const TWO_ADIC_ROOT_OF_UNITY: BigInteger = BigInteger([ 0x218077428c9942de, 0xcc49578921b60494, 0xac2e5d27b2efbee2, 0xb79fa897f2db056 ]); The 2-adicity of 32 means that there's a multiplicative subgroup of size 232 that exists in the field. The code above also defines a generator g for it, such that g232=1 and gi=1 for i∈[[1,232−1]] (so it's a primitive 232-th root of unity). Lagrange's theorem tells us that if we have a group of order n, then we'll have subgroups with orders dividing n. So in our case, we have subgroups with all the powers of 2, up to the 32-th power of 2. To find any of these groups, it is pretty straight forward as well. Notice that: let h=g2, then h231=g232=1 and so h generates a subgroup of order 31 let t=g22, then t230=g232=1 and so t generates a subgroup of order 30 and so on... In arkworks you can see how this is implemented: let size = n.next_power_of_two() as u64;\nlet log_size_of_group = ark_std::log2(usize::try_from(size).expect(\"too large\"));\nomega = Self::TWO_ADIC_ROOT_OF_UNITY;\nfor _ in log_size_of_group..Self::TWO_ADICITY { omega.square_in_place();\n} this allows you to easily find subgroups of different sizes of powers of 2, which is useful in zero-knowledge proof systems as FFT optimizations apply well on domains that are powers of 2. You can read more about that in the mina book .","breadcrumbs":"Domain » 2-adicity","id":"76","title":"2-adicity"},"77":{"body":"For each curve, we generate a domain H as the set H=1,ω,ω2,⋯,ωn−1. As we work in a multiplicative subgroup, the polynomial that vanishes in all of these points can be written and efficiently calculated as ZH​(x)=x∣H∣−1. This is because, by definition, ω∣H∣=1. If x∈H, then x=ωi for some i, and we have: x∣H∣=(ωi)∣H∣=(ω∣H∣)i=1i=1 This optimization is important, as without it calculating the vanishing polynomial would take a number of operations linear in ∣H∣ (which is not doable in a verifier circuit, for recursion). With the optimization, it takes us a logarithmic number of operation (using exponentiation by squaring ) to calculate the vanishing polynomial.","breadcrumbs":"Domain » Efficient computation of the vanishing polynomial","id":"77","title":"Efficient computation of the vanishing polynomial"},"78":{"body":"What's a lagrange base? Li​(x)=1 if x=gi, 0 otherwise.","breadcrumbs":"Lagrange Basis in Multiplicative Subgroups » Lagrange basis in multiplicative subgroups","id":"78","title":"Lagrange basis in multiplicative subgroups"},"79":{"body":"Arkworks has the formula to construct a lagrange base: Evaluate all Lagrange polynomials at τ to get the lagrange coefficients. Define the following as H: The coset we are in, with generator g and offset h m: The size of the coset H ZH​: The vanishing polynomial for H. ZH​(x)=∏i∈[m]​(x−h⋅gi)=xm−hm vi​: A sequence of values, where v0​=m∗hm−11​, and vi+1​=g⋅vi​ We then compute Li,H​(τ) as Li,H​(τ)=ZH​(τ)⋅τ−hgivi​​ However, if τ in H, both the numerator and denominator equal 0 when i corresponds to the value tau equals, and the coefficient is 0 everywhere else. We handle this case separately, and we can easily detect by checking if the vanishing poly is 0. following this, for h=1 we have: L0​(x)=m(x−1)ZH​(x)​ L1​(x)=m(x−g)ZH​(x)g​ L2​(x)=m(x−g2)ZH​(x)g2​ and so on","breadcrumbs":"Lagrange Basis in Multiplicative Subgroups » What's the formula?","id":"79","title":"What's the formula?"},"8":{"body":"Implement a type JacobianPoint and functions for computing the group operation Familiarize yourself with the types and traits in ark_ec. Specifically, todo Implement fn decompress(c: (F, bool)) -> (F, F)","breadcrumbs":"Groups » Exercises","id":"8","title":"Exercises"},"80":{"body":"https://en.wikipedia.org/wiki/Lagrange_polynomial#Barycentric_form","breadcrumbs":"Lagrange Basis in Multiplicative Subgroups » What's the logic here?","id":"80","title":"What's the logic here?"},"81":{"body":"So far we've talked about an interactive protocol between a prover and a verifier. The zero-knowledge proof was also in the honest verifier zero-knowedlge (HVZK) model, which is problematic. In practice, we want to remove the interaction and have the prover produce a proof by themselves, that anyone can verify.","breadcrumbs":"Non-Interactivity via Fiat-Shamir » Non-Interactivity via Fiat-Shamir","id":"81","title":"Non-Interactivity via Fiat-Shamir"},"82":{"body":"public-coin protocols are protocols were the messages of the verifier are simply random messages. This is important as our technique to transform an interactive protocol to a non-interactive protocol works on public-coin protocols.","breadcrumbs":"Non-Interactivity via Fiat-Shamir » Public-coin protocols","id":"82","title":"Public-coin protocols"},"83":{"body":"The whole idea is to replace the verifier by a random oracle, which in practice is a hash function. Note that by doing this, we remove potential leaks that can happen when the verifier acts dishonestly. Initially the Fiat-Shamir transformation was only applied to sigma protocols, named after the greek letter Σ due to its shape resembling the direction of messages (prover sends a commit to a verifier, verifier sends a challenge to a prover, prover sends the final proof to a verifier). A Z would have made more sense but here we are.","breadcrumbs":"Non-Interactivity via Fiat-Shamir » Fiat-Shamir trick","id":"83","title":"Fiat-Shamir trick"},"84":{"body":"As our protocol has more than three moves, where every even move is a challenge from the verifier, we need to generalize Fiat-Shamir. This is simple: every verifier move can be replaced by a hash of the transcript (every message sent and received so far) to obtain a challenge.","breadcrumbs":"Non-Interactivity via Fiat-Shamir » Generalization of Fiat-Shamir","id":"84","title":"Generalization of Fiat-Shamir"},"85":{"body":"While we use a hash function for that, a different construction called the duplex construction is particularly useful in such situations as they allow to continuously absorb the transcript and produce challenges, while automatically authenticating the fact that they produced a challenge. Merlin is a standardization of such a construction using the Strobe protocol framework (a framework to make use of a duplex construction). Note that the more recent Xoodyak (part of NIST's lightweight competition) could have been used for this as well. Note also that Mina uses none of these standards, instead it simply uses Poseidon (see section on poseidon).","breadcrumbs":"Non-Interactivity via Fiat-Shamir » In practice: a duplex construction as Merlin","id":"85","title":"In practice: a duplex construction as Merlin"},"86":{"body":"Plookup allows us to check if witness values belong to a look up table. This is usually useful for reducing the number of constraints needed for bit-wise operations. So in the rest of this document we'll use the XOR table as an example. 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For the Rust documentation, see here . Note that this book is a work in progress, not necessarily reflecting the current state of Mina. Authors: Izaak Meckler, Vanishree Rao, Matthew Ryan, Anaïs Querol, Joseph Spadavecchia, David Wong, Xiang Xie In memory of Vitaly Zelov.","breadcrumbs":"Introduction » Introduction","id":"0","title":"Introduction"},"1":{"body":"This document is intended primarily to communicate mathematical ideas to programmers with some experience of math. To that end, we will often be ambivalent about the difference between sets and types (in whatever programming language, but usually I am imagining some idealized form of Rust or OCaml). So for example we may write A is a type A is a set A is a type and these should all be interpreted the same way, assuming it makes sense in the context. 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So if we say \"let f:A→B\", we typically mean that A and B are types in some programming language which makes sense in the context (or maybe that they are sets, again depending on context) f is actually a program computing a function of the type A→B (again, in whatever sense of program makes sense in context)","breadcrumbs":"Terminology » Terminology","id":"1","title":"Terminology"},"10":{"body":"The purpose of this document is to give the reader mathematical, cryptographic, and programming context sufficient to become an effective practitioner of zero-knowledge proofs and ZK-SNARKs specifically.","breadcrumbs":"Fields » Intro","id":"10","title":"Intro"},"100":{"body":"Unlike the latest version of vanilla PlonK that implements the final check using a polynomial opening (via Maller's optimization), we implement it manually. (That is to say, Izaak implemented Maller's optimization for 5-wires.) But the check is not exactly f(ζ)=ZH​(ζ)t(ζ). This note describes how and why the implementation deviates a little.","breadcrumbs":"Final Check » Understanding the implementation of the f(ζ)=ZH​(ζ)t(ζ) check","id":"100","title":"Understanding the implementation of the f(ζ)=ZH​(ζ)t(ζ) check"},"101":{"body":"If you look at how the evaluation of f(z) is computed on the prover side (or the commitment of f is computed on the verifier side), you can see that f is missing two things: the public input part some terms in the permutation What is it missing in the permutation part? Let's look more closely. This is what we have: −1⋅⋅⋅⋅⋅⋅⋅⋅​z(ζω)⋅zkpm(ζ)⋅αPERM0(w[0](ζ)+β⋅σ[0](ζ)+γ)(w[1](ζ)+β⋅σ[1](ζ)+γ)(w[2](ζ)+β⋅σ[2](ζ)+γ)(w[3](ζ)+β⋅σ[3](ζ)+γ)(w[4](ζ)+β⋅σ[4](ζ)+γ)(w[5](ζ)+β⋅σ[5](ζ)+γ)β⋅σ[6](x)​​ In comparison, here is the list of the stuff we needed to have: the two sides of the coin: z(ζ)⋅⋅⋅⋅⋅⋅⋅⋅​zkpm(ζ)⋅αPERM0(w[0](ζ)+β⋅shift[0](ζ)+γ)(w[1](ζ)+β⋅shift[1](ζ)+γ)(w[2](ζ)+β⋅shift[2](ζ)+γ)(w[3](ζ)+β⋅shift[3](ζ)+γ)(w[4](ζ)+β⋅shift[4](ζ)+γ)(w[5](ζ)+β⋅shift[5](ζ)+γ)(w[6](ζ)+β⋅shift[6](ζ)+γ)​​ with shift[0]=1 and −1⋅⋅⋅⋅⋅⋅⋅⋅​z(ζω)⋅zkpm(ζ)⋅αPERM0(w[0](ζ)+β⋅σ[0](ζ)+γ)(w[1](ζ)+β⋅σ[1](ζ)+γ)(w[2](ζ)+β⋅σ[2](ζ)+γ)(w[3](ζ)+β⋅σ[3](ζ)+γ)(w[4](ζ)+β⋅σ[4](ζ)+γ)(w[5](ζ)+β⋅σ[5](ζ)+γ)(w[6](ζ)+β⋅σ[6](ζ)+γ)​​ the initialization: (z(ζ)−1)⋅L1​(ζ)⋅αPERM1 and the end of the accumulator: (z(ζ)−1)⋅Ln−k​(ζ)⋅αPERM2 You can read more about why it looks like that in this post . We can see clearly that the permutation stuff we have in f is clearly lacking. It's just the subtraction part (equation 2), and even that is missing some terms. It is missing exactly this: −1 ⋅⋅⋅⋅⋅⋅⋅⋅​z(ζω)⋅zkpm(ζ)⋅αPERM0(w[0](ζ)+β⋅σ[0](ζ)+γ)(w[1](ζ)+β⋅σ[1](ζ)+γ)(w[2](ζ)+β⋅σ[2](ζ)+γ)(w[3](ζ)+β⋅σ[3](ζ)+γ)(w[4](ζ)+β⋅σ[4](ζ)+γ)(w[5](ζ)+β⋅σ[5](ζ)+γ)(w[6](ζ)+γ)​​ So at the end, when we have to check for the identity f(ζ)=ZH​(ζ)t(ζ) we'll actually have to check something like this (I colored the missing parts on the left hand side of the equation): f(ζ)​+pub(ζ)+z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+βζ+γ)⋅(w[1](ζ)+β⋅shift[0](ζ)+γ)⋅(w[2](ζ)+β⋅shift[1](ζ)+γ)⋅(w[3](ζ)+β⋅shift[2](ζ)+γ)⋅(w[4](ζ)+β⋅shift[3](ζ)+γ)⋅(w[5](ζ)+β⋅shift[4](ζ)+γ)⋅(w[6](ζ)+β⋅shift[5](ζ)+γ)−z(ζω)⋅zkpm(ζ)⋅αPERM0⋅⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅⋅(w[6](ζ)+γ)+(z(ζ)−1)⋅L1​(ζ)⋅αPERM1+(z(ζ)−1)⋅Ln−k​(ζ)⋅αPERM2=t(ζ)(ζn−1)​​ This is not exactly what happens in the code. But if we move things around a bit, we end up with what's implemented. I highlight what changes in each steps. First, we just move things around: f(ζ)​+pub(ζ)+z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅shift[0]ζ+γ)⋅(w[1](ζ)+β⋅shift[1]ζ+γ)⋅(w[2](ζ)+β⋅shift[2]ζ+γ)⋅(w[3](ζ)+β⋅shift[3]ζ+γ)⋅(w[4](ζ)+β⋅shift[4]ζ+γ)⋅(w[5](ζ)+β⋅shift[5]ζ+γ)⋅(w[6](ζ)+β⋅shift[6]ζ+γ)−z(ζω)⋅zkpm(ζ)⋅αPERM0⋅⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅(w[6](ζ)+γ)−t(ζ)(ζn−1)=(1−z(ζ))L1​(ζ)αPERM1+(1−z(ζ))Ln−k​(ζ)αPERM2​​ here are the actual lagrange basis calculated with the formula here , oh and we actually use L0​ in the code, not L1​, so let's change that as well: f(ζ)++−−=​pub(ζ)z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅shift[0]ζ+γ)⋅(w[1](ζ)+β⋅shift[1]ζ+γ)⋅(w[2](ζ)+β⋅shift[2]ζ+γ)⋅(w[3](ζ)+β⋅shift[3]ζ+γ)⋅(w[4](ζ)+β⋅shift[4]ζ+γ)⋅(w[5](ζ)+β⋅shift[5]ζ+γ)⋅(w[6](ζ)+β⋅shift[6]ζ+γ)+z(ζω)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅(w[6](ζ)+γ)+t(ζ)(ζn−1)(1−z(ζ))[n(ζ−1)(ζn−1)​αPERM1+n(ζ−ωn−k)ωn−k(ζn−1)​αPERM2]​​ finally we extract some terms from the lagrange basis: ​[f(ζ)+pub(ζ)+z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅shift[0]ζ+γ)⋅(w[1](ζ)+β⋅shift[1]ζ+γ)⋅(w[2](ζ)+β⋅shift[2]ζ+γ)⋅(w[3](ζ)+β⋅shift[3]ζ+γ)⋅(w[4](ζ)+β⋅shift[4]ζ+γ)⋅(w[5](ζ)+β⋅shift[5]ζ+γ)⋅(w[6](ζ)+β⋅shift[6]ζ+γ)+−z(ζω)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅(w[6](ζ)+γ)−t(ζ)(ζn−1)]⋅(ζ−1)(ζ−ωn−k)=(1−z(ζ))[n(ζn−1)(ζ−ωn−k)​αPERM1+nωn−k(ζn−1)(ζ−1)​αPERM2]​​​ with αPERM0=α17,αPERM1=α18,αPERM2=α19 Why do we do things this way? Most likely to reduce Also, about the code: the division by n in the αPERM1 and the αPERM2 terms is missing (why?) the multiplication by wn−k in the αPERM2 term is missing (why?) As these terms are constants, and do not prevent the division by ZH​ to form t, t also omits them. In other word, they cancel one another.","breadcrumbs":"Final Check » What's f and what's missing in the final equation?","id":"101","title":"What's f and what's missing in the final equation?"},"102":{"body":"In verifier.rs you can see the following code (cleaned to remove anything not important) // compute the witness polynomials $w_0, \\cdots, w_14$ and the permutation polynomial $z$ in evaluation forms on different domains\nlet lagrange = index.cs.evaluate(&w, &z); // compute parts of the permutation stuff that is included in t\nlet (perm, bnd) = index.cs.perm_quot(&lagrange, &oracles, &z, &alpha[range::PERM])?; // divide contributions with vanishing polynomial\nlet (mut t, res) = (perm + some_other_stuff).divide_by_vanishing_poly() // add the other permutation stuff\nt += &bnd; // t is evaluated at zeta and sent... Notice that bnd is not divided by the vanishing polynomial . Also what's perm? Let's answer the latter TESTREMOVEME: perm(x)= −​aPERM0⋅zkpl(x)⋅[z(x)⋅(w[0](x)+γ+x⋅β⋅shift[0])⋅(w[1](x)+γ+x⋅β⋅shift[1])⋅…z(x⋅w)⋅(w[0](x)+γ+σ[0]⋅β) ⋅(w[1](x)+γ+σ[1]⋅β)⋅…]​​ and bnd is: bnd(x)=aPERM1⋅x−1z(x)−1​+aPERM2⋅x−sid[n−k]z(x)−1​ you can see that some terms are missing in bnd, specifically the numerator xn−1. Well, it would have been cancelled by the division by the vanishing polynomial, and this is the reason why we didn't divide that term by ZH​(x)=xn−1. Also, note that the same constant terms that were missing in f are missing in t.","breadcrumbs":"Final Check » What about t?","id":"102","title":"What about t?"},"103":{"body":"This document proposes a protocol change for kimchi .","breadcrumbs":"Maller's Optimization for Kimchi » Maller's optimization for Kimchi","id":"103","title":"Maller's optimization for Kimchi"},"104":{"body":"See the section on Maller's optimization for background.","breadcrumbs":"Maller's Optimization for Kimchi » What is Maller's optimization?","id":"104","title":"What is Maller's optimization?"},"105":{"body":"We want the verifier to form commitment to the following polynomial: L(x)=f(x)−ZH​(ζ)⋅t(x) They could do this like so: com(L)=com(f)−ZH​(ζ)⋅com(t) Since they already know f, they can produce com(f), the only thing they need is com(t). So the protocol looks like that: maller 15 1 In the rest of this document, we review the details and considerations needed to implement this change in kimchi .","breadcrumbs":"Maller's Optimization for Kimchi » Overview","id":"105","title":"Overview"},"106":{"body":"There's one challenge that prevents us from directly using this approach: com(t) is typically sent and received in several commitments (called chunks or segments throughout the codebase). As t is the largest polynomial, usually exceeding the size of the SRS (which is by default set to be the size of the domain).","breadcrumbs":"Maller's Optimization for Kimchi » How to deal with a chunked t?","id":"106","title":"How to deal with a chunked t?"},"107":{"body":"Thus, the verifier will have to produce split commitments of L and combine them with powers of ζn to verify an evaluation proof. Let's define L as: L=L0​+xnL1​+x2nL1​+⋯ where every Li​ is of degree n−1 at most. Then we have that com(L)=com(L0​)+com(xn⋅L1​)+com(x2n⋅L2​)+⋯ Which the verifier can't produce because of the powers of xn, but we can linearize it as we already know which x we're going to evaluate that polynomial with: com(L~)=com(L0​)+ζn⋅com(L1​)+ζ2n⋅com(L2​)+⋯","breadcrumbs":"Maller's Optimization for Kimchi » The verifier side","id":"107","title":"The verifier side"},"108":{"body":"This means that the prover will produce evaluation proofs on the following linearized polynomial: L~(x)=1⋅L0​(x)+ζn⋅L1​(x)+ζ2n⋅L2​(x)+⋯ which is the same as L(x) only if evaluated at ζ. As the previous section pointed out, we will need L (ζω) and L (ζω)=L(ζω).","breadcrumbs":"Maller's Optimization for Kimchi » The prover side","id":"108","title":"The prover side"},"109":{"body":"To create an evaluation proof, the prover also needs to produce the blinding factor rL​ associated with the chunked commitment of: L =f ​−(ζn−1)t~ To compute it, there are two rules to follow: when two commitment are added together, their associated blinding factors get added as well: com(a)+com(b)⟹ra​+rb​ when scaling a commitment, its blinding factor gets scalled too: n⋅com(a)⟹n⋅ra​ As such, if we know rf​ and rt​, we can compute: rL ​=rf ​​+(ζn−1)⋅rt~​ The prover knows the blinding factor of the commitment to t, as they committed to it. But what about f? They never commit to it really, and the verifier recreates it from scratch using: The commitments we sent them . In the linearization process, the verifier actually gets rid of most prover commitments, except for the commitment to the last sigma commitment Sσ6​. (TODO: link to the relevant part in the spec) As this commitment is public, it is not blinded. The public commitments . Think commitment to selector polynomials or the public input polynomial. These commitments are not blinded, and thus do not impact the calculation of the blinding factor. The evaluations we sent them . Instead of using commitments to the wires when recreating f, the verifier uses the (verified) evaluations of these in ζ. If we scale our commitment com(z) with any of these scalars, we will have to do the same with rz​. Thus, the blinding factor of L is (ζn−1)⋅rt ​.","breadcrumbs":"Maller's Optimization for Kimchi » Evaluation proof and blinding factors","id":"109","title":"Evaluation proof and blinding factors"},"11":{"body":"In this section we'll discuss the fundamental objects used in the construction of most ZK-SNARKs in use today. These objects are used extensively and without ceremony, so it's important to understand them very well. If you find you don't understand something at first, don't worry: practice is often the best teacher and in using these objects you will find they will become like the bicycle you've had for years: like an extension of yourself that you use without a thought.","breadcrumbs":"Fields » Some fundamental mathematical objects","id":"11","title":"Some fundamental mathematical objects"},"110":{"body":"The prover actually does not send a commitment to the full f polynomial. As described in the last check section . The verifier will have to compute the evaluation of L~(ζ) because it won't be zero. It should be equal to the following: ​f ​(ζ)−t (ζ)(ζn−1)=(ζ−1)(ζ−ωn−k)1−z(ζ)​[n(ζn−1)(ζ−ωn−k)​αPERM1+nωn−k(ζn−1)(ζ−1)​αPERM2]−pub(ζ)−z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅shift[0]ζ+γ)⋅(w[1](ζ)+β⋅shift[1]ζ+γ)⋅(w[2](ζ)+β⋅shift[2]ζ+γ)⋅(w[3](ζ)+β⋅shift[3]ζ+γ)⋅(w[4](ζ)+β⋅shift[4]ζ+γ)⋅(w[5](ζ)+β⋅shift[5]ζ+γ)⋅(w[6](ζ)+β⋅shift[6]ζ+γ)+z(ζω)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅(w[6](ζ)+γ)+​​ Because we use the inner product polynomial commitment , we also need: L (ζω)=f ​(ζω)−ZH​(ζ)⋅t~(ζω) Notice the ZH​(ζ). That evaluation must be sent as part of the proof as well.","breadcrumbs":"Maller's Optimization for Kimchi » The evaluation of L~","id":"110","title":"The evaluation of L~"},"111":{"body":"Now here's how we need to modify the current protocol: The evaluations f(ζ),f(ζω),t(ζ),t(ζω) (and their corresponding evaluation proofs) don't have to be part of the protocol anymore. The prover must still send the chunked commitments to t. The prover must create a linearized polynomial L by creating a linearized polynomial f ​ and a linearized polynomial t and computing: L =f ​+(ζn−1)⋅t While the verifier can compute the evaluation of L (ζ) by themselves, they don't know the evaluation of L (ζω), so the prover needs to send that. The verifier must recreate com(L ), the commitment to L , themselves so that they can verify the evaluation proofs of both L (ζ) and L (ζω). The evaluation of L~(ζω) must be absorbed in both sponges (Fq and Fr). maller 15 2 The proposal is implemented in #150 with the following details: the L~ polynomial is called ft. the evaluation of L~(ζ) is called ft_eval0. the evaluation L~(ζω) is called ft_eval1.","breadcrumbs":"Maller's Optimization for Kimchi » The actual protocol changes","id":"111","title":"The actual protocol changes"},"112":{"body":"In 2020, Plookup showed how to create lookup proofs. Proofs that some witness values are part of a lookup table . Two years later, an independent team published plonkup showing how to integrate Plookup into PlonK. This document specifies how we integrate Plookup in kimchi. It assumes that the reader understands the basics behind Plookup.","breadcrumbs":"Lookup Tables » Plookup in Kimchi","id":"112","title":"Plookup in Kimchi"},"113":{"body":"We integrate Plookup in kimchi with the following differences: we snake-ify the sorted table instead of wrapping it around (see later) we allow fixed-ahead-of-time linear combinations of columns of the queries we make we implemented different tables, like RangeCheck and XOR. we allow several lookups (or queries) to be performed within the same row zero-knowledgeness is added in a specific way (see later) The following document explains the protocol in more detail","breadcrumbs":"Lookup Tables » Overview","id":"113","title":"Overview"},"114":{"body":"As per the Plookup paper, the prover will have to compute three vectors: f, the (secret) query vector , containing the witness values that the prover wants to prove are part of the lookup table. t, the (public) lookup table . s, the (secret) concatenation of f and t, sorted by t (where elements are listed in the order they are listed in t). Essentially, Plookup proves that all the elements in f are indeed in the lookup table t if and only if the following multisets are equal: {(1+β)f,diff(t)} diff(sorted(f,t)) where diff is a new set derived by applying a \"randomized difference\" between every successive pairs of a vector, and (f,t) is the set union of f et t. More precisely, for a set S={s0​,s1​,⋯,sn​}, diff(S) is defined as the set {s0​+βs1​,s1​+βs2​,⋯,sn−1​+βsn​}. For example, with: f={5,4,1,5} t={1,4,5} we have: sorted(f,t)={1,1,4,4,5,5,5} {(1+β)f,diff(t)}={(1+β)5,(1+β)4,(1+β)1,(1+β)5,1+β4,4+β5} diff(sorted(f,t))={1+β1,1+β4,4+β4,4+β5,5+β5,5+β5} Note: This assumes that the lookup table is a single column. You will see in the next section how to address lookup tables with more than one column. The equality between the multisets can be proved with the permutation argument of PlonK, which would look like enforcing constraints on the following accumulator: init: acc0​=1 final: accn​=1 for every 0 RAM loading, disk -> RAM loading, and RAM r/w. This gives the maximum flexibility for circuit designers, and each primitive has uses for projects that we are pursuing. This design has been refined over several iterations to reduce the number of components exposed in the proof and the amount of computation that the verifier requires. The largest remaining parts represent the data itself and the method for combining them, where simplifying assumptions have been made to minimize their impact while still satisfying the design constraints.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » Why is this design the best in the space of possible designs?","id":"138","title":"Why is this design the best in the space of possible designs?"},"139":{"body":"Other designs building upon the lookup primitive reduce on some subset of this design, or on relaxing a design constraint that would remove some useful / required functionality.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What other designs have been considered and what is the rationale for not choosing them?","id":"139","title":"What other designs have been considered and what is the rationale for not choosing them?"},"14":{"body":"For a finite field Fp​ where p fits in n bits (i.e., p<2n) we have Addition, subtraction: O(n) Multiplication O(n2) Division O(n2) I believe, in practice significantly slower than multiplication. Actually, on a practical level, it's more accurate to model the complexity in terms of the number of limbs rather than the number of bits where a limb is 64 bits. Asymptotically it makes no difference but concretely it's better to think about the number of limbs for the most part. As a result you can see it's the smaller n is the better, especially with respect to multiplication, which dominates performance considerations for implementations of zk-SNARKs, since they are dominated by elliptic curve operations that consist of field operations. While still in development, Mina used to use a field of 753 bits or 12 limbs and now uses a field of 255 bits or 4 limbs. As a result, field multiplication became automatically sped up by a factor of 122/42=9, so you can see it's very useful to try to shrink the field size.","breadcrumbs":"Fields » Algorithmics of prime order fields","id":"14","title":"Algorithmics of prime order fields"},"140":{"body":"We are currently unable to expose certain useful functionality to users of Snapps / SnarkyJS, and are missing the ability to use that same functionality for important parts of the zkOracles ecosystem (such as parsing / scraping). We also are currently forced to hard-code any constraints that we need for particular uses in the verifier circuit, and users are unable to remove the parts they do not need and to replace them with other useful constraints that their circuits might benefit from. The Mina transaction model is unable to support a bytecode interpreter without this or similar extensions.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What is the impact of not doing this?","id":"140","title":"What is the impact of not doing this?"},"141":{"body":"(Elided)","breadcrumbs":"Lookup Tables » Extended Lookup Tables » Prior art","id":"141","title":"Prior art"},"142":{"body":"","breadcrumbs":"Lookup Tables » Extended Lookup Tables » Unresolved questions","id":"142","title":"Unresolved questions"},"143":{"body":"The maximum number of tables / entries of each kind that we want to support in the 'standard' protocol. Validation (or rejection) of the simplifying assumptions. Confirmation of the design constraints, or simplifications thereto. Necessity / utility of the permutation argument.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What parts of the design do you expect to resolve through the RFC process before this gets merged?","id":"143","title":"What parts of the design do you expect to resolve through the RFC process before this gets merged?"},"144":{"body":"None, this RFC has been developed in part from an initial design and iteration on implementations of parts of that design.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What parts of the design do you expect to resolve through the implementation of this feature before merge?","id":"144","title":"What parts of the design do you expect to resolve through the implementation of this feature before merge?"},"145":{"body":"Bytecode design and implementation details. Mina transaction interface for interacting with these features. Snarky / SnarkyJS interface for interacting with these features.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What related issues do you consider out of scope for this RFC that could be addressed in the future independently of the solution that comes out of this RFC?","id":"145","title":"What related issues do you consider out of scope for this RFC that could be addressed in the future independently of the solution that comes out of this RFC?"},"146":{"body":"This section explains how to design and add a custom constraint to our proof-systems library. PLONK is an AIOP. That is, it is a protocol in which the prover sends polynomials as messages and the verifier sends random challenges, and then evaluates the prover's polynomials and performs some final checks on the outputs. PLONK is very flexible. It can be customized with constraints specific to computations of interest. For example, in Mina, we use a PLONK configuration called kimchi that has custom constraints for poseidon hashing, doing elliptic curve operations, and more. A \"PLONK configuration\" specifies The set of types of constraints that you would like to be able to enforce. We will describe below how these types of constraints are specified. A number of \"eq-able\" columns W A number of \"advice\" columns A Under such configuration, a circuit is specified by A number of rows n A vector cs of constraint-types of length n. I.e., a vector that specifies, for each row, which types of constraints should be enforced on that row. A vector eqs : Vec<(Position, Position)> of equalities to enforce, where struct Position { row: usize, column: usize }. E.g., if the pair (Position { row: 0, col: 8 }, Position { row: 10, col: 2 }) is in eqs, then the circuit is saying the entries in those two positions should be equal, or in other words that they refer to the same value. This is where the distinction between \"eq-able\" and \"advice\" columns comes in. The column field of a position in the eqs array can only refer to one of the first W columns. Equalities cannot be enforced on entries in the A columns after that. Then, given such a circuit, PLONK lets you produce proofs for the statement I know W + A \"column vectors\" of field elements vs: [Vec; W + A] such that for each row index i < n, the constraint of type cs[i] holds on the values [vs[0][i], ..., vs[W+A - 1][i], vs[0][i+1], ..., vs[W+A - 1][i+1] and all the equalities in eqs hold. I.e., for (p1, p2) in eqs we have vs[p1.col][p1.row] == vs[p2.col][p2.row]. So, a constraint can check the values in two adjacent rows. Mathematically speaking, a constraint is a multivariate polynomial over the variables cCurr,i​,…,vCurr,W+A−1​,vNext,0​,…,vNext,W+A−1​. In other words, there is one variable corresponding to the value of each column in the \"current row\" and one variable correspond to the value of each column in the \"next row\". In Rust, vr,i​ is written E::cell(Column::Witness(i), r). So, for example, the variable vNext,3​ is written E::cell(Column::Witness(3), CurrOrNext::Next). let w = |i| v(Column::Witness(i)); Let's","breadcrumbs":"Custom Constraints » Specifying a constraint","id":"146","title":"Specifying a constraint"},"147":{"body":"The art in proof systems comes from knowing how to design a PLONK configuration to ensure maximal efficiency for the sorts of computations you are trying to prove. That is, how to choose the numbers of columns W and A, and how to define the set of constraint types. Let's describe the trade-offs involved here. The majority of the proving time for the PLONK prover is in committing to the W + A column polynomials, which have length equal to the number of rows n committing to the \"permutation accumulator polynomial, which has length n. committing to the quotient polynomial, which reduces to computing max(k, W) MSMs of size n, where k is the max degree of a constraint. performing the commitment opening proof, which is mostly dependent on the number of rows n. So all in all, the proving time is approximately equal to the time to perform W + A + 1 + max(k - 1, W) MSMs of size n, plus the cost of an opening proof for polynomials of degree n - 1. and maybe computing the combined constraint polynomial, which has degree k * n where k is the maximum degree of a constraint Increasing W and A increase proof size, and they potentially impact the prover-time as the prover must compute polynomial commitments to each column, and computing a polynomial commitment corresponds to doing one MSM (multi-scalar multiplication, also called a multi-exponentiation.) However, often increasing the number of columns allows you to decrease the number of rows required for a given computation. For example, if you can perform one Poseidon hash in 36 rows with 5 total columns, then you can also perform it in 12 (= 36 / 3) rows with 15 (= 5 * 3) total columns. Decreasing the number of rows (even while keeping the total number of table entries the same) is desirable because it reduces the cost of the polynomial commitment opening proof, which is dominated by a factor linear in the number of rows, and barely depends on the number of columns. Increasing the number of columns also increases verifier time, as the verifier must perform one scalar-multiplication and one hash per column. Proof length is also affected by a larger number of columns, as more polynomials need to be committed and sent along to the verifier. There is typically some interplay between these","breadcrumbs":"Custom Constraints » Defining a PLONK configuration","id":"147","title":"Defining a PLONK configuration"},"148":{"body":"The goal of this section is to draw the connections between the practical aspect of designing custom gates and the theory behind it. First, suppose we had the following layout of gates. This means that our circuit has four gates: a (single) generic gate, a (non-optimized) complete add, and then two different Example gates that we just make up for the example; ExamplePairs and ExampleTriples. row GateType:: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 Generic l r o 1 CompleteAdd x1 y1 x2 y2 x3 y3 inf same_x s inf_z x21_inv 2 ExamplePairs a e i o u y 3 ExampleTriples a b c d e f g h i j k l m n o The most important aspect to take into account when you are designing a new custom gate is writing the actual description of the new functionality. For this, we are using the Expression framework that allows us to write equations generically by representing each column of the table (wire) as a variable. Then we can write operations between them and define the gate constraints . From a mathematical perspective, these will have the shape of multivariate polynomials of as many variables as columns in the table (currently 15 for Kimchi) that is constrained to equal 0. In our example, we would define equations that represent the operations carried out by each of the gates. In the following equations we use the variable names that we wrote in the cells above, and we don't imply that the same name in a different row correspond to the same value (we are not imposing equalities between gates yet): Generic: l ⋅cl​+ r ⋅cr​+ o ⋅co​+ l ⋅ r ⋅cm​+cc​=0 CompleteAdd: x21_inv ⋅( x2 − x1 )−(1− same_x)=0 same_x ⋅( x2 − x1 )=0 same_x ⋅(2⋅ s ⋅ y1 −3⋅ x12)+(1− same_x )⋅((x2 − x1 )⋅ s − y2 + y1 )=0 x1 + x2 + x3 − s2=0 s ⋅( x1 − x3 )− y1 − y3=0 ( y2 − y1 )⋅( same_x − inf )=0 ( y2 − y1 )⋅ inf_z − inf =0 ExamplePairs: a − e =0 i − o =0 u − y =0 ExampleTriples: a ⋅ b ⋅ c =0 d ⋅ e ⋅ f =0 g ⋅ h ⋅ i =0 j ⋅ k ⋅ l =0 m ⋅ n ⋅ o =0 Nonetheless, when we write these constraints in proof-systems, we use a shared framework that helps us use the columns in each row as placeholders. For example, the i-th column of the current row Curr is represented as: E::::cell(Column::Witness(i), CurrOrNext::Curr) where the E type is a shorthand for Expr. More succinctly, we refer to the above simply as: witness_curr(i) For readability, let's just refer to them here as w0..w14. Then, the above constraints would look like the following: Generic: w0 ⋅cl​+ w1 ⋅cr​+ w2 ⋅co​+ w0 ⋅ w1 ⋅cm​+cc​=0 CompleteAdd: w10 ⋅( w2 − w0 )−(1− w7)=0 w7 ⋅( w2 − w0 )=0 w7 ⋅(2⋅ w8 ⋅ w1 −3⋅ w02)+(1− w7 )⋅((w2 − w0 )⋅ w8 − w3 + w1 )=0 w0 + w2 + w4 − w82=0 w8 ⋅( w0 − w4 )− w1 − w4=0 ( w3 − w1 )⋅( w7 − w6 )=0 ( w3 − w1 )⋅ w9 − w6 =0 ExamplePairs: w0 − w1 =0 w2 − w3 =0 w4 − w5 =0 ExampleTriples: w0 ⋅ w1 ⋅ w2 =0 w3 ⋅ w4 ⋅ w5 =0 w6 ⋅ w7 ⋅ w8 =0 w9 ⋅ w10 ⋅ w11 =0 w12 ⋅ w13 ⋅ w14 =0 Looking at the list of equations, each of them look like polynomials in the variables w0..w14, constrained to zero. But so far, the type of equations we had studied were univariate (i.e. f(X) rather than f(X0​,...,X14​)). But this will not be a problem, since these multivariate equations will soon become univariate with a simple trick: interpolation . Right now, we are describing the gate constraints row-wise (i.e. horizontally ). But as we said before, our domain will be determined by the number of rows; we will transform these equations to a vertical and univariate behaviour. The strategy to follow will be obtaining witness polynomials that evaluate to the cells of the execution trace. First , note that so far we didn't mention any values for the w0..w14 terms, but instead they are placeholders for what is yet to come . And what is this you may ask: the witness of the relation . Put it another way, the witness will be the instantiation of the cells of the table that will make all of the above constraints hold. And where do we get these values from? Well, the prover will know or will compute them . Second , we count the number of rows that we ended up with: here it's four (conveniently, a power of two). What we want to do is to create as many witness polynomials as the number of columns in the execution trace. These polynomials, that we can call w0​(X),...,w14​(X), must be design to evaluate to the cells of the trace. Formally we define each wj​(X) as: G=:wj​(gi)=trace[i,j] Meaning that wi​(x) returns the i-th column of the x-th row of the execution trace. And how can we create such polynomials? We will use interpolation, which can be efficiently computed over power-of-two-sized groups, as it is the case in the example. Recalling Lagrangian terms , they act like linearly independent selectors over a set, in particular: Let G be a multiplicative group formed by the powers of a generator g. Given any element x in the group, the i-th Lagrangian term evaluates to 1 iff x=gi and 0 iff x=gi. ∀ x∈G:LgiG​(x)=1⟺x=θ and LgiG​(x)=0⟺x=θ Given that our group has size 4, the number of rows (ignoring ZK for now), G will be the set {1,g,g2,g3}. This means that we can build such 15 witness polynomials as: ∀j∈[0..15):wj​(X)=i=0∑3​trace[i,j]⋅LgiG​(X) Then, the above constraints will become the following equations using our 15 univariate witness polynomials: Generic: w0​(X)⋅cl​+w1​(X)⋅cr​+w2​(X)⋅co​+w0​(X)⋅w1​(X)⋅cm​+cc​=0 CompleteAdd: w10​(X)⋅(w2​(X)−w0​(X))−(1−w7​(X))=0 w7​(X)⋅(w2​(X)−w0​(X))=0 w7​(X)⋅(2⋅w8​(X)⋅w1​(X)−3⋅w0​(X)2)+(1−w7​(X))⋅((w2​(X)−w0​(X))⋅w8​(X)−w3​(X)+w1​(X))=0 w0​(X)+w2​(X)+w4​(X)−w8​(X)2=0 w8​(X)⋅(w0​(X)−w4​(X))−w1​(X)−w4​(X)=0 (w3​(X)−w1​(X))⋅(w7​(X)−w6​(X))=0 (w3​(X)−w1​(X))⋅w9​(X)−w6​(X)=0 ExamplePairs: w0​(X)−w1​(X)=0 w2​(X)−w3​(X)=0 w4​(X)−w5​(X)=0 ExampleTriples: w0​(X)⋅w1​(X)⋅w2​(X)=0 w3​(X)⋅w4​(X)⋅w5​(X)=0 w6​(X)⋅w7​(X)⋅w8​(X)=0 w9​(X)⋅w10​(X)⋅w11​(X)=0 w12​(X)⋅w13​(X)⋅w14​(X)=0 These are the 16 constraints that will need to be satisfied by a correct witness for the relation. We already know how to check many equations at once (combining them with powers of alpha), and how to check that the above holds for the full domain G (by providing the quotient polynomial of the huge equation after dividing by the vanishing polynomial on G). All there's left to do is making a correspondence between the constraints that need to be satisfied, and the particular row in the execution trace to which they apply. This implies, that before even aggregating the 16 constraints into a single one, we need to impose this restriction. In proof-systems, we represent this restriction with the GateType selector. In particular, we would multiply each of the constraints above by the Expr terms: Index(GateType::YourType), which will be transformed (again) into mutually exclusive selector polynomials. This means, that each row can only have one of these polynomials being nonzero (equal to 1), whereas the evaluation of all of the other selector polynomials for the given row should be 0. These polynomials are known by the verifier, who could check that this is indeed the case. But what will these polynomials look like? I am sure the above paragraph is familiar for you already. Once again, we will be able to build them using interpolation with our 4 Lagrangian terms for G. In our example: generic(X) will equal 1 when X=g0 and 0 otherwise; add(X) will equal 1 when X=g1 and 0 otherwise; pairs(X) will equal 1 when X=g2 and 0 otherwise; and triples(X) will equal 1 when X=g3 and 0 otherwise. These polynomials are, in fact, a description of the circuit. Note that we managed to make constraints for different gates already independent from each other (thanks to the mutually exclusivity). Nonetheless, granting independence within the gate is still needed. Here's where we will be using the powers of alpha: within a gate the powers cannot be repeated, but they can be reused among gates. Then, we will need 7 different powers of alpha for this stage of the protocol, as this is the largest number of different constraints within the same gate. Putting all of the above together, this means that our LONG and single constraint will look like: gates(X)= generic(X)⋅ α0⋅(w0​(X)⋅cl​+w1​(X)⋅cr​+w2​(X)⋅co​+w0​(X)⋅w1​(X)⋅cm​+cc​) +add(X)⋅( α0⋅(w10​(X)⋅(w2​(X)−w0​(X))−(1−w7​(X))) +α1⋅(w7​(X)⋅(w2​(X)−w0​(X))) +α2⋅(w7​(X)⋅(2⋅w8​(X)⋅w1​(X)−3⋅w0​(X)2)+(1−w7​(X))⋅((w2​(X)−w0​(X))⋅w8​(X)−w3​(X)+w1​(X))) +α3⋅(w0​(X)+w2​(X)+w4​(X)−w8​(X)2) +α4⋅(w8​(X)⋅(w0​(X)−w4​(X))−w1​(X)−w4​(X)) +α5⋅((w3​(X)−w1​(X))⋅(w7​(X)−w6​(X))) +α6⋅((w3​(X)−w1​(X))⋅w9​(X)−w6​(X))) +pairs(X)⋅( α0⋅(w0​(X)−w1​(X)) +α1⋅(w2​(X)−w3​(X)) +α2⋅(w4​(X)−w5​(X))) +triples(X)⋅( α0⋅(w0​(X)⋅w1​(X)⋅w2​(X)) +α1⋅(w3​(X)⋅w4​(X)⋅w5​(X)) +α2⋅(w6​(X)⋅w7​(X)⋅w8​(X)) +α3⋅(w9​(X)⋅w10​(X)⋅w11​(X)) +α4⋅(w12​(X)⋅w13​(X)⋅w14​(X))) Finally, providing q(X)=gates(X)/vG​(X) and performing the check on a random X∈F \\ G would give the verifier the assurance of the following: The prover knows a polynomial gates(X) that equals zero on any x∈{1,g,g2,g3}. Nonetheless, it would still remain to verify that gates(X) actually corresponds to the encoding of the actual constraints. Meaning, checking that this polynomial encodes the column witnesses, and the agreed circuit. So instead of providing just gates(X) (actually, a commitment to it), the prover can send commitments to each of th 15 witness polynomials, so that the verifier can reconstruct the huge constraint using the encodings of the circuit (which is known ahead).","breadcrumbs":"Custom Gates » Custom gates","id":"148","title":"Custom gates"},"149":{"body":"This document is meant to explain the foreign field addition gate in Kimchi.","breadcrumbs":"Custom Gates » Foreign Field Addition » Foreign Field Addition RFC","id":"149","title":"Foreign Field Addition RFC"},"15":{"body":"The next object to know about is polynomials. A polynomial is a syntactic object that also defines a function. Specifically, let R be a field (or more generally it could even be a ring, which is like a field but without necessarily having division, like the integers Z). And pick a variable name like x. Then R[x] (pronounced, \"R adjoin x\" or \"polynomials in x over R\" or \"polynomials in x with R coefficients\") is the set of syntactic expressions of the form a0​+a1​x+a2​x2+⋯+ad​xd where each ai​∈R and d is any natural number. If we wanted to be very formal about it, we could define R[x] to be the set of lists of elements of R (the lists of coefficients), and the above is just a suggestive way of writing that list. An important fact about polynomials over R is that they can be interpreted as functions R→R. In other words, there is a function eval:R[x]→(R→R) defined by eval(a0​+a1​x+⋯+ad​xd)=(y:R)↦a0​+a1​y+⋯+ad​yd where x is just a variable name without a value assigned to it and eval maps it to a field element y and evaluates the function as the inner product between the list of coefficients and the powers of y. It is important to remember that polynomials are different than the functions that they compute. Polynomials are really just lists of coefficients. Over some fields, there are polynomials p1 and p2 such that eval(p1)=eval(p2), but p1=p2. For example, consider in F2​[x] the polynomials x and x2. These map to the same function F2​→F2​ (meaning for x∈R=F2​={0,1} it holds that x=x2), but are distinct polynomials. Some definitions and notation If f is a polynomial in R[x] and a∈R, we write f(a) for eval(f)(a) If f is a polynomial, the degree of f is the largest d for which the coefficient of xd is non-zero in f. For example, the degree of x3+2x10 is 10. We will use the notation R[x]n. In order to deal with this, we will divide foreign field elements into limbs that fit in our native field. We want to be compatible with the foreign field multiplication gate, and thus the parameters we will be using are the following: 3 limbs of 88 bits each (for a total of 264 bits) So f<2264, concretely: The modulus f of the foreign field Ff​ is 256 bit The modulus of our the native field Fn​ is 255 bit In other words, using 3 limbs of 88 bits each allows us to represent any foreign field element in the range [0,2264) for foreign field addition, but only up to 2259 for foreign field multiplication. Thus, with the current configuration of our limbs, our foreign field must be smaller than 2259 (because 2264⋅2255>22592+2259, more on this in Foreign Field Multiplication or the original FFmul RFC .","breadcrumbs":"Custom Gates » Foreign Field Addition » Overview","id":"150","title":"Overview"},"151":{"body":"Let's take a closer look at what we have, if we split our variables in limbs (using little endian) bits 0..............87|88...........175|176...........263 a = (-------a0-------|-------a1-------|-------a2-------)\n+/-\nb = (-------b0-------|-------b1-------|-------b2-------)\n=\nr = (-------r0-------|-------r1-------|-------r2-------) mod(f) We will perform the addition in 3 steps, one per limb. Now, when we split long additions in this way, we must be careful with carry bits between limbs. Also, even if a and b are foreign field elements, it could be the case that a+b is already larger than the modulus (in this case a+b could be at most 2f−2). But it could also be the case that the subtraction produces an underflow because a > >\n= c_0 c_1 c_2\nq = -1 | 0 | 1\n·\nf = (-------f0-------|-------f1-------|-------f2-------)\n+\nr = (-------r0-------|-------r1-------|-------r2-------) Our witness computation is currently using the BigUint library, which takes care of all of the intermediate carry limbs itself so we do not have to address all casuistries ourselves (such as propagating the low carry flag to the high limb in case the middle limb is all zeros).","breadcrumbs":"Custom Gates » Foreign Field Addition » Splitting the addition","id":"151","title":"Splitting the addition"},"152":{"body":"Last but not least, we should perform some range checks to make sure that the result r is contained in Ff​. This is important because there could be other values of the result which still fit in <2264 but are larger than f, and we must make sure that the final result is the minimum one (that we will be referring to as rmin​ in the following). Ideally, we would like to reuse some gates that we already have. In particular, we can perform range checks for 0≤X<23ℓ=23⋅88. But we want to check that 0≤rmin​ >\n= k_0 k_1\nf\n+\nu = (-------u0-------|-------u1-------|-------u2-------) Following the steps above, and representing this equation in limb form, we have: u0​u1​u2​​=r0​+0−f0​−k0​⋅288=r1​+0−f1​−k1​⋅288+k0​=r2​+288−f2​+k1​​ But now we also have to check that 0≤r. But this is implicitly covered by r being a field element of at most 264 bits (range check). When we have a chain of additions ai​+bi​=ri​ with ai​=ri−1​, we could apply the field membership check naively to every intermediate ri​, however it is sufficient to apply it only once at the end of the computations for rn​, and keep intermediate ri​∈[0,2264), in a \"raw\" form. Generally speaking, treating intermediate values lazily helps to save a lot of computation in many different FF addition and multiplication scenarios.","breadcrumbs":"Custom Gates » Foreign Field Addition » Upper bound check for foreign field membership","id":"152","title":"Upper bound check for foreign field membership"},"153":{"body":"Mathematically speaking, a subtraction within a field is no more than an addition over that field. Negative elements are not different from \"positive\" elements in finite fields (or in any modular arithmetic). Our witness computation code computes negative sums by adding the modulus to the result. To give a general example, the element −e within a field Fm​ of order m and e > > c_0 c_1 0 Inspired by the halving approach in foreign field multiplication, an optimized version of the above gate results in a reduction by 2 in the number of constraints and by 1 in the number of witness cells needed. The main idea is to condense the claims about the low and middle limbs in one single larger limb of 176 bits, which fit in our native field. This way, we can get rid of the low carry flag, its corresponding carry check, and the three result checks become just two. bits 0..............87|88...........175|176...........263\nr = (-------r0------- -------r1-------|-------r2-------)\n=\na = (-------a0------- -------a1-------|-------a2-------)\n+\ns = 1 | -1\n·\nb = (-------b0------- -------b1-------|-------b2-------)\n-\nq = -1 | 0 | 1\n·\nf = (-------f0------- -------f1-------|-------f2-------) > > c 0 These are the new equations: rbot​rtop​​=(a0​+288⋅a1​)+s⋅(b0​+288⋅b1​)−q⋅(f0​+288⋅f1​)−c⋅2176=a2​+s⋅b2​−q⋅f2​+c​ with rtop​=r2​ and c=c1​.","breadcrumbs":"Custom Gates » Foreign Field Addition » Optimization","id":"154","title":"Optimization"},"155":{"body":"The full foreign field addition/subtraction gadget will be composed of: 1 public input row containing the value 1; n rows with ForeignFieldAdd gate type: for the actual n chained additions or subtractions; 1 ForeignFieldAdd row for the bound addition; 1 Zero row for the final bound addition. 1 RangeCheck gadget for the first left input of the chain a1​:=a; Then, n of the following set of rows: 1 RangeCheck gadget for the i-th right input bi​; 1 RangeCheck gadget for the i-th result which will correspond to the (i+1)-th left input of the chain ri​=ai+1​. 1 final RangeCheck gadget for the bound check u. A total of 20 rows with 15 columns in Kimchi for 1 addition. All ranges below are inclusive. Row(s) Gate type(s) Witness 0 PublicInput 1 1..n ForeignFieldAdd +/- n+1 ForeignFieldAdd bound n+2 Zero bound n+3..n+6 multi-range-check left n+7+8i..n+10+8i multi-range-check right n+11+8i..n+14+8i multi-range-check r 9n+7..9n+10 multi-range-check u This mechanism can chain foreign field additions together. Initially, there are n foreign field addition gates, followed by a foreign field addition gate for the bound addition (whose current row corresponds to the next row of the last foreign field addition gate), and an auxiliary Zero row that holds the upper bound. At the end, an initial left input range check is performed, which is followed by a n pairs of range check gates for the right input and intermediate result (which become the left input for the next iteration). After the chained inputs checks, a final range check on the bound takes place. For example, chaining the following set of 3 instructions would result in a full gadget with 37 rows: add(add(add(a,b),c),d) Row(s) Gate type(s) Witness 0 PublicInput 1 1 ForeignFieldAdd a+b 2 ForeignFieldAdd (a+b)+c 3 ForeignFieldAdd ((a+b)+c)+d 4 ForeignFieldAdd bound 5 Zero bound 6..9 multi-range-check a 10..13 multi-range-check b 14..17 multi-range-check a+b 18..21 multi-range-check c 22..25 multi-range-check a+b+c 26..29 multi-range-check d 30..33 multi-range-check a+b+c+d 34..37 multi-range-check bound Nonetheless, such an exhaustive set of checks are not necessary for completeness nor soundness. In particular, only the very final range check for the bound is required. Thus, a shorter gadget that is equally valid and takes (8⋅n+4) fewer rows could be possible if we can assume that the inputs of each addition are correct foreign field elements. It would follow the next layout (with inclusive ranges): Row(s) Gate type(s) Witness 0 public input row for soundness of bound overflow flag 1 1..n ForeignFieldAdd n+1 ForeignFieldAdd n+2 Zero n+3..n+6 multi-range-check for bound u Otherwise, we would need range checks for each new input of the chain, but none for intermediate results; implying 4⋅n fewer rows. Row(s) Gate type(s) Witness 0 public input row for soundness of bound overflow flag 1 1..n ForeignFieldAdd ai​+bi​ n+1 ForeignFieldAdd n+2 Zero n+3..n+6 multi-range-check for first left input a1​ n+7+4i..n+10+4i multi-range-check for i-th right input bi​ 5n+7..5n+10 multi-range-check for bound u For more details see the Bound Addition section in Foreign Field Multiplication or the original Foreign Field Multiplication RFC .","breadcrumbs":"Custom Gates » Foreign Field Addition » Gadget","id":"155","title":"Gadget"},"156":{"body":"For the full mode of tests of this gate, we need to perform 4 range checks to assert that the limbs of a,b,r,u have a correct size, meaning they fit in 288 (and thus, range-checking a,b,r,u for 2264). Because each of these elements is split into 3 limbs, we will have to use 3 copy constraints between the RangeCheck gates and the ForeignFieldAdd rows (per element). That amounts to 12 copy constraints. Recalling that Kimchi only allows for the first 7 columns of each row to host a copy constraint, we necessarily have to use 2 rows for the actual addition gate. The layout of these two rows is the following: Curr Next ... Final Column ForeignFieldAdd ForeignFieldAdd Zero 0 a0​ (copy) r0​ (copy) u0​ (copy) 1 a1​ (copy) r1​ (copy) u1​ (copy) 2 a2​ (copy) r2​ (copy) u2​ (copy) 3 b0​ (copy) 4 b1​ (copy) 5 b2​ (copy) 6 q 7 c 8 9 10 11 12 13 14","breadcrumbs":"Custom Gates » Foreign Field Addition » Layout","id":"156","title":"Layout"},"157":{"body":"So far, we have pointed out the following sets of constraints: Main addition 0=a0​+b0​−r0​−q⋅f0​−288⋅c0​ 0=a1​+b1​−r1​−q⋅f1​+c0​−288⋅c1​ 0=a2​+b2​−r2​−q⋅f2​+c1​ Carry checks 0=c0​⋅(c0​+1)⋅(c0​−1) 0=c1​⋅(c0​+1)⋅(c1​−1) Sign checks TODO: decide if we really want to keep this check or leave it implicit as it is a coefficient value 0=(s+1)⋅(s−1) Overflow checks 0=q⋅(q−s)","breadcrumbs":"Custom Gates » Foreign Field Addition » Constraints","id":"157","title":"Constraints"},"158":{"body":"When we use this gate as part of a larger chained gadget, we should optimize the number of range check rows to avoid redundant checks. In particular, if the result of an addition becomes one input of another addition, there is no need to check twice that the limbs of that term have the right length. The sign is now part of the coefficients of the gate. This will allow us to remove the sign check constraint and release one witness cell element. But more importantly, it brings soundness to the gate as it is now possible to wire the 1 public input to the overflow witness of the final bound check of every addition chain.","breadcrumbs":"Custom Gates » Foreign Field Addition » Optimizations","id":"158","title":"Optimizations"},"159":{"body":"This document is an original RFC explaining how we constrain foreign field multiplication (i.e. non-native field multiplication) in the Kimchi proof system. For a more recent RFC, see FFmul RFC . Changelog Author(s) Date Details Joseph Spadavecchia and Anais Querol October 2022 Design of foreign field multiplication in Kimchi","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Foreign Field Multiplication","id":"159","title":"Foreign Field Multiplication"},"16":{"body":"An important fact for zk-SNARKs about polynomials is that we can use them to encode arrays of field elements. In other words, there is a way of converting between polynomials of degree d and arrays of field elements of length d+1. This is important for a few reasons It will allow us to translate statements about arrays of field elements into statements about polynomials, which will let us construct zk-SNARKs. It will allow us to perform certain operations on polynomials more efficiently. So let's get to defining and proving this connection between arrays of field elements. The first thing to understand is that we won't be talking about arrays directly in this section. Instead, we'll be talking about functions A→F where A is a finite subset of F. The idea is, if A has size n, and we put an ordering on A, then we can think of a function A→F as the same thing as an immutable array of length n, since such a function can be thought of as returning the value of an array at the input position. With this understanding in hand, we can start describing the \"fundamental theorem of polynomials\". If A⊆F has size d+1, this theorem will define an isomorphism between functions A→F and F[x]≤d​, the set of polynomials of degree at most d. Now let's start defining this isomorphism. One direction of it is very easy to define. It is none other than the evaluation map, restricted to A: evalA​evalA​​:F[x]≤d​→(A→F)(c0​+⋯+cd​xd)=a↦i≤d∑​ci​ai​ We would now like to construct an inverse to this map. What would that mean? It would be a function that takes as input a function φ:A→F (remember, basically an array of length ∣A∣), and returns a polynomial f which agrees with φ on the set A. In other words, we want to construct a polynomial that interpolates between the points (a,φ(a)) for a∈A. Our strategy for constructing this polynomial will be straightforward. For each a∈A, we will construct a polynomial fa​ that is equal to φ(a) when evaluated at a, and equal to 0 when evaluated anywhere else in the set A. Then, our final polynomial will be f:=∑a∈A​fa​. Then, when f is evaluated at a0​∈A, only the fa0​​ term will be non-zero (and equal to φ(a0​) as desired), all the other terms will disappear. Constructing the interpolation map requires a lemma. Lemma. (construction of vanishing polynomials) Let S⊆F. Then there is a polynomial vS​∈F[x] of degree ∣S∣ such that vS​ evaluates to 0 on S, and is non-zero off of S. vS​ is called the vanishing polynomial of S. Proof. Let vS​=∏s∈S​(x−s). Clearly vS​ has degree ∣S∣. If t∈S, then vS​(t)=0 since x−t is one of the terms in the product defining vS​, so when we evaluate at t, t−t=0 is one of the terms. If t∈/S, then all the terms in the product are non-zero, and thus vS​(t) is non-zero as well. □ Now we can define the inverse map. Define interpA​interpA​​:(A→F)→F[x]≤d​(f)=a∈A∑​vA∖{a}​(a)f(a)​⋅vA∖{a}​​ Since each vA∖{a}​ has degree d, this polynomial has degree d. Now we have, for any b∈A, evalA​(interpA​(f))(b)​=a∈A∑​vA∖{a}​(a)f(a)​⋅vA∖{a}​(b)=vA∖{b}​(b)f(b)​⋅vA∖{b}​(b)+a=b∑​vA∖{a}​(a)f(a)​⋅vA∖{a}​(b)=f(b)+a=b∑​vA∖{a}​(a)f(a)​⋅0=f(b)​ Thus evalA​∘interpA​ is the identity. So we have successfully devised a way of interpolating a set of d+1 points with a polynomial of degree d. What remains to be seen is that these two maps are inverse in the other direction. That is, that for any polynomial f∈F[x], we have interpA​(evalA​(f))=f This says that if we interpolate the function that has the values of f on A, we get back f itself. This essentially says that there is only one way to interpolate a set of d+1 points with a degree d polynomial. So let's prove it by proving that statement. Lemma: polynomials that agree on enough points are equal. Let F be a field and suppose f,g∈F[x] have degree at most d. Let A⊆F have size d+1. Suppose for all a∈A, f(a)=g(a). Then f=g as polynomials. Proof. Define h:=f−g. Our strategy will be to show that h is the zero polynomial. This would then imply that f=g as polynomials. Note that h vanishes on all of A since f and g are equal on A. To that end, let a∈A. Then we can apply the polynomial division algorithm to divide h by x−a and obtain polynomials qa​,ra​ such that h=qa​(x−a)+ra​ and ra​ has degree less than 1. I.e., ra​ is a constant in F. Now, 0=h(a)=qa​(a)⋅0+ra​=ra​ so ra​=0 and thus h=qa​(x−a). Note that qa​ is 0 on all elements b∈A with b=a since h(b)=qa​(b)(b−a)=0, but b−a=0. Thus, if we iterate the above, enumerating the elements of A as a0​,…,ad​, we find that h​=(x−a0​)qa0​​=(x−a0​)(x−a1​)qa1​​=…=(x−a0​)(x−a1​)…(x−ad​)qad​​​ Now, if qad​​ is not the zero polynomial, then h will have degree at least d+1 since it would have as a factor the d+1 linear terms (x−ai​). But since f,g both have degree at most d and h=f−g, h has degree at most d as well. Thus, qad​​=0, which means h=0 as well. This gives us the desired conclusion, that f=g as polynomials. □ Now we can easily show that interpolating the evaluations of a polynomial yields that polynomial itself. Let f∈F[x]≤d​. Then interpA​(evalA​(f)) is a polynomial of degree at most d that agrees with f on A, a set of size d+1. Thus, by the lemma, they are equal as polynomials. So indeed interpA​(evalA​(f))=f for all f∈F[x]. So far we have proven that interpA​ and evalA​ give an isomorphism of sets (i.e., a bijection) between the sets A→F and F[x]≤d​. But we can take this theorem a bit further. The set of functions A→F can be given the structure of a ring, where addition and multiplication happen pointwise. I.e., for f,g:A→F we define f+g:=a↦f(a)+g(a) and f⋅g:=a↦f(a)⋅g(a). Then we can strengthen our theorem to say Fundamental theorem of polynomials (final version) Let d∈N and let A⊆F with ∣A∣=d+1. With evalA​:F[x]≤d​→(A→F)interpA​:(A→F)→F[x]≤d​ defined as above, these two functions define an isomorphism of rings. That is is, they are mutually inverse and each one respects addition, subtraction and multiplication. The fundamental theorem of polynomials is very important when it comes to computing operations on polynomials. As we will see in the next section , the theorem will help us to compute the product of degree n polynomials in time O(nlogn), whereas the naive algorithm takes time O(n2). To put this in perspective, if n=216, n2 is 4096 times larger than nlogn and the gap only gets bigger as n grows.","breadcrumbs":"Polynomials » Fundamental theorem of polynomials","id":"16","title":"Fundamental theorem of polynomials"},"160":{"body":"This gate constrains a⋅b=cmodf where a,b,c∈Ff​, a foreign field with modulus f, using the native field Fn​ with prime modulus n.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Overview","id":"160","title":"Overview"},"161":{"body":"In foreign field multiplication the foreign field modulus f could be bigger or smaller than the native field modulus n. When the foreign field modulus is bigger, then we need to emulate foreign field multiplication by splitting the foreign field elements up into limbs that fit into the native field element size. When the foreign modulus is smaller everything can be constrained either in the native field or split up into limbs. Since our projected use cases are when the foreign field modulus is bigger (more on this below) we optimize our design around this scenario. For this case, not only must we split foreign field elements up into limbs that fit within the native field, but we must also operate in a space bigger than the foreign field. This is because we check the multiplication of two foreign field elements by constraining its quotient and remainder. That is, renaming c to r, we must constrain a⋅b=q⋅f+r, where the maximum size of q and r is f−1 and so we have a⋅b​≤q(f−1)​​⋅f+r(f−1)​​≤f2−1.​ Thus, the maximum size of the multiplication is quadratic in the size of foreign field. Naïve approach Naïvely, this implies that we have elements of size f2−1 that must split them up into limbs of size at most n−1. For example, if the foreign field modulus is 256 and the native field modulus is 255 bits, then we'd need log2​((2256)2−1)≈512 bits and, thus, require 512/255≈3 native limbs. However, each limb cannot consume all 255 bits of the native field element because we need space to perform arithmetic on the limbs themselves while constraining the foreign field multiplication. Therefore, we need to choose a limb size that leaves space for performing these computations. Later in this document (see the section entitled \"Choosing the limb configuration\") we determine the optimal number of limbs that reduces the number of rows and gates required to constrain foreign field multiplication. This results in ℓ=88 bits as our optimal limb size. In the section about intermediate products we place some upperbounds on the number of bits required when constraining foreign field multiplication with limbs of size ℓ thereby proving that the computations can fit within the native field size. Observe that by combining the naïve approach above with a limb size of 88 bits, we would require 512/88≈6 limbs for representing foreign field elements. Each limb is stored in a witness cell (a native field element). However, since each limb is necessarily smaller than the native field element size, it must be copied to the range-check gate to constrain its value. Since Kimchi only supports 7 copyable witness cells per row, this means that only one foreign field element can be stored per row. This means a single foreign field multiplication would consume at least 4 rows (just for the operands, quotient and remainder). This is not ideal because we want to limit the number of rows for improved performance. Chinese remainder theorem Fortunately, we can do much better than this, however, by leveraging the chinese remainder theorem (CRT for short) as we will now show. The idea is to reduce the number of bits required by constraining our multiplication modulo two coprime moduli: 2t and n. By constraining the multiplication with both moduli the CRT guarantees that the constraints hold with the bigger modulus 2t⋅n. The advantage of this approach is that constraining with the native modulus n is very fast, allowing us to speed up our non-native computations. This practically reduces the costs to constraining with limbs over 2t, where t is something much smaller than 512. For this to work, we must select a value for t that is big enough. That is, we select t such that 2t⋅n>f2−1. As described later on in this document, by doing so we reduce the number of bits required for our use cases to 264. With 88 bit limbs this means that each foreign field element only consumes 3 witness elements, and that means the foreign field multiplication gate now only consumes 2 rows. The section entitled \"Choosing t\" describes this in more detail. Overall approach Bringing it all together, our approach is to verify that a⋅b=q⋅f+r​​ over Z+. In order to do this efficiently we use the CRT, which means that the equation holds mod M=2t⋅n. For the equation to hold over the integers we must also check that each side of the equation is less than 2t⋅n. The overall steps are therefore Apply CRT to equation (1) Check validity with binary modulus mod2t Check validity with native modulus modn Check each side of equation (1) is less than M a⋅b<2t⋅n q⋅f+r<2t⋅n This then implies that a⋅b=cmodf. where c=r. That is, a multiplied with b is equal to c where a,b,c∈Ff​. There is a lot more to each of these steps. That is the subject of the rest of this document. Other strategies Within our overall approach, aside from the CRT, we also use a number of other strategies to reduce the number and degree of constraints. Avoiding borrows and carries Intermediate product elimination Combining multiplications The rest of this document describes each part in detail.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Approach","id":"161","title":"Approach"},"162":{"body":"This section describes important parameters that we require and how they are computed. Native field modulus n Foreign field modulus f Binary modulus 2t CRT modulus 2t⋅n Limb size in bits ℓ Choosing t Under the hood, we constrain a⋅b=q⋅f+rmod2t⋅n. Since this is a multiplication in the foreign field f, the maximum size of q and r are f−1, so this means a⋅b​≤(f−1)⋅f+(f−1)≤f2−1.​ Therefore, we need the modulus 2t⋅n such that 2t⋅n>f2−1, which is the same as saying, given f and n, we must select t such that 2t⋅nt​≥f2≥2log2​(f)−log2​(n).​ Thus, we have an lower bound on t. Instead of dynamically selecting t for every n and f combination, we fix a t that will work for the different selections of n and f relevant to our use cases. To guide us, from above we know that tmin​=2log2​(f)−log2​(n) and we know the field moduli for our immediate use cases. vesta = 2^254 + 45560315531506369815346746415080538113 (255 bits)\npallas = 2^254 + 45560315531419706090280762371685220353 (255 bits)\nsecp256k1 = 2^256 - 2^32 - 2^9 - 2^8 - 2^7 - 2^6 - 2^4 - 1 (256 bits) So we can create a table n f tmin​ vesta secp256k1 258 pallas secp256k1 258 vesta pallas 255 pallas vesta 255 and know that to cover our use cases we need t≥258. Next, given our native modulus n and t, we can compute the maximum foreign field modulus supported . Actually, we compute the maximum supported bit length of the foreign field modulus f=2m. 2t⋅nt+log2​(n)​≥f2≥(2m)2=22m>log2​(22m)=2m​ So we get mmax​=2t+log2​(n)​. With t=258,n=255 we have mmax​​=2258+255​=256.5,​ which is not enough space to handle anything larger than 256 bit moduli. Instead, we will use t=264, giving mmax​=259 bits. The above formula is useful for checking the maximum number of bits supported of the foreign field modulus, but it is not useful for computing the maximum foreign field modulus itself (because 2mmax​ is too coarse). For these checks, we can compute our maximum foreign field modulus more precisely with maxmod​=⌊2t⋅n​⌋. The max prime foreign field modulus satisfying the above inequality for both Pallas and Vesta is 926336713898529563388567880069503262826888842373627227613104999999999999999607. Choosing the limb configuration Choosing the right limb size and the right number of limbs is a careful balance between the number of constraints (i.e. the polynomial degree) and the witness length (i.e. the number of rows). Because one limiting factor that we have in Kimchi is the 12-bit maximum for range check lookups, we could be tempted to introduce 12-bit limbs. However, this would mean having many more limbs, which would consume more witness elements and require significantly more rows. It would also increase the polynomial degree by increasing the number of constraints required for the intermediate products (more on this later). We need to find a balance between the number of limbs and the size of the limbs. The limb configuration is dependent on the value of t and our maximum foreign modulus (as described in the previous section). The larger the maximum foreign modulus, the more witness rows we will require to constrain the computation. In particular, each limb needs to be constrained by the range check gate and, thus, must be in a copyable (i.e. permuteable) witness cell. We have at most 7 copyable cells per row and gates can operate on at most 2 rows, meaning that we have an upperbound of at most 14 limbs per gate (or 7 limbs per row). As stated above, we want the foreign field modulus to fit in as few rows as possible and we need to constrain operands a,b, the quotient q and remainder r. Each of these will require cells for each limb. Thus, the number of cells required for these is cells=4⋅limbs It is highly advantageous for performance to constrain foreign field multiplication with the minimal number of gates. This not only helps limit the number of rows, but also to keep the gate selector polynomial small. Because of all of this, we aim to constrain foreign field multiplication with a single gate (spanning at most 2 rows). As mentioned above, we have a maximum of 14 permuteable cells per gate, so we can compute the maximum number of limbs that fit within a single gate like this. limbsmax​​=⌊cells/4⌋=⌊14/4⌋=3​ Thus, the maximum number of limbs possible in a single gate configuration is 3. Using limbsmax​=3 and t=264 that covers our use cases (see the previous section), we are finally able to derive our limbs size ℓ​=limbsmax​t​=264/3=88​ Therefore, our limb configuration is: Limb size ℓ=88 bits Number of limbs is 3","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Parameter selection","id":"162","title":"Parameter selection"},"163":{"body":"When we constrain a⋅b−q⋅f=rmod2t we want to be as efficient as possible. Observe that the expansion of a⋅b−q⋅f into limbs would also have subtractions between limbs, requiring our constraints to account for borrowing. Dealing with this would create undesired complexity and increase the degree of our constraints. In order to avoid the possibility of subtractions we instead use a⋅b+q⋅f′ where f′​=−fmod2t=2t−f​ The negated modulus f′ becomes part of our gate coefficients and is not constrained because it is publicly auditable. Using the substitution of the negated modulus, we now must constrain a⋅b+q⋅f′=rmod2t. Observe that f′<2t since f<2t and that f′>f when f<2t−1.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Avoiding borrows","id":"163","title":"Avoiding borrows"},"164":{"body":"This section explains how we expand our constraints into limbs and then eliminate a number of extra terms. We must constrain a⋅b+q⋅f′=rmod2t on the limbs, rather than as a whole. As described above, each foreign field element x is split into three 88-bit limbs: x0​,x1​,x2​, where x0​ contains the least significant bits and x2​ contains the most significant bits and so on. Expanding the right-hand side into limbs we have ​(a0​+a1​⋅2ℓ+a2​⋅22ℓ)⋅(b0​+b1​⋅2ℓ+b2​⋅22ℓ)+(q0​+q1​⋅2ℓ+q2​⋅22ℓ)⋅(f0′​+f1′​⋅2ℓ+f2′​⋅22ℓ)= a0​⋅b0​+a0​⋅b1​⋅2ℓ+a0​⋅b2​⋅22ℓ +a1​⋅b0​⋅2ℓ+a1​⋅b1​⋅22ℓ+a1​⋅b2​⋅23ℓ +a2​⋅b0​⋅22ℓ+a2​⋅b1​⋅23ℓ+a2​⋅b2​⋅24ℓ+ q0​⋅f0′​+q0​⋅f1′​⋅2ℓ+q0​⋅f2′​⋅22ℓ +q1​⋅f0′​⋅2ℓ+q1​⋅f1′​⋅22ℓ+q1​⋅f2′​⋅23ℓ +q2​⋅f0′​⋅22ℓ+q2​⋅f1′​⋅23ℓ+q2​⋅f2′​⋅24ℓ=a0​⋅b0​+q0​⋅f0′​+2ℓ⋅(a0​⋅b1​+a1​⋅b0​+q0​⋅f1′​+q1​⋅f0′​)+22ℓ⋅(a0​⋅b2​+a2​⋅b0​+q0​⋅f2′​+q2​⋅f0′​+a1​⋅b1​+q1​⋅f1′​)+23ℓ⋅(a1​⋅b2​+a2​⋅b1​+q1​⋅f2′​+q2​⋅f1′​)+24ℓ⋅(a2​⋅b2​+q2​⋅f2′​)​ Since t=3ℓ, the terms scaled by 23ℓ and 24ℓ are a multiple of the binary modulus and, thus, congruent to zero mod2t. They can be eliminated and we don't need to compute them. So we are left with 3 intermediate products that we call p0​,p1​,p2​: Term Scale Product p0​ 1 a0​b0​+q0​f0′​ p1​ 2ℓ a0​b1​+a1​b0​+q0​f1′​+q1​f0′​ p2​ 22ℓ a0​b2​+a2​b0​+q0​f2′​+q2​f0′​+a1​b1​+q1​f1′​ So far, we have introduced these checked computations to our constraints Computation of p0​,p1​,p2​","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Intermediate products","id":"164","title":"Intermediate products"},"165":{"body":"Let's call p:=ab+qf′mod2t. Remember that our goal is to constrain that p−r=0mod2t (recall that any more significant bits than the 264th are ignored in mod2t). Decomposing that claim into limbs, that means 22ℓ(p2​−r2​)+2ℓ(p1​−r1​)+p0​−r0​=0mod2t.​​ We face two challenges Since p0​,p1​,p2​ are at least 2ℓ bits each, the right side of the equation above does not fit in Fn​ The subtraction of the remainder's limbs r0​ and r1​ could require borrowing For the moment, let's not worry about the possibility of borrows and instead focus on the first problem.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Constraining mod2t","id":"165","title":"Constraining mod2t"},"166":{"body":"The first problem is that our native field is too small to constrain 22ℓ(p2​−r2​)+2ℓ(p1​−r1​)+p0​−r0​=0mod2t. We could split this up by multiplying a⋅b and q⋅f′ separately and create constraints that carefully track borrows and carries between limbs. However, a more efficient approach is combined the whole computation together and accumulate all the carries and borrows in order to reduce their number. The trick is to assume a space large enough to hold the computation, view the outcome in binary and then split it up into chunks that fit in the native modulus. To this end, it helps to know how many bits these intermediate products require. On the left side of the equation, p0​ is at most 2ℓ+1 bits. We can compute this by substituting the maximum possible binary values (all bits set to 1) into p0​=a0​b0​+q0​f0′​ like this maxbits(p0​)​=log2​(a0​(2ℓ−1)​​b0​(2ℓ−1)​​+q0​(2ℓ−1)​​f0′​(2ℓ−1)​​)=log2​(2(22ℓ−2ℓ+1+1))=log2​(22ℓ+1−2ℓ+2+2).​ So p0​ fits in 2ℓ+1 bits. Similarly, p1​ needs at most 2ℓ+2 bits and p2​ is at most 2ℓ+3 bits. Term p0​ p1​ p2​ Bits 2ℓ+1 2ℓ+2 2ℓ+3 The diagram below shows the right hand side of the zero-sum equality from equation (2). That is, the value p−r. Let's look at how the different bits of p0​,p1​,p2​,r0​,r1​ and r2​ impact it. 0 L 2L 3L 4L\n|-------------|-------------|-------------|-------------|-------------| :\n|--------------p0-----------:-| 2L + 1 : |-------------:-p1------------| 2L + 2 p10➚ : p11➚ |----------------p2-------------| 2L + 3 :\n|-----r0------| : : |-----r1------| : |-----r2------|\n\\__________________________/ \\______________________________/ ≈ h0 ≈ h1 Within our native field modulus we can fit up to 2ℓ+δfk quickly overflows our CRT modulus 2t⋅n. For example, assuming our maximum foreign modulus of f=2259 and either of Kimchi's native moduli (i.e. Pallas or Vesta), fk>2t⋅n for k>2. That is, an overflow is possible for a chain of greater than 1 foreign field multiplication. Thus, we must check a,b,q and r are less than f for each multiplication. Fortunately, in many situations the input operands may already be checked either as inputs or as outputs of previous operations, so they may not be required for each multiplication operation. Thus, the q′ and r′ checks (or equivalently q and r) are our main focus because they must be done for every multiplication. Compute bound q′=q+f′ with addition gate (2 rows) Compute bound r′=r+f′ with addition gate (2 rows) Range check q′<2t (4 rows) Range check r′<2t (4 rows) This costs 12 rows per multiplication. In a subsequent section, we will reduce it to 8 rows.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Cost of bound checks","id":"174","title":"Cost of bound checks"},"175":{"body":"Due to the limited number of permutable cells per gate, we do not have enough cells for copy constraining q′ and r′ (or q and r) to their respective range check gadgets. To address this issue, we must decompose q′ into 2 limbs instead of 3, like so q′=q01′​+22ℓ⋅q2′​ and q01′​=q0′​+2ℓ⋅q1′​ Thus, q′ is decomposed into two limbs q01′​ (at most 2ℓ bits) and q2′​ (at most ℓ bits). Note that q′ must be range checked by a multi-range-check gadget. To do this the multi-range-check gadget must Store a copy of the limbs q0′​,q1′​ and q2′​ in its witness Range check that they are ℓ bit each Constrain that q01′​=q0′​+2ℓ⋅q1′​ (this is done with a special mode of the multi-range-check gadget) Have copy constraints for q01′​ and q2′​ as outputs of the ForeignFieldMul gate and inputs to the multi-range-check gadget Note that since the foreign field multiplication gate computes q′ from q which is already in the witness and q01′​ and q2′​ have copy constraints to a multi-range-check gadget that fully constrains their decomposition from q′, then the ForeignFieldMul gate does not need to store an additional copy of q0′​ and q1′​.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 2-limb decomposition","id":"175","title":"2-limb decomposition"},"176":{"body":"Since the q2ℓ where 2ℓ−1 is the maximum possible size of x1​ (before it overflows) and ϕ0​ is the overflow bit from the addition of the least significant limbs x0​ and f0′​. This means >2ℓ−ϕ0​−f1′​ We cannot allow x to overflow the foreign field, so we also have >x1​<(f−x0​)/22ℓ> Thus, >2ℓ−ϕ0​−f1′​<(f−x0​)/22ℓ=f/22ℓ−x0​/22ℓ> Since x0​/22ℓ=ϕ0​ we have >2ℓ−ϕ0​−f1′​ so >2ℓ−f1′​ Notice that f/22ℓ=f1​. Now we have >2ℓ−f1′​⟺>f1′​>2ℓ−f1​> However, this is a contradiction with the definition of our negated foreign field modulus limb f1′​=2ℓ−f1​. ■ We have proven that qcarry2′​ is always zero, so that allows use to simplify our constraints. We now have q0​∈[0,2ℓ) multi-range-check q1​∈[0,2ℓ) multi-range-check q01′​=q0′​+2ℓ⋅q1′​ multi-range-check qcarry01′​∈[0,2) ForeignFieldMul 22ℓ⋅qcarry01′​=s01​−q01′​ ForeignFieldMul q2​∈[0,2ℓ) multi-range-check q2′​=s2​+w01​ ForeignFieldMul In other words, we have eliminated constraint (7) and removed qcarry2′​ from the witness. Since we already needed to range-check q or q′, the total number of new constraints added is 4: 3 added to ForeignFieldMul and 1 added to multi-range-check gadget for constraining the decomposition of q01′​. This saves 2 rows per multiplication.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » An optimization","id":"176","title":"An optimization"},"177":{"body":"Due to the limited number of cells accessible to gates, we are not able to chain multiplications into multiplications. We can chain foreign field additions into foreign field multiplications, but currently do not support chaining multiplications into additions (though there is a way to do it).","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Chaining multiplications","id":"177","title":"Chaining multiplications"},"178":{"body":"Now we collect all of the checks that are required to constrain foreign field multiplication","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Constraining the computation","id":"178","title":"Constraining the computation"},"179":{"body":"If the a operand has not been constrained to [0,f) by any previous foreign field operations, then we constrain it like this Compute bound a′=a+f′ with addition gate (2 rows) ForeignFieldAdd Range check a′∈[0,2t) (4 rows) multi-range-check","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 1. Range constrain a","id":"179","title":"1. Range constrain a"},"18":{"body":"There are three common representations for polynomials used in computer implementations. Dense coefficient form. A degree d polynomial is represented as a vector of length d+1 of all the coefficients. Entry i of the vector corresponds to the coefficient ai​. This corresponds to the DensePolynomial type in arkworks. Sometimes this is also described as writing the polynomial \"in the monomial basis\", because it amounts to writing the polynomial as a linear combination of the monomials xi. Sparse coefficient form. If a polynomial does not have very many non-zero coefficients, the above representation is wasteful. In sparse coefficient form, you represent a polynomial as a vector (or potentially a hash-map) of pairs (usize, F) where F is the type of coefficients. The polynomial corresponding to the list [(i_0, b_0), ..., (i_n, b_n)] is b0​xi0​+⋯+bn​xin​ Evaluation form. We fix an index set A⊆F, with A={a0​,…,ad​}, and represent a polynomial f∈F[x]≤d​ as the vector [f(a_0), ..., f(a_d)]. By the fundamental theorem of polynomials, this is a valid way of representing the polynomials, since the coefficients form can be obtained by using the interpA​ function. The evaluation form is very important. The reason is that multiplying two polynomials in the evaluation form takes time O(n). You just multiply the two vectors entry-wise. By contrast, the coefficient forms naively require time O(n2) to multiply. Now, there is a trick. For certain sets A⊆F, we can efficiently translate between the dense coefficients form and the evaluation form. That is, for certain A, the functions interpA​ and evalA​ can be computed more efficiently than O(n2).","breadcrumbs":"Polynomials » Computer representation","id":"18","title":"Computer representation"},"180":{"body":"If the b operand has not been constrained to [0,f) by any previous foreign field operations, then we constrain it like this Compute bound b′=b+f′ with addition gate (2 rows) ForeignFieldAdd Range check b′∈[0,2t) (4 rows) multi-range-check","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 2. Range constrain b","id":"180","title":"2. Range constrain b"},"181":{"body":"The quotient q is constrained to [0,f) for each multiplication as part of the multiplication gate Compute bound q′=q+f′ with ForeignFieldMul constraints Range check q′∈[0,2t) (4 rows) multi-range-check Range check q≥0 ForeignFieldMul (implicit by storing q in witness)","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 3. Range constrain q","id":"181","title":"3. Range constrain q"},"182":{"body":"The remainder r is constrained to [0,f) for each multiplication using an external addition gate. Compute bound r′=r+f′ with addition gate (2 rows) ForeignFieldAdd Range check r′∈[0,2t) (4 rows) multi-range-check","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 4. Range constrain r","id":"182","title":"4. Range constrain r"},"183":{"body":"Compute and constrain the intermediate products p0​,p1​ and p2​ as: p0​=a0​⋅b0​+q0​⋅f0′​ ForeignFieldMul p1​=a0​⋅b1​+a1​⋅b0​+q0​⋅f1′​+q1​⋅f0′​ ForeignFieldMul p2​=a0​⋅b2​+a2​⋅b0​+a1​⋅b1​+q0​⋅f2′​+q2​⋅f0′​+q1​⋅f1′​ ForeignFieldMul where each of them is about 2ℓ-length elements.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 5. Compute intermediate products","id":"183","title":"5. Compute intermediate products"},"184":{"body":"Compute and constrain the native modulus values, which are used to check the constraints modulo n in order to apply the CRT an​=22ℓ⋅a2​+2ℓ⋅a1​+a0​modn bn​=22ℓ⋅b2​+2ℓ⋅b1​+b0​modn qn​=22ℓ⋅q2​+2ℓ⋅q1​+q0​modn rn​=22ℓ⋅r2​+2ℓ⋅r1​+r0​modn fn​=22ℓ⋅f2​+2ℓ⋅f1​+f0​modn","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 6. Native modulus checked computations","id":"184","title":"6. Native modulus checked computations"},"185":{"body":"Check that p1​=2ℓ⋅p11​+p10​: p1​=2ℓ⋅p11​+p10​ ForeignFieldMul Range check p10​∈[0,2ℓ) multi-range-check Range check p11​∈[0,2ℓ+2) p11​=p111​⋅2ℓ+p110​ ForeignFieldMul Range check p110​∈[0,2ℓ) multi-range-check Range check p111​∈[0,22) with a degree-4 constraint ForeignFieldMul","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 7. Decompose middle intermediate product","id":"185","title":"7. Decompose middle intermediate product"},"186":{"body":"Now we have to constrain the zero sum (p0​−r0​)+288(p1​−r1​)+2176(p2​−r2​)=0mod2t We constrain the first and the second halves as v0​⋅22ℓ=p0​+2ℓ⋅p10​−r0​−2ℓ⋅r1​ ForeignFieldMul v1​⋅2ℓ=(p111​⋅2ℓ+p110​)+p2​−r2​+v0​ ForeignFieldMul And some more range checks Check that v0​∈[0,22) with a degree-4 constraint ForeignFieldMul Check that v1​∈[0,2ℓ+3) Check v1​=v11​⋅288+v10​ ForeignFieldMul Check v11​∈[0,23] ForeignFieldMul Check v10​<2ℓ with range constraint multi-range-check To check that v11​∈[0,23) (i.e. that v11​ is at most 3 bits long) we first range-check v11​∈[0,212) with a 12-bit plookup. This means there can be no higher bits set beyond the 12-bits of v11​. Next, we scale v11​ by 29 in order to move the highest 12−3=9 bits beyond the 12th bit. Finally, we perform a 12-bit plookup on the resulting value. That is, we have Check v11​∈[0,212) with a 12-bit plookup (to prevent any overflow) Check scaledv11​​=29⋅v11​ Check scaledv11​​ is a 12-bit value with a 12-bit plookup Kimchi's plookup implementation is extremely flexible and supports optional scaling of the lookup target value as part of the lookup operation. Thus, we do not require two witness elements, two lookup columns, nor the scaledv11​​=29⋅v11​ custom constraint. Instead we can just store v11​ in the witness and define this column as a \"joint lookup\" comprised of one 12-bit plookup on the original cell value and another 12-bit plookup on the cell value scaled by 29, thus, yielding a 3-bit check. This eliminates one plookup column and reduces the total number of constraints.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 8. Zero sum for multiplication","id":"186","title":"8. Zero sum for multiplication"},"187":{"body":"Using the checked native modulus computations we constrain that an​⋅bn​−qn​⋅fn​−rn​=0modn.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 9. Native modulus constraint","id":"187","title":"9. Native modulus constraint"},"188":{"body":"Compute and constrain the intermediate sums s01​ and s2​ as: s01​=q01​+f01′​ s2​=q2​+f2′​ q01​=q0​+2ℓ⋅q1​ f01′​=f0′​+2ℓ⋅f1′​","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 10. Compute intermediate sums","id":"188","title":"10. Compute intermediate sums"},"189":{"body":"Check that q01′​=q0′​+2ℓ⋅q1′​. Done by (3) above with the multi-range-check on q′ q01′​=q0′​+2ℓ⋅q1′​ Range check q0′​∈[0,2ℓ) Range check q1′​∈[0,2ℓ)","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 11. Decompose the lower quotient bound","id":"189","title":"11. Decompose the lower quotient bound"},"19":{"body":"The algorithm that allows us to multiply polynomials in O(nlogn) is called the Cooley-Tukey fast Fourier transform , or FFT for short. It was discovered by Gauss 160 years earlier, but then separately rediscovered and publicized by Cooley-Tukey. The heart of Cooley-Tukey FFT is actually about converting between coefficient and evaluation representations, rather than the multiplication itself. Given polynomials p and q in dense coefficient representation, it works like this. Convert p and q from coefficient to evaluation form in O(nlogn) using Cooley-Tukey FFT Compute r=p∗q in evaluation form by multiplying their points pairwise in O(n) Convert r back to coefficient form in O(nlogn) using the inverse Cooley-Tukey FFT The key observation is that we can choose any n distinct evaluation points to represent any degree n−1 polynomial. The Cooley-Tukey FFT works by selecting points that yield an efficient FFT algorithm. These points are fixed and work for any polynomials of a given degree. The next section describes the Cooley-Tukey FFT in detail.","breadcrumbs":"Polynomials » Multiplying Polynomials » Multiplying polynomials","id":"19","title":"Multiplying polynomials"},"190":{"body":"We constrain that s01​−q01′​+22ℓ⋅(s2​−q2′​)=0mod2t by constraining the two halves 22ℓ⋅qcarry01′​=s01​−q01′​ 2ℓ⋅qcarry2′​=s2​+qcarry01′​−q2′​ We also need a couple of small range checks Check that qcarry01′​ is boolean ForeignFieldMul Check that q2′​ is boolean ForeignFieldMul","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 12. Zero sum for quotient bound addition","id":"190","title":"12. Zero sum for quotient bound addition"},"191":{"body":"Based on the constraints above, we need the following 12 values copied from the range check gates. a0, a1, a2, b0, b1, b2, q0, q1, q2, r0, r1, r2 Since we need 12 copied values for the constraints, they must span 2 rows. The q>s01​>>>​=q01​+f01′​=q0​+2ℓ⋅q1​+f0′​+2ℓ⋅f1′​=q0​+f0′​+2ℓ⋅(q1′​+f1′​)=q0′​+2ℓ⋅q1′​>​> where q0′​=q0​+f0′​ and q1′​=q1​+f1′​ can be done with checked computations. Next, for applying the CRT we compute an​=22ℓ⋅a2​+2ℓ⋅a1​+a0​modn bn​=22ℓ⋅b2​+2ℓ⋅b1​+b0​modn qn​=22ℓ⋅q2​+2ℓ⋅q1​+q0​modn rn​=22ℓ⋅r2​+2ℓ⋅r1​+r0​modn fn​=22ℓ⋅f2​+2ℓ⋅f1​+f0​modn","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Checked computations","id":"192","title":"Checked computations"},"193":{"body":"In total we require the following checks p111​∈[0,22) v10​∈[0,2ℓ) multi-range-check p110​∈[0,2ℓ) multi-range-check p10​∈[0,2ℓ) multi-range-check p11​=2ℓ⋅p111​+p110​ p1​=2ℓ⋅p11​+p10​ v0​∈[0,22) 22ℓ⋅v0​=p0​+2ℓ⋅p10​−r0​−2ℓ⋅r1​ v11​∈[0,23) v1​=2ℓ⋅v11​+v10​ 2ℓ⋅v1​=v0​+p11​+p2​−r2​ an​⋅bn​−qn​⋅fn​=rn​ q0′​∈[0,2ℓ) multi-range-check q1′​∈[0,2ℓ) multi-range-check q2′​∈[0,2ℓ) multi-range-check q01′​=q0′​+2ℓ⋅q1′​ multi-range-check qcarry01′​∈[0,2) 22ℓ⋅qcarry01′​=s01​−q01′​ q2′​=s2​+qcarry01′​","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Checks","id":"193","title":"Checks"},"194":{"body":"These checks can be condensed into the minimal number of constraints as follows. First, we have the range-check corresponding to (1) as a degree-4 constraint C1: p111​⋅(p111​−1)⋅(p111​−2)⋅(p111​−3) Checks (2) - (4) are all handled by a single multi-range-check gadget. C2: multi-range-check v10​,p10​,p110​ Next (5) and (6) can be combined into C3: 2ℓ⋅(2ℓ⋅p111​+p110​)+p10​=p1​ Now we have the range-check corresponding to (7) as another degree-4 constraint C4: v0​⋅(v0​−1)⋅(v0​−2)⋅(v0​−3) Next we have check (8) C5: 22ℓ⋅v0​=p0​+2ℓ⋅p10​−r0​−2ℓ⋅r1​ Up next, check (9) is a 3-bit range check C6: Plookup v11​ (12-bit plookup scaled by 29) Now checks (10) and (11) can be combined into C7: 2ℓ⋅(v11​⋅2ℓ+v10​)=p2​+p11​+v0​−r2​ Next, for our use of the CRT, we must constrain that a⋅b=q⋅f+rmodn. Thus, check (12) is C8: an​⋅bn​−qn​⋅fn​=rn​ Next we must constrain the quotient bound addition. Checks (13) - (16) are all combined into multi-range-check gadget C9: multi-range-check q0′​,q1′​,q2′​ and q01′​=q0′​+2ℓ⋅q1′​. Check (17) is a carry bit boolean check C10: qcarry01′​⋅(qcarry01′​−1) Next, check (18) is C11: 22ℓ⋅qcarry10′​=s01​−q01′​ Finally, check (19) is C12: q2′​=s2​+qcarry01′​ The Zero gate has no constraints and is just used to hold values required by the ForeignFieldMul gate.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Constraints","id":"194","title":"Constraints"},"195":{"body":"The following checks must be done with other gates to assure the soundness of the foreign field multiplication Range check input multi-range-check a multi-range-check b Range check witness data multi-range-check q′ and check q01′​=q0′​+2ℓq1′​ multi-range-check ℓ-bit limbs: p10​,p110​,p111​ Range check output multi-range-check either r or r′ Compute and constrain r′ the bound on r ForeignFieldAdd r+2t=1⋅f+r′ Copy constraints must connect the above witness cells to their respective input cells within the corresponding external check gates witnesses.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » External checks","id":"195","title":"External checks"},"196":{"body":"The Keccak gadget is comprised of 3 circuit gates (Xor16, Rot64, and Zero) Keccak works with 64-bit words. The state is represented using 5×5 matrix of 64 bit words. Each compression step of Keccak consists of 24 rounds. Let us denote the state matrix with A (indexing elements as A[x,y]), from which we derive further states as follows in each round. Each round then consists of the following 5 steps: C[x]D[x]E[x,y]B[y,2x+3y]F[x,y]Fp[0,0]​=A[x,0]⊕A[x,1]⊕A[x,2]⊕A[x,3]⊕A[x,4]=C[x−1]⊕ROT(C[x+1],1)=A[x,y]⊕D[x]=ROT(E[x,y],ρ[x,y])=B[x,y]⊕((NOT B[x+1,y])AND B[x+2,y])=F[0,0]⊕RC​​ for 0≤x,y≤4 and ρ[x,y] is the rotation offset defined for Keccak. The values are in the table below extracted from the Keccak reference https://keccak.team/files/Keccak-reference-3.0.pdf x = 3 x = 4 x = 0 x = 1 x = 2 y = 2 155 231 3 10 171 y = 1 55 276 36 300 6 y = 0 28 91 0 1 190 y = 4 120 78 210 66 253 y = 3 21 136 105 45 15","breadcrumbs":"Custom Gates » Keccak » Keccak Gate","id":"196","title":"Keccak Gate"},"197":{"body":"The atomic operations are XOR, ROT, NOT, AND. In the sections below, we will describe the gates for these operations. Below are some common approaches followed in their design. To fit within 15 wires, we first decompose each word into its lower and upper 32-bit components. A gate for an atomic operation works with those 32-bit components at a time. Before we describe the specific gate design approaches, below are some constraints in the Kimchi framework that dictated those approaches. only 4 lookups per row only first 7 columns are available to the permutation polynomial","breadcrumbs":"Custom Gates » Keccak » Design Approach:","id":"197","title":"Design Approach:"},"198":{"body":"It is clear from the definition of the rotation gate that its constraints are complete (meaning that honest instances always satisfy the constraints). It is left to be proven that the proposal is sound. In this section, we will give a proof that as soon as we perform the range checks on the excess and shifted parts of the input, only one possible assignment satisfies the constraints. This means that there is no dishonest instance that can make the constraints pass. We will also give an example where one could find wrong rotation witnesses that would satisfy the constraints if we did not check the range.","breadcrumbs":"Custom Gates » Keccak » Rot64","id":"198","title":"Rot64"},"199":{"body":"First of all, we will illustrate the necessity of range-checks with a simple example. For the sake of readability, we will use some toy field lengths. In particular, let us assume that our words have 4 bits, meaning all of the elements between 0x0 and 0xF. Next, we will be using the native field F32​. As we will later see, this choice of field lengths is not enough to perform any 4-bit rotation, since the operations in the constraints would overflow the native field. Nonetheless, it will be sufficient for our example where we will only rotate by 1 bit. Assume we want to rotate the word 0b1101 (meaning 13) by 1 bit to the left. This gives us the rotated word 0b1011 (meaning 11). The excess part of the word is 0b1, whereas the shifted part corresponds to 0b1010. We recall the constraints for the rotation gate: word⋅2rotrotated​=excess⋅2len+shifted=excess+shifted​ Applied to our example, this results in the following equations: 13⋅211​=excess⋅16+shifted=excess+shifted​ We can easily check that the proposed values of the shifted 0b1010=10 and the excess 0b1=1 values satisfy the above constraint because 26=1⋅16+10 and 11=1+10. Now, the question is: can we find another value for excess and shifted, such that their addition results in an incorrect rotated word? The answer to this question is yes, due to diophantine equations . We basically want to find x,y such that 26=x⋅16+y( mod 32). The solution to this equation is: ∀k∈[0…31]:xy​=k ∧=26−16⋅k​ We chose these word and field lengths to better understand the behaviour of the solution. Here, we can see two \"classes\" of evaluations. If we choose an even k, then y will have the following shape: 26−16⋅(2⋅n)⟺26−32n≡32​26 Meaning, if x=2n then y=26. If on the other hand, we chose an odd k, then y will have the following shape instead: 26−16⋅(2⋅n+1)⟺26−32n−16≡32​26−16=10 Meaning, if x=2n+1 then y=10. Thus, possible solutions to the diophantine equation are: xy​=0,1,2,3,4,5…=26,10,26,10,26,10…​ Note that our valid witness is part of that set of solutions, meaning x=1 and y=10. Of course, we can also find another dishonest instantiation such as x=0 and y=26. Perhaps one could think that we do not need to worry about this case, because the resulting rotation word would be 0+26=26, and if we later use that result as an input to a subsequent gate such as XOR, the value 26 would not fit and at some point the constraint system would complain. Nonetheless, we still have other solutions to worry about, such as (x=3,y=10) or (x=5,y=10), since they would result in a rotated word that would fit in the word length of 4 bits, yet would be incorrect (not equal to 11). All of the above incorrect solutions differ in one thing: they have different bit lengths. This means that we need to range check the values for the excess and shifted witnesses to make sure they have the correct length.","breadcrumbs":"Custom Gates » Keccak » Necessity of range checks","id":"199","title":"Necessity of range checks"},"2":{"body":"Groups are an algebraic structure which are used in most widely deployed signature schemes are used in most widely deployed zk-SNARKs generalize the notion of a collection of symmetries of an object generalize certain aspects of numbers I will not touch on the last two too much. First let's do a quick definition and then some examples. Definition. A group is a set G with a binary operation ∗:G×G→G, an identity e∈G for that operation, and an inverse map ()−1:G→G such that ∗ is associative: x∗(y∗z)=(x∗y)∗z for all x,y,z. e is an identity for ∗. I.e., e∗x=x∗e=x for all x. ()−1 is an inverse map for ∗ with identity e. I.e., x∗x−1=x−1∗x=e for all x. So basically, an invertible binary operation. Definition in hand, we can see some examples: The integers Z with +, 0 as identity, and negation as inverse. For any natural number n, the integers mod n. This can be thought of as the set of numbers {0,…,n−1} with the operation being + followed by taking remainder mod n. It can also be thought of as the group of rotations of an n-gon. For any set X, the set of invertible functions X→X (a.k.a permutations) with function composition as the operation, the identity function as the identity, and function inverse as the inverse operation. The set of translations of a plane R×R with composition of translations (since the composition of translations is a translation). The identity is translating by (0,0) and inverse is reversing the translation. This is equivalent to the group R×R with coordinate-wise addition as the operation and coordinate-wise negation as the inverse. The set of rotations of a sphere with composition as the operation, doing nothing as the identity, and performing the rotation in reverse as the inverse map. For any field F, the set F×:=F∖{0}, with field multiplication as the operation, 1 as the identity, and x↦1/x as the inverse map. This is called the group of units of F. Sometimes, instead of x∗y we simply write xy when ∗ is clear from context.","breadcrumbs":"Groups » Groups","id":"2","title":"Groups"},"20":{"body":"This section describes how the Cooley-Tukey fast Fourier transform works. As we learned in the previous section, the key is to select evaluation points that yield an efficient FFT algorithm. Specifically, say we have ω∈F such that ωn=1, and ωr=1 for any 00, then we will split f into two polynomials, recursively call FFT on them, and reconstruct the result from the recursive calls. To that end, define f0​ to be the polynomial whose coefficients are all the even-index coefficients of f and f1​ the polynomial whose coefficients are all the odd-index coefficients of f. In terms of the array representation, this just means splitting out every other entry into two arrays. So that can be done in time O(n). Write f=∑i<2k​ci​xi, so that f0​=∑i<2k−1​c2i​xi and f1​=∑i<2k−1​c2i+1​xi. Then f(x)​=i<2k∑​ci​xi=i<2k−1∑​c2i​x2i+i<2k−1∑​c2i+1​x2i+1=i<2k−1∑​c2i​(x2)i+i<2k−1∑​c2i+1​x⋅(x2)i=i<2k−1∑​c2i​(x2)i+xi<2k−1∑​c2i+1​(x2)i=f0​(x2)+xf1​(x2)​ Now, notice that if ω is a 2kth root of unity, then ω2 is a 2k−1th root of unity. Thus we can recurse with FFT(k−1,ω2,f0​) and similarly for f1​. Let [e0,0​,…,e0,2k−1−1​][e1,0​,…,e1,2k−1−1​]​=FFT(k−1,ω2,f0​)=FFT(k−1,ω2,f1​)​ By assumption ei,j​=fi​((ω2)j). So, for any j we have f(ωj)​=f0​((ω2)j)+ωjf1​((ω2)j)​ Now, since j may be larger than 2k−1−1, we need to reduce it mod 2k−1, relying on the fact that if τ is an nth root of unity then τj=τjmodn since τn=1. Thus, (ω2)j=(ω2)jmod2k−1 and so we have f(ωj)​=f0​((ω2)jmod2k−1)+ωjf1​((ω2)jmod2k−1)=e0,jmod2k−1​+ωje1,jmod2k−1​​ We can compute the array W=[1,ω,…,ω2k−1] in time O(n) (since each entry is the previous entry times ω). Then we can compute each entry of the output in O(1) as f(ωj)​=e0,jmod2k−1​+W[j]⋅e1,jmod2k−1​​ There are n such entries, so this takes time O(n). This concludes the recursive definition of the algorithm FFT(k,ω,f). Algorithm: computing evalAk​​ Input f=[c0​,…,c2k−1​] the coefficients of polynomial f(x)=∑i<2k​ci​xi Compute W←[1,ω,ω2,...,ω2k−1] FFT(k,ω,f)→[f(1),f(ω),f(ω2)…,f(ω2k−1)] if k==0 return f else Compute f0​=[c0​,c2​,...,c2k−2​] the even coefficients of f, corresponding to f0​(x)=∑i<2k−1​c2i​xi Compute f1​=[c1​,c3​,...,c2k−1​] the odd coefficients of f, corresponding to f1​(x)=∑i<2k−1​c2i+1​xi e0​←FFT(k−1,ω2,f0​) e1​←FFT(k−1,ω2,f1​) for j∈[0,2k−1] Fj​←e0,jmod2k−1​+W[j]⋅e1,jmod2k−1​ return F Now let's analyze the time complexity. Let T(n) be the complexity on an instance of size n (that is, for n=2k). Looking back at what we have done, we have done O(n) for computing f0​ and f1​ two recursive calls, each of size n/2 O(n) for computing the powers of ω O(n) for combining the results of the recursive calls In total, this is O(n)+2T(n/2). Solving this recurrence yields T(n)=O(n)⋅logn=O(nlogn). Basically, there are logn recursions before we hit the base case, and each step takes time O(n). □ Now, in practice there are ways to describe this algorithm non-recursively that have better concrete performance, but that's out of scope for this document. Read the code if you are interested. Using the FFT algorithm to compute interpAk​​ So far we have a fast way to compute evalAk​​(f) all at once, where Ak​ is the set of powers of a 2kth root of unity ω. For convenience let n=2k. Now we want to go the other way and compute a polynomial given an array of evaluations. Specifically, n evaluations [f(x0​),f(x1​),...,f(xn−1​)] uniquely define a degree n−1 polynomial. This can be written as a system of n equations f(x0​)f(x1​)⋮f(xn−1​)​=c0​+c1​x0​+…+cn−1​x0n−1​=c0​+c1​x1​+…+cn−1​x1n−1​=c0​+c1​xn−1​+…+cn−1​xn−1n−1​,​ which can be rewritten as a matrix vector product. ​f(x0​)f(x1​)⋮f(xn−1​)​​=​11⋮1​x0​x1​⋮xn−1​​⋯⋯⋱⋯​x0n−1​x1n−1​⋮xn−1n−1​​​×​c0​c1​⋮cn−1​​​ This n×n matrix is a Vandermonde matrix and it just so happens that square Vandermonde matrices are invertible, iff the xi​ are unique. Since we purposely selected our xi​ to be the powers of ω, a primitive n-th root of unity, by definition xi​=ωi are unique. Therefore, to compute the polynomial given the corresponding array of evaluations (i.e. interpolation) we can solve for the polynomial's coefficients using the inverse of the matrix. ​c0​c1​⋮cn−1​​​=​11⋮1​1ω⋮ωn−1​⋯⋯⋱⋯​1n−1ωn−1⋮ω(n−1)(n−1)​​−1×​f(1)f(ω)⋮f(ωn−1)​​ All we need now is the inverse of this matrix, which is slightly complicated to compute. I'm going to skip it for now, but if you have the details please make a pull request. Substituting in the inverse matrix we obtain the equation for interpolation. ​c0​c1​⋮cn−1​​​=n1​​11⋮1​1ω−1⋮ω−(n−1)​⋯⋯⋱⋯​1n−1ω−(n−1)⋮ω−(n−1)(n−1)​​×​f(1)f(ω)⋮f(ωn−1)​​ Observe that this equation is nearly identical to the original equation for evaluation, except with the following substitution. ωi⇒n1​ω−1i Consequently and perhaps surprisingly, we can reuse the FFT algorithm evalAk​​ in order to compute the inverse-- interpAk​​. So, suppose we have an array [a0​,…,an−1​] of field elements (which you can think of as a function Ak​→F) and we want to compute the coefficients of a polynomial f with f(ωi)=ai​. To this end, define a polynomial g by g=∑j264, then b′ cannot be smaller than 264 as it was assumed. CONTRADICTION. b=b′ and a=a′: then 264(a−a′)≡n​0 and this can only happen if a′=a+kn. But since n>2r, then a′ cannot be smaller than 2r as it was assumed. CONTRADICTION. a=a′ and b=b′: then we have something like 264α+β≡n​0. This means β≡n​−264α=k⋅n−264α for any k. According to the assumption, both 0≤a<2r and 0≤a′<2r. This means, the difference α:=(a−a′) lies anywhere in between the following interval: 1−2r≤α≤2r−1 We plug in this interval to the above equation to obtain the following interval for β: k⋅n−264(1−2r)≤β≤k⋅n−264(2r−1) We look at this interval from both sides of the inequality: β≥kn−264+264+r and β≤kn+264−264+r and we wonder if kn−264+264+r≤kn+264−264+r is at all possible. We rewrite as follows: 264+r−2642⋅264+r264+r​≤264−264+r≤2⋅264≤264​ But this can only happen if r≤0, which is impossible since we assume 0p. We have a proof system (Kimchi) over Fp​ and Fq​, where commitments to public inputs are: Pp​=⟨e,H⟩∈Ep​(Fq​)Pq​=⟨e,G⟩∈Eq​(Fp​) Respectively. See Pasta Curves for more details. When referring to the Fq​-side we mean the proof system for circuit over the field Fq​.","breadcrumbs":"Deferred Computation » Deferred Computation","id":"217","title":"Deferred Computation"},"218":{"body":"In pickles-rs we have the notion of \"passing\" a variable (including the transcript) from one side to the other. e.g. when a field element α∈Fp​ needs to be used as a scalar on Fq​. This document explains what goes on \"under the hood\". Let us start by understanding why: Let v∈Fq​ be a scalar which we want to use to do both: Field arithmetic in Fq​ Scalar operations on Eq​(Fp​) In order to do so efficiently, we need to split these operations across two circuits (and therefore proofs) because: Emulating Fq​ arithmetic in Fp​ is very expensive, e.g. computing v⋅w requires O(log(q)2) multiplications over Fp​: 100's of gates for a single multiplication. Since Eq​(Fp​)⊆Fp​×Fp​ we cannot compute [v]⋅H∈Eq​(Fp​) over Fq​ efficiently, because, like before, emulating Fp​ arithmetic in Fq​ is very expensive...","breadcrumbs":"Deferred Computation » Public Inputs / Why Passing","id":"218","title":"Public Inputs / Why Passing"},"219":{"body":"The solution is to \"pass\" a value v between the two proofs, in other words to have two values v ∈Fp​ and v∈Fq​ which are equal as integers i.e. lift(v)=lift(v )∈Z: they represent \"the same number\". A naive first attempt would be to simply add v~∈Fp​ to the witness on the Fp​-side, however this has two problems: Insufficient Field Size: pq; always the case for any cycle of curves, p is only ≈q​ smaller than q, by Hasse. Now \"v \" is \"represented\" by the two values h ,l~∈Fp​. Note that no decomposition is nessary if the \"original value\" v was in Fp​, since Fq​ is big enough to hold the lift of any element in Fp​. Problem 2: Compute Commitment to the Public Input of other side To solve the binding issue we will add l,h to the public inputs on the Fp​-side, for simplicity we will describe the case where l,h are the only public inputs in the Fp​-side, which means that the commitment Pp​∈Ep​(Fq​) to the public inputs on the Fp​ side is: Pp​=[h]⋅Gh​+[l]⋅Gl​∈Ep​(Fq​) At this point it is important to note that Ep​ is defined over Fq​! Which means that we can compute Pp​∈Ep​(Fq​) efficiently on the Fq​-side! Therefore to enforce the binding, we: Add a sub-circuit which checks: Pp​=[h]⋅Gh​+[l]⋅Gl​∈Ep​(Fq​),∣h∣=⌊logp⌋∣l∣=1 Add Pp​=(x,y)∈Fq​×Fq​ to the public input on the Fq​-side.","breadcrumbs":"Deferred Computation » Solution","id":"219","title":"Solution"},"22":{"body":"Implement types DensePolynomial and Evaluations that wrap a Vec and implement the FFT algorithms described above for converting between them Familiarize yourself with the types and functions provided by ark_poly","breadcrumbs":"Polynomials » Fast Fourier Transform » Exercises","id":"22","title":"Exercises"},"220":{"body":"At this point the statement of the proof in Fq​-side is: the Fq​-proof is sound, condition on providing an opening of Pp​ that satisifies the Fp​-relation. At this point you can stop and verify the proof (in the case of a \"step proof\" you would), by recomputing Pp​ outside the circuit while checking the Fp​-relation manually \"in the clear\". However, when recursing (e.g. wrapping in a \"wrap proof\") we need to \"ingest\" this public input Pp​; after all, to avoid blowup in the proof size everything (proofs/accumulators/public inputs etc.) must eventually become part of the witness and every computation covered by a circuit... To this end, the wrap proof is a proof for the Fp​-relation with the public input Pp​ which additionally verifies the Fq​-proof. The next \"step\" proof then verifies this wrap proof which means that Pp​ then becomes part of the witness!","breadcrumbs":"Deferred Computation » We recurse onwards...","id":"220","title":"We recurse onwards..."},"221":{"body":"We can arbitrarily choose which side should compute the public input of the other, in pickles we let \"wrap\" compute the commitment to the public input.","breadcrumbs":"Deferred Computation » In Pickles","id":"221","title":"In Pickles"},"222":{"body":"Enforces that the public input of the proof verified on the Fr side is equal to the Fp input computed on Fr side.","breadcrumbs":"Deferred Computation » Enforcing Equality","id":"222","title":"Enforcing Equality"},"223":{"body":"This section contains a series of diagrams giving an overview of implementation of pickles, closely following the structures and abstractions in the code. However, they are very technical and are primarily intended for developer use. The primary source of all the images in this section is pickles_structure.drawio file, and if one edits it one must re-generate the .svg renders by doing \"export -> selection\". The legend of the diagrams is quite straightforward: The black boxes are data structures that have names and labels following the implementation. MFNStep/MFNWrap is an abbreviation from MessagesForNextStep and MessagesForNextWrap that is used for brevity. Most other datatypes are exactly the same as in the codebase. The blue boxes are computations. Sometimes, when the computation is trivial or only vaguely indicated, it is denoted as a text sign directly on an arrow. Arrows are blue by default and denote moving a piece of data from one place to another with no (or very little) change. Light blue arrows are denoting witness query that is implemented through the handler mechanism. The \"chicken foot\" connector means that this arrow accesses just one field in an array: such an arrow could connect e.g. a input field of type old_a: A in a structure Vec<(A,B)> to an output new_a: A, which just means that we are inside a for loop and this computation is done for all the elemnts in the vector/array. Colour of the field is sometimes introduced and denotes how many steps ago was this piece of data created. The absense of the colour means either that (1) the data structure contains different subfields of different origin, or that (2) it was not coloured but it could be. The colours are assigned according to the following convention:","breadcrumbs":"Technical Diagrams » Pickles Technical Diagrams","id":"223","title":"Pickles Technical Diagrams"},"224":{"body":"The following is the diagram that explains the Wrap computation. The left half of it corresponds to the wrap.ml file and the general logic of proof creation, while the right part of it is wrap_main.ml/wrap_verifier.ml and explains in-circuit computation. This diagram explains the deferred_values computation which is in the heart of wrap.ml represented as a blue box in the middle of the left pane of the main Wrap diagram. Deferred values for Wrap are computed as follows:","breadcrumbs":"Technical Diagrams » Wrap Computatiton and Deferred Values","id":"224","title":"Wrap Computatiton and Deferred Values"},"225":{"body":"The following is the diagram that explains the Step computation, similarly to Wrap. The left half of it corresponds to the general logic in step.ml, and the right part of it is step_main.ml and explains in-circuit computation. We provide no deferred_values computation diagram for Step, but it is very conceptually similar to the one already presented for Wrap.","breadcrumbs":"Technical Diagrams » Step Computation","id":"225","title":"Step Computation"},"226":{"body":"Poseidon is a hash function that can efficiently run in a zk circuit. (See poseidon-hash.info ) It is based on the sponge function , with a state composed of field elements and a permutation based on field element operation (addition and exponentiation). The permutation contains an S-box (exponentiation of a group element), adding constants to the state, and matrix multiplication of the state (multiplications and additions) with an MDS matrix . Since a field element is around 255-bit, a single field element is enough as the capacity of the sponge to provide around 116-bit security. Note We might want to piggy back on the zcash poseidon spec at some point (perhaps by making this an extension of the zcash poseidon spec).","breadcrumbs":"Poseidon hash » Poseidon hash function","id":"226","title":"Poseidon hash function"},"227":{"body":"We define a base sponge, and a scalar sponge. Both must be instantiated when verifying a proof (this is due to recursion-support baked in Kimchi). External users of kimchi (or pickles) are most likely to interact with a wrap proof (see the pickles specification ). As such, the sponges they need to instantiate are most likely to be instantiated with: Poseidon-Fp for base sponge Poseidon-Fq for the scalar sponge","breadcrumbs":"Poseidon hash » APIs","id":"227","title":"APIs"},"228":{"body":"new(params) -> BaseSponge. Creates a new base sponge. BaseSponge.absorb(field_element). Absorbs a field element by calling the underlying sponge absorb function. BaseSponge.absorb_point(point). Absorbs an elliptic curve point. If the point is the point at infinity, absorb two zeros. Otherwise, absorb the x and y coordinates with two calls to absorb. BaseSponge.absorb_scalar(field_element_of_scalar_field). Absorbs a scalar. If the scalar field is smaller than the base field (e.g. Fp is smaller than Fq), then the scalar is casted to a field element of the base field and absorbed via absorb. Otherwise, the value is split between its least significant bit and the rest. Then both values are casted to field elements of the base field and absorbed via absorb (the high bits first, then the low bit). BaseSponge.digest() -> field_element. The squeeze function of the underlying sponge function is called and the first field element is returned. BaseSponge.digest_scalar() -> field_element_of_scalar_field. BaseSponge.challenge // TODO: specify. BaseSponge.challenge_fq // TODO: specify.","breadcrumbs":"Poseidon hash » Base Sponge","id":"228","title":"Base Sponge"},"229":{"body":"new(params) -> ScalarSponge ScalarSponge.absorb(scalar_field_element) ScalarSponge.digest() -> scalar_field_element ScalarSponge.challenge // TODO: specify","breadcrumbs":"Poseidon hash » Scalar Sponge","id":"229","title":"Scalar Sponge"},"23":{"body":"A \"commitment scheme\" is a cryptographic scheme that lets us provide a \"commitment\" to a given piece of data, in such a way that we can later \"open\" that commitment. I recommend checking out section 2.4.1 of David's book Real-World Cryptography. There are two properties we typically expect of our commitment schemes: Hiding. A commitment to a piece of data should not reveal anything about that data. Binding. There should be no ambiguity about what data is committed to. That is, it should not be possible for the committer to open the commitment to a piece of data other than the one they originally committed to. There are various kinds of commitments that allow opening in different ways, revealing only part of the data, or some function of the data. Sometimes it is even useful to elide the hiding property entirely--- so-called non-hiding commitments.","breadcrumbs":"Commitments » Commitments","id":"23","title":"Commitments"},"230":{"body":"Note that every operation is done in the field of the sponge. In this section we define the high-level algorithm behind the permutation and the sponge. The permutation is never used directly by users, it is used only by the sponge function.","breadcrumbs":"Poseidon hash » Algorithms","id":"230","title":"Algorithms"},"231":{"body":"In pseudo-code: def sbox(field_element): # modular exponentiation return field_element^7 # apply MDS matrix\ndef apply_mds(state): n = [0, 0, 0] n[0] = state[0] * mds[0][0] + state[1] * mds[0][1] + state[2] * mds[0][2] n[1] = state[0] * mds[1][0] + state[1] * mds[1][1] + state[2] * mds[1][2] n[2] = state[0] * mds[2][0] + state[1] * mds[2][1] + state[2] * mds[2][2] return n # a round in the permutation\ndef apply_round(round, state): # sbox state[0] = sbox(state[0]) state[1] = sbox(state[1]) state[2] = sbox(state[2]) # apply MDS matrix state = apply_mds(state) # add round constant state[0] += round_constants[round][0] state[1] += round_constants[round][1] state[2] += round_constants[round][2] # the permutation\ndef permutation(state): round_offset = 0 if ARK_INITIAL: constant = round_constants[0] state[0] += constant[0] state[1] += constant[1] state[2] += constant[2] round_offset = 1 for round in range(round_offset, ROUNDS + round_offset): apply_round(round, state)","breadcrumbs":"Poseidon hash » Permutation","id":"231","title":"Permutation"},"232":{"body":"In pseudo-code: def new(): return { \"state\": [0] * RATE, # `RATE` field elements set to 0 \"mode\": \"absorbing\", \"offset\": 0, } def absorb(sponge, field_element): # if we're changing mode, reset the offset if sponge.mode == \"squeezing\": sponge.mode = \"absorbing\" sponge.offset = 0 # we reached the end of the rate, permute elif sponge.offset == RATE: sponge.state = permutation(sponge.state) sponge.offset = 0 # absorb by adding to the state sponge.state[sponge.offset] += field_element sponge.offset += 1 def squeeze(sponge): # permute when changing mode or when we already squeezed everything if sponge.mode == \"absorbing\" or sponge.offset == RATE: sponge.mode = \"squeezing\" sponge.state = permutation(sponge.state) sponge.offset = 0 result = sponge.state[sponge.offset] sponge.offset += 1 return result","breadcrumbs":"Poseidon hash » Sponge","id":"232","title":"Sponge"},"233":{"body":"We instantiate two versions of Poseidon, one for the field Fp, and one for the field Fq (see the pasta specification ). Both share the following sponge configuration: capacity 1 rate: 2 and the following permutation configuration: number of full rounds: 55 sbox: 7 ARK_INITIAL: false","breadcrumbs":"Poseidon hash » Instantiations","id":"233","title":"Instantiations"},"234":{"body":"You can find the MDS matrix and round constants we use in /poseidon/src/pasta/fp_kimchi.rs .","breadcrumbs":"Poseidon hash » Poseidon-Fp","id":"234","title":"Poseidon-Fp"},"235":{"body":"You can find the MDS matrix and round constants we use in /poseidon/src/pasta/fp_kimchi.rs .","breadcrumbs":"Poseidon hash » Poseidon-Fq","id":"235","title":"Poseidon-Fq"},"236":{"body":"We have test vectors contained in /poseidon/src/tests/test_vectors/kimchi.json .","breadcrumbs":"Poseidon hash » Test vectors","id":"236","title":"Test vectors"},"237":{"body":"our ocaml implementation: https://github.com/minaprotocol/mina/blob/develop/src/lib/random_oracle/random_oracle.mli relies on random_oracle_input: https://github.com/minaprotocol/mina/blob/develop/src/lib/random_oracle_input/random_oracle_input.ml is instantiated with two types of fields: https://github.com/minaprotocol/mina/blob/develop/src/nonconsensus/snark_params/snark_params_nonconsensus.ml","breadcrumbs":"Poseidon hash » Pointers to the OCaml code","id":"237","title":"Pointers to the OCaml code"},"238":{"body":"Our polynomial commitment scheme is a bootleproof-type of commitment scheme, which is a mix of: The polynomial commitment scheme described in appendix A.1 of the PCD paper . The zero-knowledge opening described in section 3.1 of the HALO paper .","breadcrumbs":"Polynomial Commitment » Kimchi Polynomial Commitment","id":"238","title":"Kimchi Polynomial Commitment"},"239":{"body":"The two curves pallas and vesta (pa(llas ve)sta) created by the Zcash team. Each curve's scalar field is the other curve's base field, which is practical for recursion (see Pickles). supporting evidence mina parameters: https://github.com/o1-labs/proof-systems/tree/master/curves/src/pasta arkworks ark-pallas , pallas and vesta Note that in general Fq refers to the base field (in which the curve is defined over), while Fr refers to the scalar field (defined by the order of the curve). But in our code, because of the cycles: Fp refers to the base field of Pallas, and the scalar field of Vesta Fq refers to the base field of Vesta, and the scalar field of Pallas Note that ∣Fq​∣>∣Fp​∣. In pickles: Vesta is also referred to as the step curve, or the tick curve. Pallas is also referred to as the wrap curve, or the tock curve.","breadcrumbs":"Pasta Curves » Pasta Curves","id":"239","title":"Pasta Curves"},"24":{"body":"The simplest kind of a commitment is one in which opening reveals all of the underlying data. Let's give a simple construction of such a scheme for committing to field elements. Suppose we have Fp​ a prime order field, with p being rather large (say on the order of 2256). A hash function h:List(Fp​)→Fp​. Then we define commit(x,r)=h([x,r])open(c)=(x,r)verify(c,x,r)=true/false The argument r of the commit function is data that must only be known to the committer (until the commitment is opened). When a committer wants to commit to a field element x, they sample a random \"blinder\" r∈Fp​ and hash it together with x to form the commitment. To open the commitment c, they simply provide the committed value together with the blinder. Alternatively, if the verifier already knows x, they can just provide r, i.e. open(x,c)=r. Finally, given the commitment and the opening, we can verify whether the input was the value originally committed to using the verify function. If the hash function is collision-resistant, then this function is binding (because there's no way the committer could find another preimage of h([x,r])). If the hash function is one-way, then this commitment is also hiding (assuming x is revealed only as part of the opening). Commitments are often used in protocols between provers and verifiers. The following illustration provides an example with a prover named Peggy and a verifier named Victor. Peggyinputblinder​ }→commit​ commitment→open​ ​ ⟶⋮ ⟶⋮ ​Victorcommitment input,blinder commitmentinputblinder​ }→verify​→true/false​ ​ Here Peggy commits to an input using a blinder, obtains the commitment and sends it to Victor. The interlocutors continue their protocol, but eventually to convince Victor of her claims, Peggy must send the opening proof to her earlier commitment. Victor verifies the opening (i.e. the input and blinder) against the commitment. If the verification fails, then Victor knows that Peggy was trying to trick him, otherwise Victor has sufficient assurances that Peggy was telling the truth.","breadcrumbs":"Commitments » Simple commitments","id":"24","title":"Simple commitments"},"240":{"body":"curve equation: y2=x3+5 base field: 28948022309329048855892746252171976963363056481941560715954676764349967630337 scalar field: 28948022309329048855892746252171976963363056481941647379679742748393362948097 mina generator: (1,12418654782883325593414442427049395787963493412651469444558597405572177144507) arkworks generator: (−1,2) endo: endo_q = 2D33357CB532458ED3552A23A8554E5005270D29D19FC7D27B7FD22F0201B547\nendo_r = 397E65A7D7C1AD71AEE24B27E308F0A61259527EC1D4752E619D1840AF55F1B1 You can use sage to test this: Fp = GF(28948022309329048855892746252171976963363056481941560715954676764349967630337)\nPallas = EllipticCurve(Fp, [0, 5])\nPallas.count_points()","breadcrumbs":"Pasta Curves » Pallas","id":"240","title":"Pallas"},"241":{"body":"curve equation: y2=x3+5 base field: 28948022309329048855892746252171976963363056481941647379679742748393362948097 scalar field: 28948022309329048855892746252171976963363056481941560715954676764349967630337 mina generator: (1,11426906929455361843568202299992114520848200991084027513389447476559454104162) arkworks generator: (−1,2) endo: endo_q = 06819A58283E528E511DB4D81CF70F5A0FED467D47C033AF2AA9D2E050AA0E4F\nendo_r = 12CCCA834ACDBA712CAAD5DC57AAB1B01D1F8BD237AD31491DAD5EBDFDFE4AB9 You can use sage to test this: Fq = GF(28948022309329048855892746252171976963363056481941647379679742748393362948097)\nVesta = EllipticCurve(Fq, [0, 5])\nVesta.count_points()","breadcrumbs":"Pasta Curves » Vesta","id":"241","title":"Vesta"},"242":{"body":"This document specifies kimchi , a zero-knowledge proof system that's a variant of PLONK. This document does not specify how circuits are created or executed, but only how to convert a circuit and its execution into a proof. Table of content: Overview Tables used to describe a circuit Tables produced during proof creation Dependencies Polynomial Commitments Poseidon hash function Pasta Constraints Linearization Permutation Lookup The Lookup Tables The Lookup Selectors Producing the sorted table as the prover Gates Double Generic Gate Poseidon Elliptic Curve Addition Endo Scalar Endo Scalar Multiplication Scalar Multiplication Range Check Foreign Field Addition Foreign Field Multiplication Rotation Xor Gadgets Not And Setup Common Index Lookup Index Prover Index Verifier Index Proof Construction & Verification Proof Structure Proof Creation Proof Verification Fiat-Shamir argument Partial verification Batch verification of proofs Optimizations Security Considerations","breadcrumbs":"Kimchi » Kimchi","id":"242","title":"Kimchi"},"243":{"body":"There are three main algorithms to kimchi: Setup : takes a circuit and produces a prover index, and a verifier index. Proof creation : takes the prover index, and the execution trace of the circuit to produce a proof. Proof verification : takes the verifier index and a proof to verify. As part of these algorithms, a number of tables are created (and then converted into polynomials) to create a proof.","breadcrumbs":"Kimchi » Overview","id":"243","title":"Overview"},"244":{"body":"The following tables are created to describe the circuit: Gates . A circuit is described by a series of gates, that we list in a table. The columns of the tables list the gates, while the rows are the length of the circuit. For each row, only a single gate can take a value 1 while all other gates take the value 0. row Generic Poseidon CompleteAdd VarBaseMul EndoMul EndoMulScalar 0 1 0 0 0 0 0 1 0 1 0 0 0 0 Coefficients . The coefficient table has 15 columns, and is used to tweak the gates. Currently, only the Generic and the Poseidon gates use it (refer to their own sections to see how). All other gates set their values to 0. row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 / / / / / / / / / / / / / / / Wiring (or Permutation, or sigmas) . For gates to take the outputs of other gates as inputs, we use a wiring table to wire registers together. To learn about registers, see the next section. It is defined at every row, but only for the first 7 registers. Each cell specifies a (row, column) tuple that it should be wired to. Cells that are not connected to another cell are wired to themselves. Note that if three or more registers are wired together, they must form a cycle. For example, if register (0, 4) is wired to both registers (80, 6) and (90, 0) then you would have the following table: row 0 1 2 3 4 5 6 0 0,0 0,1 0,2 0,3 80,6 0,5 0,6 ... 80 80,0 80,1 80,2 80,3 80,4 80,5 90,0 ... 90 0,4 90,1 90,2 90,3 90,4 90,5 90,6 The lookup feature is currently optional, as it can add some overhead to the protocol. In the case where you would want to use lookups, the following tables would be needed: Lookup Tables . The different lookup tables that are used in the circuit. For example, the XOR lookup table: l r o 1 0 1 0 1 1 1 1 0 0 0 0 Lookup selectors . A lookup selector is used to perform a number of queries in different lookup tables. Any gate can advertise its use of a lookup selector (so a lookup selector can be associated to several gates), and on which rows they want to use them (current and/or next). In cases where a gate need to use lookups in its current row only, and is the only one performing a specific combination of queries, then its gate selector can be used in place of a lookup selector. As with gates, lookup selectors (including gates used as lookup selectors) are mutually exclusives (only one can be used on a given row). For example, suppose we have two lookup selectors: row ChaChaQuery ChaChaFinalQuery 0 0 0 1 1 0 Where each applies 4 queries. A query is a table describing which lookup table it queries, and the linear combination of the witness to use in the query. For example, the following table describes a query into the XOR table made out of linear combinations of registers (checking that r0​⊕r2​=2⋅r1​): table_id l r o XOR 1, r0 1, r2 2, r1","breadcrumbs":"Kimchi » Tables used to describe a circuit","id":"244","title":"Tables used to describe a circuit"},"245":{"body":"The following tables are created by the prover at runtime: Registers (or Witness) . Registers are also defined at every row, and are split into two types: the IO registers from 0 to 6 usually contain input or output of the gates (note that a gate can output a value on the next row as well). I/O registers can be wired to each other (they'll be forced to have the same value), no matter what row they're on (for example, the register at row:0, col:4 can be wired to the register at row:80, col:6). The rest of the registers, 7 through 14, are called advice registers as they can store values that useful only for the row's active gate. Think of them as intermediary or temporary values needed in the computation when the prover executes a circuit. row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 / / / / / / / / / / / / / / / Wiring (Permutation) trace . You can think of the permutation trace as an extra register that is used to enforce the wiring specified in the wiring table. It is a single column that applies on all the rows as well, which the prover computes as part of a proof. row pt 0 / Queries trace . These are the actual values made by queries, calculated by the prover at runtime, and used to construct the proof. Table trace . Represents the concatenation of all the lookup tables, combined into a single column at runtime by both the prover and the verifier. Sorted trace . Represents the processed (see the lookup section) concatenation of the queries trace and the table trace. It is produced at runtime by the prover. The sorted trace is long enough that it is split in several columns. Lookup (aggregation, or permutation) trace . This is a one column table that is similar to the wiring (permutation) trace we talked above. It is produced at runtime by the prover.","breadcrumbs":"Kimchi » Tables produced during proof creation","id":"245","title":"Tables produced during proof creation"},"246":{"body":"To specify kimchi, we rely on a number of primitives that are specified outside of this specification. In this section we list these specifications, as well as the interfaces we make use of in this specification.","breadcrumbs":"Kimchi » Dependencies","id":"246","title":"Dependencies"},"247":{"body":"Refer to the specification on polynomial commitments . We make use of the following functions from that specification: PolyCom.non_hiding_commit(poly) -> PolyCom::NonHidingCommitment PolyCom.commit(poly) -> PolyCom::HidingCommitment PolyCom.evaluation_proof(poly, commitment, point) -> EvaluationProof PolyCom.verify(commitment, point, evaluation, evaluation_proof) -> bool","breadcrumbs":"Kimchi » Polynomial Commitments","id":"247","title":"Polynomial Commitments"},"248":{"body":"Refer to the specification on Poseidon . We make use of the following functions from that specification: Poseidon.init(params) -> FqSponge Poseidon.update(field_elem) Poseidon.finalize() -> FieldElem specify the following functions on top: Poseidon.produce_challenge() (TODO: uses the endomorphism) Poseidon.to_fr_sponge() -> state_of_fq_sponge_before_eval, FrSponge With the current parameters: S-Box alpha: 7 Width: 3 Rate: 2 Full rounds: 55 Round constants: fp_kimchi , fq_kimchi MDS matrix: fp_kimchi , fq_kimchi","breadcrumbs":"Kimchi » Poseidon hash function","id":"248","title":"Poseidon hash function"},"249":{"body":"Kimchi is made to work on cycles of curves, so the protocol switch between two fields Fq and Fr, where Fq represents the base field and Fr represents the scalar field. See the Pasta curves specification .","breadcrumbs":"Kimchi » Pasta","id":"249","title":"Pasta"},"25":{"body":"Instead of a cryptographic hash function, we can use elliptic curve scalar multiplication to construct a commitment scheme. Here scalar multiplication is used like a one-way function based on the hardness assumption of the elliptic curve discrete logarithm problem (ECDLP). Suppose we have Fp​ a prime order field, with p being large (e.g. something like 2256). Publicly agreed generator point G over an elliptic curve E(Fp​) Another publicly agreed curve point H for which no one knows the discrete logarithm commit(x,r)=xG+rHopen(c)=(x,r) where x∈Fp​ is the value being committed to, r∈Fp​ is a random blinding factor and the commitment c=commit(x,r) is a curve point. These commitments are algebraic (i.e. they do not use a boolean-based cryptographic hash function) and have homomorphic properties: you can add commitments together to form another commitment of the added committed values. For example, if you have commitments A and B, you can perform: A+B​=commit(xa​,ra​)+commit(xb​,rb​)=xa​G+ra​H+xb​G+rb​H=(xa​+xb​)G+(ra​+rb​)H=commit(xa​+xb​,ra​+rb​)​ In other words, the sum of commitments A and B is equal to the commitment of the sum of the two committed values xa​ and xb​ and blinders ra​ and rb​. This is possible because in such a scheme scaling is commutative with adding scalars. As a cryptographic primitive, the ability to find a public curve point H for which no one knows the discrete logarithm may, at first, seem rather mind-blowing and powerful. Actually, it's as easy as it is awesome to find such a point--- simply perform rejection sampling by cryptographically hashing G (or, respectively, the hash output), using the output as the x-coordinate of a candidate point on E and checking whether it's valid. The first valid curve point obtained is H and by the hardness assumption of the ECDLP, no one knows it. Since approximately half of the hash outputs will be valid curve points on E, sampling will terminate very quickly. Indeed, as we will see later, this process can be used to sample many public curve points G1​,…,Gn​ for which the discrete logarithms are unknown; the so-called hash to curve algorithm.","breadcrumbs":"Commitments » Algebraic and homomorphic commitments","id":"25","title":"Algebraic and homomorphic commitments"},"250":{"body":"Kimchi enforces the correct execution of a circuit by creating a number of constraints and combining them together. In this section, we describe all the constraints that make up the main polynomial f once combined. We define the following functions: combine_constraints(range_alpha, constraints), which takes a range of contiguous powers of alpha and a number of constraints. It returns the sum of all the constraints, where each constraint has been multiplied by a power of alpha. In other words it returns: i∑​αi⋅constrainti​ The different ranges of alpha are described as follows: gates . Offset starts at 0 and 21 powers of α are used Permutation . Offset starts at 21 and 3 powers of α are used Note As gates are mutually exclusive (a single gate is used on each row), we can reuse the same range of powers of alpha across all the gates.","breadcrumbs":"Kimchi » Constraints","id":"250","title":"Constraints"},"251":{"body":"TODO","breadcrumbs":"Kimchi » Linearization","id":"251","title":"Linearization"},"252":{"body":"The permutation constraints are the following 4 constraints: The two sides of the coin (with shift0​=1): ​z(x)⋅zkpm(x)⋅αPERM0⋅(w0​(x)+β⋅shift0​x+γ)⋅(w1​(x)+β⋅shift1​x+γ)⋅(w2​(x)+β⋅shift2​x+γ)⋅(w3​(x)+β⋅shift3​x+γ)⋅(w4​(x)+β⋅shift4​x+γ)⋅(w5​(x)+β⋅shift5​x+γ)⋅(w6​(x)+β⋅shift6​x+γ)​​ and ​−1⋅z(xω)⋅zkpm(x)⋅αPERM0⋅(w0​(x)+β⋅σ0​(x)+γ)⋅(w1​(x)+β⋅σ1​(x)+γ)⋅(w2​(x)+β⋅σ2​(x)+γ)⋅(w3​(x)+β⋅σ3​(x)+γ)⋅(w4​(x)+β⋅σ4​(x)+γ)⋅(w5​(x)+β⋅σ5​(x)+γ)⋅(w6​(x)+β⋅σ6​(x)+γ)⋅​​ the initialization of the accumulator: (z(x)−1)L1​(x)αPERM1 and the accumulator's final value: (z(x)−1)Ln−k​(x)αPERM2 You can read more about why it looks like that in this post . The quotient contribution of the permutation is split into two parts perm and bnd. They will be used by the prover. perm(x)=​aPERM0⋅zkpl(x)⋅[z(x)⋅(w0​(x)+γ+x⋅β⋅shift0​)⋅(w1​(x)+γ+x⋅β⋅shift1​)⋅(w2​(x)+γ+x⋅β⋅shift2​)⋅(w3​(x)+γ+x⋅β⋅shift3​)⋅(w4​(x)+γ+x⋅β⋅shift4​)⋅(w5​(x)+γ+x⋅β⋅shift5​)⋅(w6​(x)+γ+x⋅β⋅shift6​)⋅−z(x⋅w)⋅(w0​(x)+γ+σ0​⋅β)⋅(w1​(x)+γ+σ1​⋅β)⋅(w2​(x)+γ+σ2​⋅β)⋅(w3​(x)+γ+σ3​⋅β)⋅(w4​(x)+γ+σ4​⋅β)⋅(w5​(x)+γ+σ5​⋅β)⋅(w6​(x)+γ+σ6​⋅β)⋅]​​ and bnd: bnd(x)=aPERM1⋅x−1z(x)−1​+aPERM2⋅x−sid[n−k]z(x)−1​ The linearization: scalar⋅σ6​(x) where scalar is computed as: z(ζω)βαPERM0zkpl(ζ)⋅(γ+βσ0​(ζ)+w0​(ζ))⋅(γ+βσ1​(ζ)+w1​(ζ))⋅(γ+βσ2​(ζ)+w2​(ζ))⋅(γ+βσ3​(ζ)+w3​(ζ))⋅(γ+βσ4​(ζ)+w4​(ζ))⋅(γ+βσ5​(ζ)+w5​(ζ))⋅​​ To compute the permutation aggregation polynomial, the prover interpolates the polynomial that has the following evaluations. The first evaluation represents the initial value of the accumulator: z(g0)=1 For i=0,⋅,n−4, where n is the size of the domain, evaluations are computed as: z(gi+1)=z1​/z2​ with z1​=​ (w0​(gi+sid(gi)⋅beta⋅shift0​+γ)⋅ (w1​(gi)+sid(gi)⋅beta⋅shift1​+γ)⋅ (w2​(gi)+sid(gi)⋅beta⋅shift2​+γ)⋅ (w3​(gi)+sid(gi)⋅beta⋅shift3​+γ)⋅ (w4​(gi)+sid(gi)⋅beta⋅shift4​+γ)⋅ (w5​(gi)+sid(gi)⋅beta⋅shift5​+γ)⋅ (w6​(gi)+sid(gi)⋅beta⋅shift6​+γ)​​ and z2​=​ (w0​(gi)+σ0​⋅beta+γ)⋅ (w1​(gi)+σ1​⋅beta+γ)⋅ (w2​(gi)+σ2​⋅beta+γ)⋅ (w3​(gi)+σ3​⋅beta+γ)⋅ (w4​(gi)+σ4​⋅beta+γ)⋅ (w5​(gi)+σ5​⋅beta+γ)⋅ (w6​(gi)+σ6​⋅beta+γ)​​ We randomize the evaluations at n - zk_rows + 1 and n - zk_rows + 2 in order to add zero-knowledge to the protocol. For a valid witness, we then have have z(gn−zkr​ows)=1.","breadcrumbs":"Kimchi » Permutation","id":"252","title":"Permutation"},"253":{"body":"Lookups in kimchi allows you to check if a single value, or a series of values, are part of a table. The first case is useful to check for checking if a value belongs to a range (from 0 to 1,000, for example), whereas the second case is useful to check truth tables (for example, checking that three values can be found in the rows of an XOR table) or write and read from a memory vector (where one column is an index, and the other is the value stored at that index). Note Similarly to the generic gate, each values taking part in a lookup can be scaled with a fixed field element. The lookup functionality is an opt-in feature of kimchi that can be used by custom gates. From the user's perspective, not using any gates that make use of lookups means that the feature will be disabled and there will be no overhead to the protocol. Please refer to the lookup RFC for an overview of the lookup feature. In this section, we describe the tables kimchi supports, as well as the different lookup selectors (and their associated queries) The Lookup Tables Kimchi currently supports two lookup tables: /// The table ID associated with the XOR lookup table.\npub const XOR_TABLE_ID: i32 = 0; /// The range check table ID.\npub const RANGE_CHECK_TABLE_ID: i32 = 1; XOR The lookup table for 4-bit xor. Note that it is constructed so that (0, 0, 0) is the last position in the table. This is because tables are extended to the full size of a column (essentially) by padding them with their final value. And, having the value (0, 0, 0) here means that when we commit to this table and use the dummy value in the lookup_sorted columns, those entries that have the dummy value of 0=0+j∗0+j2∗0 will translate into a scalar multiplication by 0, which is free. 12-bit Check The range check table is a single-column table containing the numbers from 0 to 2^12 (excluded). This is used to check that the value fits in 12 bits. Runtime tables Another type of lookup tables has been suggested in the Extended Lookup Tables . The Lookup Selectors XorSelector . Performs 4 queries to the XOR lookup table. l r o - l r o - l r o - l r o 1, r3 1, r7 1, r11 - 1, r4 1, r8 1, r12 - 1, r5 1, r9 1, r13 - 1, r6 1, r10 1, r14 Producing the sorted table as the prover Because of our ZK-rows, we can't do the trick in the plookup paper of wrapping around to enforce consistency between the sorted lookup columns. Instead, we arrange the LookupSorted table into columns in a snake-shape. Like so, _ _\n| | | | |\n| | | | |\n|_| |_| | or, imagining the full sorted array is [ s0, ..., s8 ], like s0 s4 s4 s8\ns1 s3 s5 s7\ns2 s2 s6 s6 So the direction (\"increasing\" or \"decreasing\" (relative to LookupTable) is if i % 2 = 0 { Increasing } else { Decreasing } Then, for each i < max_lookups_per_row, if i % 2 = 0, we enforce that the last element of LookupSorted(i) = last element of LookupSorted(i + 1), and if i % 2 = 1, we enforce that the first element of LookupSorted(i) = first element of LookupSorted(i + 1).","breadcrumbs":"Kimchi » Lookup","id":"253","title":"Lookup"},"254":{"body":"A circuit is described as a series of gates. In this section we describe the different gates currently supported by kimchi, the constraints associated to them, and the way the register table, coefficient table, and permutation can be used in conjunction. TODO: for each gate describe how to create it? Double Generic Gate The double generic gate contains two generic gates. A generic gate is simply the 2-fan in gate specified in the vanilla PLONK protocol that allows us to do operations like: addition of two registers (into an output register) or multiplication of two registers equality of a register with a constant More generally, the generic gate controls the coefficients ci​ in the equation: c0​⋅l+c1​⋅r+c2​⋅o+c3​⋅(l×r)+c4​ The layout of the gate is the following: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 l1 r1 o1 l2 r2 o2 where l1, r1, and o1 (resp. l2, r2, o2) are the left, right, and output registers of the first (resp. second) generic gate. The selectors are stored in the coefficient table as: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 l1 r1 o1 m1 c1 l2 r2 o2 m2 c2 with m1 (resp. m2) the mul selector for the first (resp. second) gate, and c1 (resp. c2) the constant selector for the first (resp. second) gate. The constraints: w0​⋅c0​+w1​⋅c1​+w2​⋅c2​+w0​⋅w1​⋅c3​+c4​ w3​⋅c5​+w4​⋅c6​+w5​⋅c7​+w3​w4​c8​+c9​ where the ci​ are the coefficients. Poseidon The poseidon gate encodes 5 rounds of the poseidon permutation. A state is represents by 3 field elements. For example, the first state is represented by (s0, s0, s0), and the next state, after permutation, is represented by (s1, s1, s1). Below is how we store each state in the register table: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 s0 s0 s0 s4 s4 s4 s1 s1 s1 s2 s2 s2 s3 s3 s3 s5 s5 s5 The last state is stored on the next row. This last state is either used: with another Poseidon gate on that next row, representing the next 5 rounds. or with a Zero gate, and a permutation to use the output elsewhere in the circuit. or with another gate expecting an input of 3 field elements in its first registers. Note As some of the poseidon hash variants might not use 5k rounds (for some k), the result of the 4-th round is stored directly after the initial state. This makes that state accessible to the permutation. We define Mr,c​ as the MDS matrix at row r and column c. We define the S-box operation as wS for S the SPONGE_BOX constant. We store the 15 round constants ri​ required for the 5 rounds (3 per round) in the coefficient table: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 The initial state, stored in the first three registers, are not constrained. The following 4 states (of 3 field elements), including 1 in the next row, are constrained to represent the 5 rounds of permutation. Each of the associated 15 registers is associated to a constraint, calculated as: first round: w6​−(r0​+(M0,0​w0S​+M0,1​w1S​+M0,2​w2S​)) w7​−(r1​+(M1,0​w0S​+M1,1​w1S​+M1,2​w2S​)) w8​−(r2​+(M2,0​w0S​+M2,1​w1S​+M2,2​w2S​)) second round: w9​−(r3​+(M0,0​w6S​+M0,1​w7S​+M0,2​w8S​)) w10​−(r4​+(M1,0​w6S​+M1,1​w7S​+M1,2​w8S​)) w11​−(r5​+(M2,0​w6S​+M2,1​w7S​+M2,2​w8S​)) third round: w12​−(r6​+(M0,0​w9S​+M0,1​w10S​+M0,2​w11S​)) w13​−(r7​+(M1,0​w9S​+M1,1​w10S​+M1,2​w11S​)) w14​−(r8​+(M2,0​w9S​+M2,1​w10S​+M2,2​w11S​)) fourth round: w3​−(r9​+(M0,0​w12S​+M0,1​w13S​+M0,2​w14S​)) w4​−(r10​+(M1,0​w12S​+M1,1​w13S​+M1,2​w14S​)) w5​−(r11​+(M2,0​w12S​+M2,1​w13S​+M2,2​w14S​)) fifth round: w0,next​−(r12​+(M0,0​w3S​+M0,1​w4S​+M0,2​w5S​)) w1,next​−(r13​+(M1,0​w3S​+M1,1​w4S​+M1,2​w5S​)) w2,next​−(r14​+(M2,0​w3S​+M2,1​w4S​+M2,2​w5S​)) where wi,next​ is the polynomial wi​(ωx) which points to the next row. Elliptic Curve Addition The layout is 0 1 2 3 4 5 6 7 8 9 10 x1 y1 x2 y2 x3 y3 inf same_x s inf_z x21_inv where (x1, y1), (x2, y2) are the inputs and (x3, y3) the output. inf is a boolean that is true iff the result (x3, y3) is the point at infinity. The rest of the values are inaccessible from the permutation argument, but same_x is a boolean that is true iff x1 == x2. The following constraints are generated: constraint 1: x0​=w2​−w0​ (w10​⋅x0​−F(1)−w7​) constraint 2: x0​=w2​−w0​ w7​⋅x0​ constraint 3: x0​=w2​−w0​ x1​=w3​−w1​ x2​=w0​⋅w0​ w7​⋅(2⋅w8​⋅w1​−2⋅x2​−x2​)+(F(1)−w7​)⋅(x0​⋅w8​−x1​) constraint 4: w0​+w2​+w4​−w8​⋅w8​ constraint 5: w8​⋅(w0​−w4​)−w1​−w5​ constraint 6: x1​=w3​−w1​ x1​⋅(w7​−w6​) constraint 7: x1​=w3​−w1​ x1​⋅w9​−w6​ Endo Scalar We give constraints for the endomul scalar computation. Each row corresponds to 8 iterations of the inner loop in \"Algorithm 2\" on page 29 of the Halo paper . The state of the algorithm that's updated across iterations of the loop is (a, b). It's clear from that description of the algorithm that an iteration of the loop can be written as (a, b, i) -> ( 2 * a + c_func(r_{2 * i}, r_{2 * i + 1}), 2 * b + d_func(r_{2 * i}, r_{2 * i + 1}) ) for some functions c_func and d_func. If one works out what these functions are on every input (thinking of a two-bit input as a number in {0,1,2,3}), one finds they are given by c_func(x), defined by c_func(0) = 0 c_func(1) = 0 c_func(2) = -1 c_func(3) = 1 d_func(x), defined by d_func(0) = -1 d_func(1) = 1 d_func(2) = 0 d_func(3) = 0 One can then interpolate to find polynomials that implement these functions on {0,1,2,3}. You can use sage , as R = PolynomialRing(QQ, 'x')\nc_func = R.lagrange_polynomial([(0, 0), (1, 0), (2, -1), (3, 1)])\nd_func = R.lagrange_polynomial([(0, -1), (1, 1), (2, 0), (3, 0)]) Then, c_func is given by 2/3 * x^3 - 5/2 * x^2 + 11/6 * x and d_func is given by 2/3 * x^3 - 7/2 * x^2 + 29/6 * x - 1 <=> c_func + (-x^2 + 3x - 1) We lay it out the witness as 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Type n0 n8 a0 b0 a8 b8 x0 x1 x2 x3 x4 x5 x6 x7 ENDO where each xi is a two-bit \"crumb\". We also use a polynomial to check that each xi is indeed in {0,1,2,3}, which can be done by checking that each xi​ is a root of the polyunomial below: crumb(x)\n= x (x - 1) (x - 2) (x - 3)\n= x^4 - 6*x^3 + 11*x^2 - 6*x\n= x *(x^3 - 6*x^2 + 11*x - 6) Each iteration performs the following computations Update n: ni+1​=2⋅ni​+xi​ Update a: ai+1​=2⋅ai​+ci​ Update b: bi+1​=2⋅bi​+di​ Then, after the 8 iterations, we compute expected values of the above operations as: expected_n8 := 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * (2 * n0 + x0) + x1 ) + x2 ) + x3 ) + x4 ) + x5 ) + x6 ) + x7 expected_a8 := 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * (2 * a0 + c0) + c1 ) + c2 ) + c3 ) + c4 ) + c5 ) + c6 ) + c7 expected_b8 := 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * (2 * b0 + d0) + d1 ) + d2 ) + d3 ) + d4 ) + d5 ) + d6 ) + d7 Putting together all of the above, these are the 11 constraints for this gate Checking values after the 8 iterations: Constrain n: 0 = expected_n8 - n8 Constrain a: 0 = expected_a8 - a8 Constrain b: 0 = expected_b8 - b8 Checking the crumbs, meaning each x is indeed in the range {0,1,2,3}: Constrain x0​: 0 = x0 * ( x0^3 - 6 * x0^2 + 11 * x0 - 6 ) Constrain x1​: 0 = x1 * ( x1^3 - 6 * x1^2 + 11 * x1 - 6 ) Constrain x2​: 0 = x2 * ( x2^3 - 6 * x2^2 + 11 * x2 - 6 ) Constrain x3​: 0 = x3 * ( x3^3 - 6 * x3^2 + 11 * x3 - 6 ) Constrain x4​: 0 = x4 * ( x4^3 - 6 * x4^2 + 11 * x4 - 6 ) Constrain x5​: 0 = x5 * ( x5^3 - 6 * x5^2 + 11 * x5 - 6 ) Constrain x6​: 0 = x6 * ( x6^3 - 6 * x6^2 + 11 * x6 - 6 ) Constrain x7​: 0 = x7 * ( x7^3 - 6 * x7^2 + 11 * x7 - 6 ) Endo Scalar Multiplication We implement custom gate constraints for short Weierstrass curve endomorphism optimised variable base scalar multiplication. Given a finite field Fq​ of order q, if the order is not a multiple of 2 nor 3, then an elliptic curve over Fq​ in short Weierstrass form is represented by the set of points (x,y) that satisfy the following equation with a,b∈Fq​ and 4a3+27b2=Fq​​0: E(Fq​):y2=x3+ax+b If P=(xp​,yp​) and T=(xt​,yt​) are two points in the curve E(Fq​), the goal of this operation is to perform the operation 2P±T efficiently as (P±T)+P. S = (P + (b ? T : −T)) + P The same algorithm can be used to perform other scalar multiplications, meaning it is not restricted to the case 2⋅P, but it can be used for any arbitrary k⋅P. This is done by decomposing the scalar k into its binary representation. Moreover, for every step, there will be a one-bit constraint meant to differentiate between addition and subtraction for the operation (P±T)+P: In particular, the constraints of this gate take care of 4 bits of the scalar within a single EVBSM row. When the scalar is longer (which will usually be the case), multiple EVBSM rows will be concatenated. Row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Type i xT yT Ø Ø xP yP n xR yR s1 s3 b1 b2 b3 b4 EVBSM i+1 = = xS yS n' xR' yR' s1' s3' b1' b2' b3' b4' EVBSM The layout of this gate (and the next row) allows for this chained behavior where the output point of the current row S gets accumulated as one of the inputs of the following row, becoming P in the next constraints. Similarly, the scalar is decomposed into binary form and n (n′ respectively) will store the current accumulated value and the next one for the check. For readability, we define the following variables for the constraints: endo := EndoCoefficient xq1 :=(1+(endo−1)⋅b1​)⋅xt​ xq2 :=(1+(endo−1)⋅b3​)⋅xt​ yq1 :=(2⋅b2​−1)⋅yt​ yq2 :=(2⋅b4​−1)⋅yt​ These are the 11 constraints that correspond to each EVBSM gate, which take care of 4 bits of the scalar within a single EVBSM row: First block: (xq1 - xp) * s1 = yq1 - yp (2 * xp – s1^2 + xq1) * ((xp – xr) * s1 + yr + yp) = (xp – xr) * 2 * yp (yr + yp)^2 = (xp – xr)^2 * (s1^2 – xq1 + xr) Second block: (xq2 - xr) * s3 = yq2 - yr (2*xr – s3^2 + xq2) * ((xr – xs) * s3 + ys + yr) = (xr – xs) * 2 * yr (ys + yr)^2 = (xr – xs)^2 * (s3^2 – xq2 + xs) Booleanity checks: Bit flag b1​: 0 = b1 * (b1 - 1) Bit flag b2​: 0 = b2 * (b2 - 1) Bit flag b3​: 0 = b3 * (b3 - 1) Bit flag b4​: 0 = b4 * (b4 - 1) Binary decomposition: Accumulated scalar: n_next = 16 * n + 8 * b1 + 4 * b2 + 2 * b3 + b4 The constraints above are derived from the following EC Affine arithmetic equations: (1) => (xq1​​−xp​)⋅s1​=yq1​​−yp​ (2&3) => (xp​–xr​)⋅s2​=yr​+yp​ (2) => (2⋅xp​+xq1​​–s12​)⋅(s1​+s2​)=2⋅yp​ <=> (2⋅xp​–s12​+xq1​​)⋅((xp​–xr​)⋅s1​+yr​+yp​)=(xp​–xr​)⋅2⋅yp​ (3) => s12​−s22​=xq1​​−xr​ <=> (yr​+yp​)2=(xp​–xr​)2⋅(s12​–xq1​​+xr​) (4) => (xq2​​−xr​)⋅s3​=yq2​​−yr​ (5&6) => (xr​–xs​)⋅s4​=ys​+yr​ (5) => (2⋅xr​+xq2​​–s32​)⋅(s3​+s4​)=2⋅yr​ <=> (2⋅xr​–s32​+xq2​​)⋅((xr​–xs​)⋅s3​+ys​+yr​)=(xr​–xs​)⋅2⋅yr​ (6) => s32​–s42​=xq2​​−xs​ <=> (ys​+yr​)2=(xr​–xs​)2⋅(s32​–xq2​​+xs​) Defining s2​ and s4​ as s2​:=2∗xP​+xT​−s12​2⋅yP​​−s1​ s4​:=2∗xR​+xT​−s32​2⋅yR​​−s3​ Gives the following equations when substituting the values of s2​ and s4​: (xq1 - xp) * s1 = (2 * b1 - 1) * yt - yp (2 * xp – s1^2 + xq1) * ((xp – xr) * s1 + yr + yp) = (xp – xr) * 2 * yp (yr + yp)^2 = (xp – xr)^2 * (s1^2 – xq1 + xr) (xq2 - xr) * s3 = (2 * b2 - 1) * yt - yr (2 * xr – s3^2 + xq2) * ((xr – xs) * s3 + ys + yr) = (xr – xs) * 2 * yr (ys + yr)^2 = (xr – xs)^2 * (s3^2 – xq2 + xs) Scalar Multiplication We implement custom Plonk constraints for short Weierstrass curve variable base scalar multiplication. Given a finite field Fq​ of order q, if the order is not a multiple of 2 nor 3, then an elliptic curve over Fq​ in short Weierstrass form is represented by the set of points (x,y) that satisfy the following equation with a,b∈Fq​ and 4a3+27b2=Fq​​0: E(Fq​):y2=x3+ax+b If P=(xp​,yp​) and Q=(xq​,yq​) are two points in the curve E(Fq​), the algorithm we represent here computes the operation 2P+Q (point doubling and point addition) as (P+Q)+Q. Info Point Q=(xq​,yq​) has nothing to do with the order q of the field Fq​. The original algorithm that is being used can be found in the Section 3.1 of https://arxiv.org/pdf/math/0208038.pdf , which can perform the above operation using 1 multiplication, 2 squarings and 2 divisions (one more squaring) if P=Q), thanks to the fact that computing the Y-coordinate of the intermediate addition is not required. This is more efficient to the standard algorithm that requires 1 more multiplication, 3 squarings in total and 2 divisions. Moreover, this algorithm can be applied not only to the operation 2P+Q, but any other scalar multiplication kP. This can be done by expressing the scalar k in biwise form and performing a double-and-add approach. Nonetheless, this requires conditionals to differentiate 2P from 2P+Q. For that reason, we will implement the following pseudocode from https://github.com/zcash/zcash/issues/3924 (where instead, they give a variant of the above efficient algorithm for Montgomery curves b⋅y2=x3+a⋅x2+x). Acc := [2]T\nfor i = n-1 ... 0: Q := (k_{i + 1} == 1) ? T : -T Acc := Acc + (Q + Acc)\nreturn (k_0 == 0) ? Acc - P : Acc The layout of the witness requires 2 rows. The i-th row will be a VBSM gate whereas the next row will be a ZERO gate. Row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Type i xT yT x0 y0 n n' x1 y1 x2 y2 x3 y3 x4 y4 VBSM i+1 x5 y5 b0 b1 b2 b3 b4 s0 s1 s2 s3 s4 ZERO The gate constraints take care of 5 bits of the scalar multiplication. Each single bit consists of 4 constraints. There is one additional constraint imposed on the final number. Thus, the VarBaseMul gate argument requires 21 constraints. For every bit, there will be one constraint meant to differentiate between addition and subtraction for the operation (P±T)+P: S = (P + (b ? T : −T)) + P We follow these criteria: If the bit is positive, the sign should be a subtraction If the bit is negative, the sign should be an addition Then, paraphrasing the above, we will represent this behavior as: S = (P - (2 * b - 1) * T ) + P Let us call Input the point with coordinates (xI, yI) and Target is the point being added with coordinates (xT, yT). Then Output will be the point with coordinates (xO, yO) resulting from O = ( I ± T ) + I Info Do not confuse our Output point (xO, yO) with the point at infinity that is normally represented as O. In each step of the algorithm, we consider the following elliptic curves affine arithmetic equations: s1​:=xi​−xt​yi​−(2⋅b−1)⋅yt​​ s2​:=2∗xi​+xt​−s12​2⋅yi​​−s1​ xo​:=xt​+s22​−s12​ yo​:=s2​⋅(xi​−xo​)−yi​ For readability, we define the following 3 variables in such a way that s2​ can be expressed as u / t: rx :=s12​−xi​−xt​ t :=xi​− rx ⟺2⋅xi​−s12​+xt​ u :=2⋅yi​− t ⋅s1​⟺2⋅yi​−s1​⋅(2⋅xi​−s12​+xt​) Next, for each bit in the algorithm, we create the following 4 constraints that derive from the above: Booleanity check on the bit b: 0 = b * b - b Constrain s1​: (xI - xT) * s1 = yI – (2b - 1) * yT Constrain Output X-coordinate xo​ and s2​: 0 = u^2 - t^2 * (xO - xT + s1^2) Constrain Output Y-coordinate yo​ and s2​: 0 = (yO + yI) * t - (xI - xO) * u When applied to the 5 bits, the value of the Target point (xT, yT) is maintained, whereas the values for the Input and Output points form the chain: [(x0, y0) -> (x1, y1) -> (x2, y2) -> (x3, y3) -> (x4, y4) -> (x5, y5)] Similarly, 5 different s0..s4 are required, just like the 5 bits b0..b4. Finally, the additional constraint makes sure that the scalar is being correctly expressed into its binary form (using the double-and-add decomposition) as: n′=25⋅n+24⋅b0​+23⋅b1​+22⋅b2​+21⋅b3​+b4​ This equation is translated as the constraint: Binary decomposition: 0 = n' - (b4 + 2 * (b3 + 2 * (b2 + 2 * (b1 + 2 * (b0 + 2*n))))) Range Check The multi range check gadget is comprised of three circuit gates (RangeCheck0, RangeCheck1 and Zero) and can perform range checks on three values (v0​,v1​ and v2​) of up to 88 bits each. Values can be copied as inputs to the multi range check gadget in two ways: (Standard mode) With 3 copies, by copying v0​,v1​ and v2​ to the first cells of the first 3 rows of the gadget. In this mode the first gate coefficient is set to 0. (Compact mode) With 2 copies, by copying v2​ to the first cell of the first row and copying v10​=v0​+2ℓ⋅v1​ to the 2nd cell of row 2. In this mode the first gate coefficient is set to 1. The RangeCheck0 gate can also be used on its own to perform 64-bit range checks by constraining witness cells 1-2 to zero. Byte-order: Each cell value is in little-endian byte order Limbs are mapped to columns in big-endian order (i.e. the lowest columns contain the highest bits) We also have the highest bits covered by copy constraints and plookups, so that we can copy the highest two constraints to zero and get a 64-bit lookup, which are envisioned to be a common case The values are decomposed into limbs as follows: L is a 12-bit lookup (or copy) limb, C is a 2-bit \"crumb\" limb (we call half a nybble a crumb). <----6----> <------8------> v0 = L L L L L L C C C C C C C C v1 = L L L L L L C C C C C C C C <2> <--4--> <---------------18----------------> v2 = C C L L L L C C C C C C C C C C C C C C C C C C Witness structure: Row Contents 0 v0​ 1 v1​ 2 v2​ 3 v0​,v1​,v2​ The first 2 rows contain v0​ and v1​ and their respective decompositions into 12-bit and 2-bit limbs The 3rd row contains v2​ and part of its decomposition: four 12-bit limbs and the 1st 10 crumbs The final row contains v0​'s and v1​'s 5th and 6th 12-bit limbs as well as the remaining 10 crumbs of v2​ Note Because we are constrained to 4 lookups per row, we are forced to postpone some lookups of v0 and v1 to the final row. Constraints: For efficiency, the limbs are constrained differently according to their type: 12-bit limbs are constrained with plookups 2-bit crumbs are constrained with degree-4 constraints x(x−1)(x−2)(x−3) Layout: This is how the three 88-bit inputs v0​,v1​ and v2​ are laid out and constrained. vipj is the jth 12-bit limb of value vi​ vicj is the jth 2-bit crumb limb of value vi​ Gates RangeCheck0 RangeCheck0 RangeCheck1 Zero Rows 0 1 2 3 Cols 0 v0 v1 v2 crumb v2c9 MS:1 copy v0p0 copy v1p0 optional v12 crumb v2c10 2 copy v0p1 copy v1p1 crumb v2c0 crumb v2c11 3 plookup v0p2 plookup v1p2 plookup v2p0 plookup v0p0 4 plookup v0p3 plookup v1p3 plookup v2p1 plookup v0p1 5 plookup v0p4 plookup v1p4 plookup v2p2 plookup v1p0 6 plookup v0p5 plookup v1p5 plookup v2p3 plookup v1p1 7 crumb v0c0 crumb v1c0 crumb v2c1 crumb v2c12 8 crumb v0c1 crumb v1c1 crumb v2c2 crumb v2c13 9 crumb v0c2 crumb v1c2 crumb v2c3 crumb v2c14 10 crumb v0c3 crumb v1c3 crumb v2c4 crumb v2c15 11 crumb v0c4 crumb v1c4 crumb v2c5 crumb v2c16 12 crumb v0c5 crumb v1c5 crumb v2c6 crumb v2c17 13 crumb v0c6 crumb v1c6 crumb v2c7 crumb v2c18 LS:14 crumb v0c7 crumb v1c7 crumb v2c8 crumb v2c19 The 12-bit chunks are constrained with plookups and the 2-bit crumbs are constrained with degree-4 constraints of the form x(x−1)(x−2)(x−3). Note that copy denotes a plookup that is deferred to the 4th gate (i.e. Zero). This is because of the limitation that we have at most 4 lookups per row. The copies are constrained using the permutation argument. Gate types: Different rows are constrained using different CircuitGate types Row CircuitGate Purpose 0 RangeCheck0 Partially constrain v0​ 1 RangeCheck0 Partially constrain v1​ 2 RangeCheck1 Fully constrain v2​ (and trigger plookups constraints on row 3) 3 Zero Complete the constraining of v0​ and v1​ using lookups Note Each CircuitGate type corresponds to a unique polynomial and thus is assigned its own unique powers of alpha RangeCheck0 - Range check constraints This circuit gate is used to partially constrain values v0​ and v1​ Optionally, it can be used on its own as a single 64-bit range check by constraining columns 1 and 2 to zero The rest of v0​ and v1​ are constrained by the lookups in the Zero gate row This gate operates on the Curr row It uses three different types of constraints: copy - copy to another cell (12-bits) plookup - plookup (12-bits) crumb - degree-4 constraint (2-bits) Given value v the layout looks like this Column Curr 0 v 1 copy vp0 2 copy vp1 3 plookup vp2 4 plookup vp3 5 plookup vp4 6 plookup vp5 7 crumb vc0 8 crumb vc1 9 crumb vc2 10 crumb vc3 11 crumb vc4 12 crumb vc5 13 crumb vc6 14 crumb vc7 where the notation vpi and vci defined in the \"Layout\" section above. RangeCheck1 - Range check constraints This circuit gate is used to fully constrain v2​ It operates on the Curr and Next rows It uses two different types of constraints: plookup - plookup (12-bits) crumb - degree-4 constraint (2-bits) Given value v2 the layout looks like this Column Curr Next 0 v2 crumb v2c9 1 optional v12 crumb v2c10 2 crumb v2c0 crumb v2c11 3 plookup v2p0 (ignored) 4 plookup v2p1 (ignored) 5 plookup v2p2 (ignored) 6 plookup v2p3 (ignored) 7 crumb v2c1 crumb v2c12 8 crumb v2c2 crumb v2c13 9 crumb v2c3 crumb v2c14 10 crumb v2c4 crumb v2c15 11 crumb v2c5 crumb v2c16 12 crumb v2c6 crumb v2c17 13 crumb v2c7 crumb v2c18 14 crumb v2c8 crumb v2c19 where the notation v2ci and v2pi defined in the \"Layout\" section above. Foreign Field Addition These circuit gates are used to constrain that left_input +/- right_input = field_overflow * foreign_modulus + result Documentation For more details please see the Foreign Field Addition chapter. Mapping To make things clearer, the following mapping between the variable names used in the code and those of the RFC document can be helpful. left_input_lo -> a0 right_input_lo -> b0 result_lo -> r0 bound_lo -> u0 left_input_mi -> a1 right_input_mi -> b1 result_mi -> r1 bound_mi -> u1 left_input_hi -> a2 right_input_hi -> b2 result_hi -> r2 bound_hi -> u2 field_overflow -> q sign -> s carry_lo -> c0 carry_mi -> c1 bound_carry_lo -> k0 bound_carry_mi -> k1 Note: Our limbs are 88-bit long. We denote with: lo the least significant limb (in little-endian, this is from 0 to 87) mi the middle limb (in little-endian, this is from 88 to 175) hi the most significant limb (in little-endian, this is from 176 to 263) Let left_input_lo, left_input_mi, left_input_hi be 88-bit limbs of the left element Let right_input_lo, right_input_mi, right_input_hi be 88-bit limbs of the right element Let foreign_modulus_lo, foreign_modulus_mi, foreign_modulus_hi be 88-bit limbs of the foreign modulus Then the limbs of the result are result_lo = left_input_lo +/- right_input_lo - field_overflow * foreign_modulus_lo - 2^{88} * result_carry_lo result_mi = left_input_mi +/- right_input_mi - field_overflow * foreign_modulus_mi - 2^{88} * result_carry_mi + result_carry_lo result_hi = left_input_hi +/- right_input_hi - field_overflow * foreign_modulus_hi + result_carry_mi field_overflow =0 or 1 or −1 handles overflows in the field result_carry_i =−1,0,1 are auxiliary variables that handle carries between limbs Apart from the range checks of the chained inputs, we need to do an additional range check for a final bound to make sure that the result is less than the modulus, by adding 2^{3*88} - foreign_modulus to it. (This can be computed easily from the limbs of the modulus) Note that 2^{264} as limbs represents: (0, 0, 0, 1) then: The upper-bound check can be calculated as: bound_lo = result_lo - foreign_modulus_lo - bound_carry_lo * 2^{88} bound_mi = result_mi - foreign_modulus_mi - bound_carry_mi * 2^{88} + bound_carry_lo bound_hi = result_hi - foreign_modulus_hi + 2^{88} + bound_carry_mi Which is equivalent to another foreign field addition with right input 2^{264}, q = 1 and s = 1 bound_lo = result_lo + s * 0 - q * foreign_modulus_lo - bound_carry_lo * 2^{88} bound_mi = result_mi + s * 0 - q * foreign_modulus_mi - bound_carry_mi * 2^{88} + bound_carry_lo bound_hi = result_hi + s * 2^{88} - q * foreign_modulus_hi + bound_carry_mi bound_carry_i =0 or 1 or −1 are auxiliary variables that handle carries between limbs The range check of bound can be skipped until the end of the operations and result is an intermediate value that is unused elsewhere (since the final result must have had the right amount of moduli subtracted along the way, meaning a multiple of the modulus). In other words, intermediate results could potentially give a valid witness that satisfies the constraints but where the result is larger than the modulus (yet smaller than 2^{264}). The reason that we have a final bound check is to make sure that the final result (min_result) is indeed the minimum one (meaning less than the modulus). A more optimized version of these constraints is able to reduce by 2 the number of constraints and by 1 the number of witness cells needed. The idea is to condense the low and middle limbs in one longer limb of 176 bits (which fits inside our native field) and getting rid of the low carry flag. With this idea in mind, the sole carry flag we need is the one located between the middle and the high limbs. Layout The sign of the operation (whether it is an addition or a subtraction) is stored in the fourth coefficient as a value +1 (for addition) or -1 (for subtraction). The first 3 coefficients are the 3 limbs of the foreign modulus. One could lay this out as a double-width gate for chained foreign additions and a final row, e.g.: col ForeignFieldAdd chain ForeignFieldAdd final ForeignFieldAdd final Zero 0 left_input_lo (copy) result_lo (copy) min_result_lo (copy) bound_lo (copy) 1 left_input_mi (copy) result_mi (copy) min_result_mi (copy) bound_mi (copy) 2 left_input_hi (copy) result_hi (copy) min_result_hi (copy) bound_hi (copy) 3 right_input_lo (copy) 0 (check) 4 right_input_mi (copy) 0 (check) 5 right_input_hi (copy) 2^88 (check) 6 field_overflow (copy?) 1 (check) 7 carry bound_carry 8 9 10 11 12 13 14 We reuse the foreign field addition gate for the final bound check since this is an addition with a specific parameter structure. Checking that the correct right input, overflow, and overflow are used shall be done by copy constraining these values with a public input value. One could have a specific gate for just this check requiring less constrains, but the cost of adding one more selector gate outweights the savings of one row and a few constraints of difference. Integration Copy final overflow bit from public input containing value 1 * Range check the final bound Foreign Field Multiplication This gadget is used to constrain that left_input * right_input = quotient * foreign_field_modulus + remainder Documentation For more details please see the Foreign Field Multiplication chapter or the original Foreign Field Multiplication RFC Notations For clarity, we use more descriptive variable names in the code than in the RFC, which uses mathematical notations. In order to relate the two documents, the following mapping between the variable names used in the code and those of the RFC can be helpful. left_input0 => a0 right_input0 => b0 quotient0 => q0 remainder01 => r01\nleft_input1 => a1 right_input1 => b1 quotient1 => q1\nleft_input2 => a2 right_input2 => b2 quotient2 => q2 remainder2 => r2 product1_lo => p10 product1_hi_0 => p110 product1_hi_1 => p111 carry0 => v0 carry1_lo => v10 carry1_hi => v11 quotient_hi_bound => q'2 Suffixes The variable names in this code uses descriptive suffixes to convey information about the positions of the bits referred to. When a word is split into up to n parts we use: 0, 1 ... n (where n is the most significant). For example, if we split word x into three limbs, we'd name them x0, x1 and x2 or x[0], x[1] and x[2]. Continuing in this fashion, when one of those words is subsequently split in half, then we add the suffixes _lo and _hi, where hi corresponds to the most significant bits. For our running example, x1 would become x1_lo and x1_hi. If we are splitting into more than two things, then we pick meaningful names for each. So far we've explained our conventions for a splitting depth of up to 2. For splitting deeper than two, we simply cycle back to our depth 1 suffixes again. So for example, x1_lo would be split into x1_lo_0 and x1_lo_1. Parameters hi_foreign_field_modulus := high limb of foreign field modulus $f$ (stored in gate coefficient 0) neg_foreign_field_modulus := negated foreign field modulus $f'$ (stored in gate coefficients 1-3) n := the native field modulus is obtainable from F, the native field's trait bound Witness left_input := left foreign field element multiplicand $ ~\\in F_f$ right_input := right foreign field element multiplicand $ ~\\in F_f$ quotient := foreign field quotient $ ~\\in F_f$ remainder := foreign field remainder $ ~\\in F_f$ carry0 := 2 bit carry carry1_lo := low 88 bits of carry1 carry1_hi := high 3 bits of carry1 product1_lo := lowest 88 bits of middle intermediate product product1_hi_0 := lowest 88 bits of middle intermediate product's highest 88 + 2 bits product1_hi_1 := highest 2 bits of middle intermediate product quotient_hi_bound := quotient high bound for checking q2 ≤ f2 Layout The foreign field multiplication gate's rows are laid out like this col ForeignFieldMul Zero 0 left_input0 (copy) remainder01 (copy) 1 left_input1 (copy) remainder2 (copy) 2 left_input2 (copy) quotient0 (copy) 3 right_input0 (copy) quotient1 (copy) 4 right_input1 (copy) quotient2 (copy) 5 right_input2 (copy) quotient_hi_bound (copy) 6 product1_lo (copy) product1_hi_0 (copy) 7 carry1_0 (plookup) product1_hi_1 (dummy) 8 `carry1_12 (plookup) carry1_48 (plookup) 9 carry1_24 (plookup) carry1_60 (plookup) 10 carry1_36 (plookup) carry1_72 (plookup) 11 carry1_84 carry0 12 carry1_86 13 carry1_88 14 carry1_90 Rotation Rotation of a 64-bit word by a known offset Rot64 onstrains known-length rotation of 64-bit words: This circuit gate is used to constrain that a 64-bit word is rotated by $r < 64$ bits to the \"left\". The rotation is performed towards the most significant side (thus, the new LSB is fed with the old MSB). This gate operates on the Curr and Next rows. The idea is to split the rotation operation into two parts: Shift to the left Add the excess bits to the right We represent shifting with multiplication modulo $2^{64}$. That is, for each word to be rotated, we provide in the witness a quotient and a remainder, similarly to ForeignFieldMul such that the following operation holds: $$word \\cdot 2^{rot} = quotient \\cdot 2^{64} + remainder$$ Then, the remainder corresponds to the shifted word, and the quotient corresponds to the excess bits. $$word \\cdot 2^{rot} = excess \\cdot 2^{64} + shifted$$ Thus, in order to obtain the rotated word, we need to add the quotient and the remainder as follows: $$rotated = shifted + excess$$ The input word is known to be of length 64 bits. All we need for soundness is check that the shifted and excess parts of the word have the correct size as well. That means, we need to range check that: $$ \\begin{aligned} excess &< 2^{rot}\\ shifted &< 2^{64} \\end{aligned} $$ The latter can be obtained with a RangeCheck0 gate setting the two most significant limbs to zero. The former is equivalent to the following check: $$bound = excess - 2^{rot} + 2^{64} < 2^{64}$$ which is doable with the constraints in a RangeCheck0 gate. Since our current row within the Rot64 gate is almost empty, we can use it to perform the range check within the same gate. Then, using the following layout and assuming that the gate has a coefficient storing the value $2^{rot}$, which is publicly known Gate Rot64 RangeCheck0 gadgets (designer's duty) Column Curr Next ------ ------------------- ---------------- 0 copy word shifted 1 copy rotated 0 2 excess 0 3 bound_limb0 shifted_limb0 4 bound_limb1 shifted_limb1 5 bound_limb2 shifted_limb2 6 bound_limb3 shifted_limb3 7 bound_crumb0 shifted_crumb0 8 bound_crumb1 shifted_crumb1 9 bound_crumb2 shifted_crumb2 10 bound_crumb3 shifted_crumb3 11 bound_crumb4 shifted_crumb4 12 bound_crumb5 shifted_crumb5 13 bound_crumb6 shifted_crumb6 14 bound_crumb7 shifted_crumb7 In Keccak, rotations are performed over a 5x5 matrix state of w-bit words each cell. The values used to perform the rotation are fixed, public, and known in advance, according to the following table, depending on the coordinate of each cell within the 5x5 matrix state: y \\ x 0 1 2 3 4 0 0 36 3 105 210 1 1 300 10 45 66 2 190 6 171 15 253 3 28 55 153 21 120 4 91 276 231 136 78 But since we will always be using 64-bit words in our Keccak usecase ($w = 64$), we can have an equivalent table with these values modulo 64 to avoid needing multiple passes of the rotation gate (a single step would cause overflows otherwise): y \\ x 0 1 2 3 4 0 0 36 3 41 18 1 1 44 10 45 2 2 62 6 43 15 61 3 28 55 25 21 56 4 27 20 39 8 14 Since there is one value of the coordinates (x, y) where the rotation is 0 bits, we can skip that step in the gadget. This will save us one gate, and thus the whole 25-1=24 rotations will be performed in just 48 rows. Xor Xor16 - Chainable XOR constraints for words of multiples of 16 bits. This circuit gate is used to constrain that in1 xored with in2 equals out The length of in1, in2 and out must be the same and a multiple of 16bits. This gate operates on the Curr and Next rows. It uses three different types of constraints: copy - copy to another cell (32-bits) plookup - xor-table plookup (4-bits) decomposition - the constraints inside the gate The 4-bit nybbles are assumed to be laid out with 0 column being the least significant nybble. Given values in1, in2 and out, the layout looks like this: Column Curr Next 0 copy in1 copy in1' 1 copy in2 copy in2' 2 copy out copy out' 3 plookup0 in1_0 4 plookup1 in1_1 5 plookup2 in1_2 6 plookup3 in1_3 7 plookup0 in2_0 8 plookup1 in2_1 9 plookup2 in2_2 10 plookup3 in2_3 11 plookup0 out_0 12 plookup1 out_1 13 plookup2 out_2 14 plookup3 out_3 One single gate with next values of in1', in2' and out' being zero can be used to check that the original in1, in2 and out had 16-bits. We can chain this gate 4 times as follows to obtain a gadget for 64-bit words XOR: Row CircuitGate Purpose 0 Xor16 Xor 2 least significant bytes of the words 1 Xor16 Xor next 2 bytes of the words 2 Xor16 Xor next 2 bytes of the words 3 Xor16 Xor 2 most significant bytes of the words 4 Generic Zero values, can be reused as generic gate Info We could halve the number of rows of the 64-bit XOR gadget by having lookups for 8 bits at a time, but for now we will use the 4-bit XOR table that we have. Rough computations show that if we run 8 or more Keccaks in one circuit we should use the 8-bit XOR table.","breadcrumbs":"Kimchi » Gates","id":"254","title":"Gates"},"255":{"body":"Here we describe basic gadgets that we build using a combination of the gates described above. Not We implement NOT, i.e. bitwise negation, as a gadget in two different ways, needing no new gate type for it. Instead, it reuses the XOR gadget and the Generic gate. The first version of the NOT gadget reuses Xor16 by making the following observation: the bitwise NOT operation is equivalent to the bitwise XOR operation with the all one words of a certain length . In other words, $$\\neg x = x \\oplus 1^ $$ where $1^ $ denotes a bitstring of all ones of length $|x|$. Let $x_i$ be the $i$-th bit of $x$, the intuition is that if $x_i = 0$ then XOR with $1$ outputs $1$, thus negating $x_i$. Similarly, if $x_i = 1$ then XOR with 1 outputs 0, again negating $x_i$. Thus, bitwise XOR with $1^*$ is equivalent to bitwise negation (i.e. NOT). Then, if we take the XOR gadget with a second input to be the all one word of the same length, that gives us the NOT gadget. The correct length can be imposed by having a public input containing the 2^bits - 1 value and wiring it to the second input of the XOR gate. This approach needs as many rows as an XOR would need, for a single negation, but it comes with the advantage of making sure the input is of a certain length. The other approach can be more efficient if we already know the length of the inputs. For example, the input may be the input of a range check gate, or the output of a previous XOR gadget (which will be the case in our Keccak usecase). In this case, we simply perform the negation as a subtraction of the input word from the all one word (which again can be copied from a public input). This comes with the advantage of holding up to 2 word negations per row (an eight-times improvement over the XOR approach), but it requires the user to know the length of the input. NOT Layout using XOR Here we show the layout of the NOT gadget using the XOR approach. The gadget needs a row with a public input containing the all-one word of the given length. Then, a number of XORs follow, and a final Zero row is needed. In this case, the NOT gadget needs $\\ceil(\\frac{|x|}{16})$ Xor16 gates, that means one XOR row for every 16 bits of the input word. Row CircuitGate Purpose pub Generic Leading row with the public $1^*$ value i...i+n-1 Xor16 Negate every 4 nybbles of the word, from least to most significant i+n Generic Constrain that the final row is all zeros for correctness of Xor gate NOT Layout using Generic gates Here we show the layout of the NOT gadget using the Generic approach. The gadget needs a row with a public input containing the all-one word of the given length, exactly as above. Then, one Generic gate reusing the all-one word as left inputs can be used to negate up to two words per row. This approach requires that the input word is known (or constrained) to have a given length. Row CircuitGate Purpose pub Generic Leading row with the public $1^*$ value i Generic Negate one or two words of the length given by the length of the all-one word And We implement the AND gadget making use of the XOR gadget and the Generic gate. A new gate type is not needed, but we could potentially add an And16 gate type reusing the same ideas of Xor16 so as to save one final generic gate, at the cost of one additional AND lookup table that would have the same size as that of the Xor. For now, we are willing to pay this small overhead and produce AND gadget as follows: We observe that we can express bitwise addition as follows: $$A + B = (A \\oplus B) + 2 \\cdot (A \\wedge B)$$ where $\\oplus$ is the bitwise XOR operation, $\\wedge$ is the bitwise AND operation, and $+$ is the addition operation. In other words, the value of the addition is nothing but the XOR of its operands, plus the carry bit if both operands are 1. Thus, we can rewrite the above equation to obtain a definition of the AND operation as follows: $$A \\wedge B = \\frac{A + B - (A \\oplus B)}{2}$$ Let us define the following operations for better readability: a + b = sum\na x b = xor\na ^ b = and Then, we can rewrite the above equation as follows: $$ 2 \\cdot and = sum - xor $$ which can be expressed as a double generic gate. Then, our AND gadget for $n$ bytes looks as follows: $n/8$ Xor16 gates 1 (single) Generic gate to check that the final row of the XOR chain is all zeros. 1 (double) Generic gate to check sum $a + b = sum$ and the conjunction equation $2\\cdot and = sum - xor$. Finally, we connect the wires in the following positions (apart from the ones already connected for the XOR gates): Column 2 of the first Xor16 row (the output of the XOR operation) is connected to the right input of the second generic operation of the last row. Column 2 of the first generic operation of the last row is connected to the left input of the second generic operation of the last row. Meaning, the xor in a x b = xor is connected to the xor in 2 \\cdot and = sum - xor the sum in a + b = sum is connected to the sum in 2 \\cdot and = sum - xor","breadcrumbs":"Kimchi » Gadgets","id":"255","title":"Gadgets"},"256":{"body":"In this section we specify the setup that goes into creating two indexes from a circuit: A prover index , necessary for the prover to create proofs. A verifier index , necessary for the verifier to verify proofs. Note The circuit creation part is not specified in this document. It might be specified in a separate document, or we might want to specify how to create the circuit description tables. As such, the transformation of a circuit into these two indexes can be seen as a compilation step. Note that the prover still needs access to the original circuit to create proofs, as they need to execute it to create the witness (register table).","breadcrumbs":"Kimchi » Setup","id":"256","title":"Setup"},"257":{"body":"In this section we describe data that both the prover and the verifier index share. URS (Uniform Reference String) The URS is a set of parameters that is generated once, and shared between the prover and the verifier. It is used for polynomial commitments, so refer to the poly-commitment specification for more details. Note Kimchi currently generates the URS based on the circuit, and attach it to the index. So each circuit can potentially be accompanied with a different URS. On the other hand, Mina reuses the same URS for multiple circuits ( see zkapps for more details ). Domain . A domain large enough to contain the circuit and the zero-knowledge rows (used to provide zero-knowledge to the protocol). Specifically, the smallest subgroup in our field that has order greater or equal to n + ZK_ROWS, with n is the number of gates in the circuit. TODO: what if the domain is larger than the URS? Ordering of elements in the domain Note that in this specification we always assume that the first element of a domain is $1$. Shifts . As part of the permutation, we need to create PERMUTS shifts. To do that, the following logic is followed (in pseudo code): (TODO: move shift creation within the permutation section?) shifts[0] = 1 # first shift is identity for i in 0..7: # generate 7 shifts i = 7 shift, i = sample(domain, i) while shifts.contains(shift) do: shift, i = sample(domain, i) shift[i] = shift def sample(domain, i): i += 1 shift = Field(Blake2b512(to_be_bytes(i))) while is_not_quadratic_non_residue(shift) || domain.contains(shift): i += 1 shift = Field(Blake2b512(to_be_bytes(i))) return shift, i Public . This variable simply contains the number of public inputs. (TODO: actually, it's not contained in the verifier index) The compilation steps to create the common index are as follow: If the circuit is less than 2 gates, abort. Compute the number of zero-knowledge rows (zk_rows) that will be required to achieve zero-knowledge. The following constraints apply to zk_rows: The number of chunks c results in an evaluation at zeta and zeta * omega in each column for 2*c evaluations per column, so zk_rows >= 2*c + 1. The permutation argument interacts with the c chunks in parallel, so it is possible to cross-correlate between them to compromise zero knowledge. We know that there is some c >= 1 such that zk_rows = 2*c + k from the above. Thus, attempting to find the evaluation at a new point, we find that: the evaluation of every witness column in the permutation contains k unknowns; the evaluations of the permutation argument aggregation has k-1 unknowns; the permutation argument applies on all but zk_rows - 3 rows; and thus we form the equation zk_rows - 3 < 7 * k + (k - 1) to ensure that we can construct fewer equations than we have unknowns. This simplifies to k > (2 * c - 2) / 7, giving zk_rows > (16 * c - 2) / 7. We can derive c from the max_poly_size supported by the URS, and thus we find zk_rows and domain_size satisfying the fixpoint zk_rows = (16 * (domain_size / max_poly_size) + 5) / 7\ndomain_size = circuit_size + zk_rows Create a domain for the circuit. That is, compute the smallest subgroup of the field that has order greater or equal to n + zk_rows elements. Pad the circuit: add zero gates to reach the domain size. sample the PERMUTS shifts.","breadcrumbs":"Kimchi » Common Index","id":"257","title":"Common Index"},"258":{"body":"If lookup is used, the following values are added to the common index: LookupSelectors . The list of lookup selectors used. In practice, this tells you which lookup tables are used. TableIds . This is a list of table ids used by the Lookup gate. MaxJointSize . This is the maximum number of columns appearing in the lookup tables used by the lookup selectors. For example, the XOR lookup has 3 columns. To create the index, follow these steps: If no lookup is used in the circuit, do not create a lookup index Get the lookup selectors and lookup tables that are specified implicitly Concatenate explicit runtime lookup tables with the ones (implicitly) used by gates. Get the highest number of columns max_table_width that a lookup table can have. Create the concatenated table of all the fixed lookup tables. It will be of height the size of the domain, and of width the maximum width of any of the lookup tables. In addition, create an additional column to store all the tables' table IDs. For example, if you have a table with ID 0 1 2 3 5 6 7 0 0 0 and another table with ID 1 8 9 the concatenated table in a domain of size 5 looks like this: 1 2 3 5 6 7 0 0 0 8 9 0 0 0 0 with the table id vector: table id 0 0 0 1 0 To do this, for each table: Update the corresponding entries in a table id vector (of size the domain as well) with the table ID of the table. Copy the entries from the table to new rows in the corresponding columns of the concatenated table. Fill in any unused columns with 0 (to match the dummy value) Pad the end of the concatened table with the dummy value. Pad the end of the table id vector with 0s. pre-compute polynomial and evaluation form for the look up tables pre-compute polynomial and evaluation form for the table IDs, only if a table with an ID different from zero was used.","breadcrumbs":"Kimchi » Lookup Index","id":"258","title":"Lookup Index"},"259":{"body":"Both the prover and the verifier index, besides the common parts described above, are made out of pre-computations which can be used to speed up the protocol. These pre-computations are optimizations, in the context of normal proofs, but they are necessary for recursion. pub struct ProverIndex> { /// constraints system polynomials #[serde(bound = \"ConstraintSystem: Serialize + DeserializeOwned\")] pub cs: ConstraintSystem, /// The symbolic linearization of our circuit, which can compile to concrete types once certain values are learned in the protocol. #[serde(skip)] pub linearization: Linearization>, Column>, /// The mapping between powers of alpha and constraints #[serde(skip)] pub powers_of_alpha: Alphas, /// polynomial commitment keys #[serde(skip)] #[serde(bound(deserialize = \"OpeningProof::SRS: Default\"))] pub srs: Arc, /// maximal size of polynomial section pub max_poly_size: usize, #[serde(bound = \"ColumnEvaluations: Serialize + DeserializeOwned\")] pub column_evaluations: ColumnEvaluations, /// The verifier index corresponding to this prover index #[serde(skip)] pub verifier_index: Option>, /// The verifier index digest corresponding to this prover index #[serde_as(as = \"Option\")] pub verifier_index_digest: Option,\n}","breadcrumbs":"Kimchi » Prover Index","id":"259","title":"Prover Index"},"26":{"body":"The homomorphic commitment commit(x,r)=xG+rH described above is known as a Pedersen commitment. If you remove the rH term you get a non-hiding commitment, called a Pedersen hash . Both rely on the ECDLP hardness assumption. This means that, at least theoretically, you might be lucky (or have a quantum computer) and figure out that H=hG, which would allow you to find different values x′ and h′ to open the commitment. We say that pedersen commitments are computationally binding and not unconditionally binding. For example, you could express c=xG+rH alternatively as c=xG+rhG=(x+rh)G and compute a satisfying opening pair x′=rh and r′=hx​. On the other hand, Pedersen commitments are unconditionally hiding , as there is no way (even with a magic computer) to reveal what x is without knowing r. Lack of perfect binding is the reason why most of the \"proofs\" we will see later in this book are not referred to as proofs, but instead are referred to as arguments of knowledge (although we may care little about this distinction). Just remember that you need perfect binding to be called a proof. Interestingly, it is impossible to have a commitment scheme that has both perfect hiding and perfect binding. To recap, in cryptography the following distinctions are important Perfect. The property that an algorithm is statistically sound without hardness assumptions, also known as unconditional or statistical soundness. Computational. The algorithm relies on a hardness assumption or computational limitation for soundness. Thus, said another way, Pedersen commitments provide perfect hiding and computational binding.","breadcrumbs":"Commitments » Pedersen commitments","id":"26","title":"Pedersen commitments"},"260":{"body":"Same as the prover index, we have a number of pre-computations as part of the verifier index. #[serde_as]\n#[derive(Serialize, Deserialize, Debug, Clone)]\npub struct LookupVerifierIndex { pub joint_lookup_used: bool, #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub lookup_table: Vec>, #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub lookup_selectors: LookupSelectors>, /// Table IDs for the lookup values. /// This may be `None` if all lookups originate from table 0. #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub table_ids: Option>, /// Information about the specific lookups used pub lookup_info: LookupInfo, /// An optional selector polynomial for runtime tables #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub runtime_tables_selector: Option>,\n} #[serde_as]\n#[derive(Serialize, Deserialize, Debug, Clone)]\npub struct VerifierIndex> { /// evaluation domain #[serde_as(as = \"o1_utils::serialization::SerdeAs\")] pub domain: D, /// maximal size of polynomial section pub max_poly_size: usize, /// the number of randomized rows to achieve zero knowledge pub zk_rows: u64, /// polynomial commitment keys #[serde(skip)] #[serde(bound(deserialize = \"OpeningProof::SRS: Default\"))] pub srs: Arc, /// number of public inputs pub public: usize, /// number of previous evaluation challenges, for recursive proving pub prev_challenges: usize, // index polynomial commitments /// permutation commitment array #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub sigma_comm: [PolyComm; PERMUTS], /// coefficient commitment array #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub coefficients_comm: [PolyComm; COLUMNS], /// coefficient commitment array #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub generic_comm: PolyComm, // poseidon polynomial commitments /// poseidon constraint selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub psm_comm: PolyComm, // ECC arithmetic polynomial commitments /// EC addition selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub complete_add_comm: PolyComm, /// EC variable base scalar multiplication selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub mul_comm: PolyComm, /// endoscalar multiplication selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub emul_comm: PolyComm, /// endoscalar multiplication scalar computation selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub endomul_scalar_comm: PolyComm, /// RangeCheck0 polynomial commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub range_check0_comm: Option>, /// RangeCheck1 polynomial commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub range_check1_comm: Option>, /// Foreign field addition gates polynomial commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub foreign_field_add_comm: Option>, /// Foreign field multiplication gates polynomial commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub foreign_field_mul_comm: Option>, /// Xor commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub xor_comm: Option>, /// Rot commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub rot_comm: Option>, /// wire coordinate shifts #[serde_as(as = \"[o1_utils::serialization::SerdeAs; PERMUTS]\")] pub shift: [G::ScalarField; PERMUTS], /// zero-knowledge polynomial #[serde(skip)] pub permutation_vanishing_polynomial_m: OnceCell>, // TODO(mimoo): isn't this redundant with domain.d1.group_gen ? /// domain offset for zero-knowledge #[serde(skip)] pub w: OnceCell, /// endoscalar coefficient #[serde(skip)] pub endo: G::ScalarField, #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub lookup_index: Option>, #[serde(skip)] pub linearization: Linearization>, Column>, /// The mapping between powers of alpha and constraints #[serde(skip)] pub powers_of_alpha: Alphas,\n}","breadcrumbs":"Kimchi » Verifier Index","id":"260","title":"Verifier Index"},"261":{"body":"Originally, kimchi is based on an interactive protocol that was transformed into a non-interactive one using the Fiat-Shamir transform. For this reason, it can be useful to visualize the high-level interactive protocol before the transformation: sequenceDiagram participant Prover participant Verifier Note over Prover,Verifier: Prover produces commitments to secret polynomials Prover->>Verifier: public input & witness commitment Verifier->>Prover: beta & gamma Prover->>Verifier: permutation commitment opt lookup Prover->>Verifier: sorted Prover->>Verifier: aggreg end Note over Prover,Verifier: Prover produces commitment to quotient polynomial Verifier->>Prover: alpha Prover->>Verifier: quotient commitment Note over Prover,Verifier: Verifier produces an evaluation point Verifier->>Prover: zeta Note over Verifier: change of verifier (change of sponge) opt recursion Prover->>Verifier: recursion challenges end Note over Prover,Verifier: Prover provides needed evaluations for the linearization Prover->>Verifier: ft(zeta * omega) Prover->>Verifier: negated public input p(zeta) & p(zeta * omega) Prover->>Verifier: the 15 registers w_i(zeta) & w_i(zeta * omega) Prover->>Verifier: permutation poly z(zeta) & z(zeta * omega) Prover->>Verifier: the 6 sigmas s_i(zeta) & s_i(zeta * omega) Prover->>Verifier: the 15 coefficients c_i(zeta) & c_i(zeta * omega) opt lookup Prover->>Verifier: sorted(zeta) & sorted(zeta * omega) Prover->>Verifier: aggreg(zeta) & aggreg(zeta * omega) Prover->>Verifier: table(zeta) & table(zeta * omega) end Prover->>Verifier: the generic selector gen(zeta) & gen(zeta * omega) Prover->>Verifier: the poseidon selector pos(zeta) & pos(zeta * omega) Note over Prover,Verifier: Batch verification of evaluation proofs Verifier->>Prover: v, u Note over Verifier: change of verifier (change of sponge) Prover->>Verifier: aggregated evaluation proof (involves more interaction) The Fiat-Shamir transform simulates the verifier messages via a hash function that hashes the transcript of the protocol so far before outputing verifier messages. You can find these operations under the proof creation and proof verification algorithms as absorption and squeezing of values with the sponge.","breadcrumbs":"Kimchi » Proof Construction & Verification","id":"261","title":"Proof Construction & Verification"},"262":{"body":"A proof consists of the following data structures: /// Evaluations of a polynomial at 2 points\n#[serde_as]\n#[derive(Copy, Clone, Serialize, Deserialize, Default, Debug)]\n#[cfg_attr( feature = \"ocaml_types\", derive(ocaml::IntoValue, ocaml::FromValue, ocaml_gen::Struct)\n)]\n#[serde(bound( serialize = \"Vec: serde_with::SerializeAs\", deserialize = \"Vec: serde_with::DeserializeAs<'de, Evals>\"\n))]\npub struct PointEvaluations { /// Evaluation at the challenge point zeta. #[serde_as(as = \"Vec\")] pub zeta: Evals, /// Evaluation at `zeta . omega`, the product of the challenge point and the group generator. #[serde_as(as = \"Vec\")] pub zeta_omega: Evals,\n} // TODO: this should really be vectors here, perhaps create another type for chunked evaluations?\n/// Polynomial evaluations contained in a `ProverProof`.\n/// - **Chunked evaluations** `Field` is instantiated with vectors with a length\n/// that equals the length of the chunk\n/// - **Non chunked evaluations** `Field` is instantiated with a field, so they\n/// are single-sized#[serde_as]\n#[serde_as]\n#[derive(Debug, Clone, Serialize, Deserialize)]\npub struct ProofEvaluations { /// public input polynomials pub public: Option, /// witness polynomials pub w: [Evals; COLUMNS], /// permutation polynomial pub z: Evals, /// permutation polynomials /// (PERMUTS-1 evaluations because the last permutation is only used in /// commitment form) pub s: [Evals; PERMUTS - 1], /// coefficient polynomials pub coefficients: [Evals; COLUMNS], /// evaluation of the generic selector polynomial pub generic_selector: Evals, /// evaluation of the poseidon selector polynomial pub poseidon_selector: Evals, /// evaluation of the elliptic curve addition selector polynomial pub complete_add_selector: Evals, /// evaluation of the elliptic curve variable base scalar multiplication /// selector polynomial pub mul_selector: Evals, /// evaluation of the endoscalar multiplication selector polynomial pub emul_selector: Evals, /// evaluation of the endoscalar multiplication scalar computation selector /// polynomial pub endomul_scalar_selector: Evals, // Optional gates /// evaluation of the RangeCheck0 selector polynomial pub range_check0_selector: Option, /// evaluation of the RangeCheck1 selector polynomial pub range_check1_selector: Option, /// evaluation of the ForeignFieldAdd selector polynomial pub foreign_field_add_selector: Option, /// evaluation of the ForeignFieldMul selector polynomial pub foreign_field_mul_selector: Option, /// evaluation of the Xor selector polynomial pub xor_selector: Option, /// evaluation of the Rot selector polynomial pub rot_selector: Option, // lookup-related evaluations /// evaluation of lookup aggregation polynomial pub lookup_aggregation: Option, /// evaluation of lookup table polynomial pub lookup_table: Option, /// evaluation of lookup sorted polynomials pub lookup_sorted: [Option; 5], /// evaluation of runtime lookup table polynomial pub runtime_lookup_table: Option, // lookup selectors /// evaluation of the runtime lookup table selector polynomial pub runtime_lookup_table_selector: Option, /// evaluation of the Xor range check pattern selector polynomial pub xor_lookup_selector: Option, /// evaluation of the Lookup range check pattern selector polynomial pub lookup_gate_lookup_selector: Option, /// evaluation of the RangeCheck range check pattern selector polynomial pub range_check_lookup_selector: Option, /// evaluation of the ForeignFieldMul range check pattern selector /// polynomial pub foreign_field_mul_lookup_selector: Option,\n} /// Commitments linked to the lookup feature\n#[serde_as]\n#[derive(Debug, Clone, Serialize, Deserialize)]\n#[serde(bound = \"G: ark_serialize::CanonicalDeserialize + ark_serialize::CanonicalSerialize\")]\npub struct LookupCommitments { /// Commitments to the sorted lookup table polynomial (may have chunks) pub sorted: Vec>, /// Commitment to the lookup aggregation polynomial pub aggreg: PolyComm, /// Optional commitment to concatenated runtime tables pub runtime: Option>,\n} /// All the commitments that the prover creates as part of the proof.\n#[serde_as]\n#[derive(Debug, Clone, Serialize, Deserialize)]\n#[serde(bound = \"G: ark_serialize::CanonicalDeserialize + ark_serialize::CanonicalSerialize\")]\npub struct ProverCommitments { /// The commitments to the witness (execution trace) pub w_comm: [PolyComm; COLUMNS], /// The commitment to the permutation polynomial pub z_comm: PolyComm, /// The commitment to the quotient polynomial pub t_comm: PolyComm, /// Commitments related to the lookup argument pub lookup: Option>,\n} /// The proof that the prover creates from a\n/// [ProverIndex](super::prover_index::ProverIndex) and a `witness`.\n#[serde_as]\n#[derive(Debug, Clone, Serialize, Deserialize)]\n#[serde(bound = \"G: ark_serialize::CanonicalDeserialize + ark_serialize::CanonicalSerialize\")]\npub struct ProverProof { /// All the polynomial commitments required in the proof pub commitments: ProverCommitments, /// batched commitment opening proof #[serde(bound( serialize = \"OpeningProof: Serialize\", deserialize = \"OpeningProof: Deserialize<'de>\" ))] pub proof: OpeningProof, /// Two evaluations over a number of committed polynomials pub evals: ProofEvaluations>>, /// Required evaluation for [Maller's /// optimization](https://o1-labs.github.io/proof-systems/kimchi/maller_15.html#the-evaluation-of-l) #[serde_as(as = \"o1_utils::serialization::SerdeAs\")] pub ft_eval1: G::ScalarField, /// The challenges underlying the optional polynomials folded into the proof pub prev_challenges: Vec>,\n} /// A struct to store the challenges inside a `ProverProof`\n#[serde_as]\n#[derive(Debug, Clone, Deserialize, Serialize)]\n#[serde(bound = \"G: ark_serialize::CanonicalDeserialize + ark_serialize::CanonicalSerialize\")]\npub struct RecursionChallenge\nwhere G: AffineRepr,\n{ /// Vector of scalar field elements #[serde_as(as = \"Vec\")] pub chals: Vec, /// Polynomial commitment pub comm: PolyComm,\n} The following sections specify how a prover creates a proof, and how a verifier validates a number of proofs.","breadcrumbs":"Kimchi » Proof Structure","id":"262","title":"Proof Structure"},"263":{"body":"To create a proof, the prover expects: A prover index, containing a representation of the circuit (and optionaly pre-computed values to be used in the proof creation). The (filled) registers table, representing parts of the execution trace of the circuit. Note The public input is expected to be passed in the first Public rows of the registers table. The following constants are set: EVAL_POINTS = 2. This is the number of points that the prover has to evaluate their polynomials at. ($\\zeta$ and $\\zeta\\omega$ where $\\zeta$ will be deterministically generated.) ZK_ROWS = 3. This is the number of rows that will be randomized to provide zero-knowledgeness. Note that it only needs to be greater or equal to the number of evaluations (2) in the protocol. Yet, it contains one extra row to take into account the last constraint (final value of the permutation accumulator). (TODO: treat the final constraint separately so that ZK_ROWS = 2) The prover then follows the following steps to create the proof: Ensure we have room in the witness for the zero-knowledge rows. We currently expect the witness not to be of the same length as the domain, but instead be of the length of the (smaller) circuit. If we cannot add zk_rows rows to the columns of the witness before reaching the size of the domain, abort. Pad the witness columns with Zero gates to make them the same length as the domain. Then, randomize the last zk_rows of each columns. Setup the Fq-Sponge. Absorb the digest of the VerifierIndex. Absorb the commitments of the previous challenges with the Fq-sponge. Compute the negated public input polynomial as the polynomial that evaluates to $-p_i$ for the first public_input_size values of the domain, and $0$ for the rest. Commit (non-hiding) to the negated public input polynomial. Absorb the commitment to the public polynomial with the Fq-Sponge. Note: unlike the original PLONK protocol, the prover also provides evaluations of the public polynomial to help the verifier circuit. This is why we need to absorb the commitment to the public polynomial at this point. Commit to the witness columns by creating COLUMNS hidding commitments. Note: since the witness is in evaluation form, we can use the commit_evaluation optimization. Absorb the witness commitments with the Fq-Sponge. Compute the witness polynomials by interpolating each COLUMNS of the witness. As mentioned above, we commit using the evaluations form rather than the coefficients form so we can take advantage of the sparsity of the evaluations (i.e., there are many 0 entries and entries that have less-than-full-size field elemnts.) If using lookup: if using runtime table: check that all the provided runtime tables have length and IDs that match the runtime table configuration of the index we expect the given runtime tables to be sorted as configured, this makes it easier afterwards calculate the contribution to the second column of the lookup table (the runtime vector) If queries involve a lookup table with multiple columns then squeeze the Fq-Sponge to obtain the joint combiner challenge $j'$, otherwise set the joint combiner challenge $j'$ to $0$. Derive the scalar joint combiner $j$ from $j'$ using the endomorphism (TODO: specify) If multiple lookup tables are involved, set the table_id_combiner as the $j^i$ with $i$ the maximum width of any used table. Essentially, this is to add a last column of table ids to the concatenated lookup tables. Compute the dummy lookup value as the combination of the last entry of the XOR table (so (0, 0, 0)). Warning: This assumes that we always use the XOR table when using lookups. Compute the lookup table values as the combination of the lookup table entries. Compute the sorted evaluations. Randomize the last EVALS rows in each of the sorted polynomials in order to add zero-knowledge to the protocol. Commit each of the sorted polynomials. Absorb each commitments to the sorted polynomials. Sample $\\beta$ with the Fq-Sponge. Sample $\\gamma$ with the Fq-Sponge. If using lookup: Compute the lookup aggregation polynomial. Commit to the aggregation polynomial. Absorb the commitment to the aggregation polynomial with the Fq-Sponge. Compute the permutation aggregation polynomial $z$. Commit (hidding) to the permutation aggregation polynomial $z$. Absorb the permutation aggregation polynomial $z$ with the Fq-Sponge. Sample $\\alpha'$ with the Fq-Sponge. Derive $\\alpha$ from $\\alpha'$ using the endomorphism (TODO: details) TODO: instantiate alpha? Compute the quotient polynomial (the $t$ in $f = Z_H \\cdot t$). The quotient polynomial is computed by adding all these polynomials together: the combined constraints for all the gates the combined constraints for the permutation TODO: lookup the negated public polynomial and by then dividing the resulting polynomial with the vanishing polynomial $Z_H$. TODO: specify the split of the permutation polynomial into perm and bnd? commit (hiding) to the quotient polynomial $t$ Absorb the commitment of the quotient polynomial with the Fq-Sponge. Sample $\\zeta'$ with the Fq-Sponge. Derive $\\zeta$ from $\\zeta'$ using the endomorphism (TODO: specify) If lookup is used, evaluate the following polynomials at $\\zeta$ and $\\zeta \\omega$: the aggregation polynomial the sorted polynomials the table polynonial Chunk evaluate the following polynomials at both $\\zeta$ and $\\zeta \\omega$: $s_i$ $w_i$ $z$ lookup (TODO, see this issue ) generic selector poseidon selector By \"chunk evaluate\" we mean that the evaluation of each polynomial can potentially be a vector of values. This is because the index's max_poly_size parameter dictates the maximum size of a polynomial in the protocol. If a polynomial $f$ exceeds this size, it must be split into several polynomials like so: $$f(x) = f_0(x) + x^n f_1(x) + x^{2n} f_2(x) + \\cdots$$ And the evaluation of such a polynomial is the following list for $x \\in {\\zeta, \\zeta\\omega}$: $$(f_0(x), f_1(x), f_2(x), \\ldots)$$ TODO: do we want to specify more on that? It seems unnecessary except for the t polynomial (or if for some reason someone sets that to a low value) Evaluate the same polynomials without chunking them (so that each polynomial should correspond to a single value this time). Compute the ft polynomial. This is to implement Maller's optimization . construct the blinding part of the ft polynomial commitment see this section Evaluate the ft polynomial at $\\zeta\\omega$ only. Setup the Fr-Sponge Squeeze the Fq-sponge and absorb the result with the Fr-Sponge. Absorb the previous recursion challenges. Compute evaluations for the previous recursion challenges. Absorb the unique evaluation of ft: $ft(\\zeta\\omega)$. Absorb all the polynomial evaluations in $\\zeta$ and $\\zeta\\omega$: the public polynomial z generic selector poseidon selector the 15 register/witness 6 sigmas evaluations (the last one is not evaluated) Sample $v'$ with the Fr-Sponge Derive $v$ from $v'$ using the endomorphism (TODO: specify) Sample $u'$ with the Fr-Sponge Derive $u$ from $u'$ using the endomorphism (TODO: specify) Create a list of all polynomials that will require evaluations (and evaluation proofs) in the protocol. First, include the previous challenges, in case we are in a recursive prover. Then, include: the negated public polynomial the ft polynomial the permutation aggregation polynomial z polynomial the generic selector the poseidon selector the 15 registers/witness columns the 6 sigmas the optional gates optionally, the runtime table if using lookup: add the lookup sorted polynomials add the lookup aggreg polynomial add the combined table polynomial if present, add the runtime table polynomial the lookup selectors Create an aggregated evaluation proof for all of these polynomials at $\\zeta$ and $\\zeta\\omega$ using $u$ and $v$.","breadcrumbs":"Kimchi » Proof Creation","id":"263","title":"Proof Creation"},"264":{"body":"TODO: we talk about batch verification, but is there an actual batch operation? It seems like we're just verifying an aggregated opening proof We define two helper algorithms below, used in the batch verification of proofs. Fiat-Shamir argument We run the following algorithm: Setup the Fq-Sponge. This sponge mostly absorbs group Absorb the digest of the VerifierIndex. Absorb the commitments of the previous challenges with the Fq-sponge. Absorb the commitment of the public input polynomial with the Fq-Sponge. Absorb the commitments to the registers / witness columns with the Fq-Sponge. If lookup is used: If it involves queries to a multiple-column lookup table, then squeeze the Fq-Sponge to obtain the joint combiner challenge $j'$, otherwise set the joint combiner challenge $j'$ to $0$. Derive the scalar joint combiner challenge $j$ from $j'$ using the endomorphism. (TODO: specify endomorphism) absorb the commitments to the sorted polynomials. Sample the first permutation challenge $\\beta$ with the Fq-Sponge. Sample the second permutation challenge $\\gamma$ with the Fq-Sponge. If using lookup, absorb the commitment to the aggregation lookup polynomial. Absorb the commitment to the permutation trace with the Fq-Sponge. Sample the quotient challenge $\\alpha'$ with the Fq-Sponge. Derive $\\alpha$ from $\\alpha'$ using the endomorphism (TODO: details). Enforce that the length of the $t$ commitment is of size 7. Absorb the commitment to the quotient polynomial $t$ into the argument. Sample $\\zeta'$ with the Fq-Sponge. Derive $\\zeta$ from $\\zeta'$ using the endomorphism (TODO: specify). Setup the Fr-Sponge. This sponge absorbs elements from Squeeze the Fq-sponge and absorb the result with the Fr-Sponge. Absorb the previous recursion challenges. Compute evaluations for the previous recursion challenges. Evaluate the negated public polynomial (if present) at $\\zeta$ and $\\zeta\\omega$. NOTE: this works only in the case when the poly segment size is not smaller than that of the domain. Absorb the unique evaluation of ft: $ft(\\zeta\\omega)$. Absorb all the polynomial evaluations in $\\zeta$ and $\\zeta\\omega$: the public polynomial z generic selector poseidon selector the 15 register/witness 6 sigmas evaluations (the last one is not evaluated) Sample the \"polyscale\" $v'$ with the Fr-Sponge. Derive $v$ from $v'$ using the endomorphism (TODO: specify). Sample the \"evalscale\" $u'$ with the Fr-Sponge. Derive $u$ from $u'$ using the endomorphism (TODO: specify). Create a list of all polynomials that have an evaluation proof. Compute the evaluation of $ft(\\zeta)$. Partial verification For every proof we want to verify, we defer the proof opening to the very end. This allows us to potentially batch verify a number of partially verified proofs. Essentially, this steps verifies that $f(\\zeta) = t(\\zeta) * Z_H(\\zeta)$. Check the length of evaluations inside the proof. Commit to the negated public input polynomial. Run the Fiat-Shamir argument . Combine the chunked polynomials' evaluations (TODO: most likely only the quotient polynomial is chunked) with the right powers of $\\zeta^n$ and $(\\zeta * \\omega)^n$. Compute the commitment to the linearized polynomial $f$. To do this, add the constraints of all of the gates, of the permutation, and optionally of the lookup. (See the separate sections in the constraints section.) Any polynomial should be replaced by its associated commitment, contained in the verifier index or in the proof, unless a polynomial has its evaluation provided by the proof in which case the evaluation should be used in place of the commitment. Compute the (chuncked) commitment of $ft$ (see Maller's optimization ). List the polynomial commitments, and their associated evaluations, that are associated to the aggregated evaluation proof in the proof: recursion public input commitment ft commitment (chunks of it) permutation commitment index commitments that use the coefficients witness commitments coefficient commitments sigma commitments optional gate commitments lookup commitments Batch verification of proofs Below, we define the steps to verify a number of proofs (each associated to a verifier index ). You can, of course, use it to verify a single proof. If there's no proof to verify, the proof validates trivially. Ensure that all the proof's verifier index have a URS of the same length. (TODO: do they have to be the same URS though? should we check for that?) Validate each proof separately following the partial verification steps. Use the PolyCom.verify to verify the partially evaluated proofs.","breadcrumbs":"Kimchi » Proof Verification","id":"264","title":"Proof Verification"},"265":{"body":"commit_evaluation: TODO","breadcrumbs":"Kimchi » Optimizations","id":"265","title":"Optimizations"},"266":{"body":"TODO","breadcrumbs":"Kimchi » Security Considerations","id":"266","title":"Security Considerations"},"267":{"body":"Note Note that the URS is called SRS in our codebase at the moment. SRS is incorrect, as it is used in the presence of a trusted setup (which kimchi does not have). This needs to be fixed. The URS comprises of: Gs: an arbitrary ordered list of curve points that can be used to commit to a polynomial in a non-hiding way. H: a blinding curve point that can be used to add hidding to a polynomial commitment scheme. The URS is generated deterministically, and thus can be rederived if not stored. As different circuits of different size do not share the same domain, and the size of the domain is directly equal to the number of Gs curve points we use to commit to polynomials, we often want to share the same URS of maximum size across many circuits (constraint systems). Circuit associated with smaller domains will simply use a truncation of the shared URS. Note A circuit's domain is generated as the first power of 2 that is equal, or larger, to the number of gates used in the circuit (including zero-knowledge gates).","breadcrumbs":"Universal Reference String (URS) » Universal Reference String (URS)","id":"267","title":"Universal Reference String (URS)"},"268":{"body":"TODO: specify this part","breadcrumbs":"Universal Reference String (URS) » Group map","id":"268","title":"Group map"},"269":{"body":"The following pseudo-code specifies how the Gs curve points, as well as the H curve point, are generated. def create(depth): # generators Gs = [] for i in range(0, depth): digest = Blake2b512.hash(be_encode(i)) Gs.push(point_of_random_bytes(digest)) # blinding point digest = (b\"srs_misc\" || be_encode(0)) H = point_of_random_bytes(digest) # return (Gs, H) def point_of_random_bytes(random_bytes): # packing in bit-representation const N: usize = 31 let mut bits = [false; 8 * N] for i in range(0, N) { for j in range(0, 8) { bits[8 * i + j] = (random_bytes[i] >> j) & 1 == 1; } } let n = BigInt::from_bits_be(&bits); let t = G::BaseField::from_repr(n) return map.to_group(t) TODO: specify map.to_group in the previous section.","breadcrumbs":"Universal Reference String (URS) » Curve points to commit to a polynomial","id":"269","title":"Curve points to commit to a polynomial"},"27":{"body":"We can commit to several values x1​,⋯,xn​ by sending separate Pedersen commitments to all of these values as such: x1​G+r1​H,⋮xn​G+rn​H But we can instead batch/aggregate all of these commitments together as a single commitment: x1​G1​+⋯+xn​Gn​+rH with G1​,⋯,Gn​,H independent bases with unknown discrete logarithms. If you represent xs and the Gs as two vectors x=(x1​,⋯,xn​) and G=(G1​,⋯,Gn​), we can quickly write the previous statement as an inner product xG+rH Vector commitments (sometimes referred to as multi-commitments) are a powerful construction because an arbitrarily large vector can be committed with a single curve point. The naive approach to constructing an opening proof for a length n vector commitment has size O(n). It is simply the tuple (x1​,…,xn​,r). As we will see later, opening proofs for vector commitments is an interesting topic and there is a much more efficient algorithm.","breadcrumbs":"Commitments » Vector commitments","id":"27","title":"Vector commitments"},"270":{"body":"As there is no limit to the number of commitment curve points you can generate, we only provide the first three ones to serve as test vectors. TODO: specify the encoding","breadcrumbs":"Universal Reference String (URS) » URS values in practice","id":"270","title":"URS values in practice"},"271":{"body":"Gs . G0 = (x=121C4426885FD5A9701385AAF8D43E52E7660F1FC5AFC5F6468CC55312FC60F8, y=21B439C01247EA3518C5DDEB324E4CB108AF617780DDF766D96D3FD8AB028B70)\nG1 = (x=26C9349FF7FB4AB230A6F6AEF045F451FBBE9B37C43C3274E2AA4B82D131FD26, y=1996274D67EC0464C51F79CCFA1F511C2AABB666ABE67733EE8185B71B27A504)\nG2 = (x=26985F27306586711466C5B2C28754AA62FE33516D75CEF1F7751F1A169713FD, y=2E8930092FE6A18B331CE0E6E27B413AA18E76394F18A2835DA9FAE10AA3229D) H : H = (x=092060386301C999AAB4F263757836369CA27975E28BC7A8E5B2CE5B26262201, y=314FC4D83AE66A509F9D41BE6165F2606A209A9B5805EE85CE20249C5EBCBE26)","breadcrumbs":"Universal Reference String (URS) » Vesta","id":"271","title":"Vesta"},"272":{"body":"G0 = (x363D83141FD1E0540718FADBA7278ABAEEDB46D7A3F050F2CFF1DF4F300C9C30, y=034C68F4079B4F338A19BE2D7BFA44B395C65B9790DD273F361327446C778764)\nG1 = (x2CC40B77D87665244AE5EB5304E8744004C80061AD08476A0F0656C13134EA45, y=28146EC860159DB55CB5EA5B14F0AA2F8751DEDFE0DDAFD1C313B15575C4B4AC)\nG2 = (x2808BC21BEB90314377BF6130285FABE6CE4B8A4457FB25BC95EBA0083DF27E3, y=1E04E53DD6395FAB8018D7FE98F9C7FAB39C40BFBE48589626A7B8532728B002) H : H = (x221B959DACD2052AAE26193FCA36B53279866A4FBBAB0D5A2F828B5FD7778201, y=058C8F1105CAE57F4891EADC9B85C8954E5067190E155E61D66855ACE69C16C0)","breadcrumbs":"Universal Reference String (URS) » Pallas","id":"272","title":"Pallas"},"273":{"body":"Pickles is Mina’s inductive zk-SNARK composition system. It lets you construct proofs with zk-SNARKs and combine them in flexible ways to deliver incremental verifiable computation. Pickles uses a pair of amicable curves called Pasta in order to deliver incremental verifiable computation efficiently. These curves are referred to as \"tick\" and \"tock\" within the Mina source code. Tick - Vesta (a.k.a. Step), constraint domain size 218 (block and transaction proofs) Tock - Pallas (a.k.a. Wrap), constraint domain size 217 (signatures) The Tock prover does less (only performs recursive verifications and no other logic), so it requires fewer constraints and has a smaller domain size. Internally Pickles refers to Tick and Tock as Step and Wrap, respectively. Tock is used to prove the verification of a Tick proof and outputs a Tick proof. Tick is used to prove the verification of a Tock proof and outputs a Tock proof. In other words, Provetock​(Verify(Tick))=Tickproof​ Provetick​(Verify(Tock))=Tockproof​ Both Tick and Tock can verify at most 2 proofs of the opposite kind, though, theoretically more is possible. Currently, in Mina we have the following situation. Every Tock always wraps 1 Tick proof, such as Tick proofs can verify 2 Tock proofs Blockchain SNARK takes previous blockchain SNARK proof and a transaction proof Verifying two Tock transaction proofs","breadcrumbs":"Pickles » Pickles","id":"273","title":"Pickles"},"274":{"body":"vrf is appendix C of https://eprint.iacr.org/2017/573.pdf","breadcrumbs":"Consensus » Consensus","id":"274","title":"Consensus"},"28":{"body":"To construct SNARKs we use polynomial commitments. A polynomial commitment scheme for a field F (or it could even be a ring) is a way of committing to a polynomial f∈F[x] to get a commitment c, in such a way that for any α∈F, you can provide y=f(α), along with an \"opening proof\" π that proves that the polynomial committed to in c equals y when evaluated at α. In other words, it is a type of commitment C, a type of randomness R, a type of opening proof P along with algorithms commit:F[x]×R→Copen:C×F×(F[x]×R)→F×Pverify:C×(F×P)→Bool such that for any f∈F[x],r∈R,α∈F , we have c:=commit(f,r)verify(c,open(c,α,(f,r)))=true and if b=f(α) then it is not possible to compute π∈P such that verify(c,(b,π))=true In other words, if b=f(α) then every π which is feasible to compute results in verify(c,(b,π))=false. One thing that's pretty cool is that because polynomial commitment schemes let you construct zk-SNARKs, polynomial commitment schemes imply commitment schemes with arbitrary opening functionality. TODO","breadcrumbs":"Commitments » Polynomial commitments","id":"28","title":"Polynomial commitments"},"29":{"body":"All known constructions of polynomial commitment schemes are a bit complicated. The easiest to describe is called the Kate (pronounced \"kah-TAY\") scheme, also known as \"KZG\". It requires a \"prime-order group with a pairing\", which is three groups G1​,G2​,GT​ of prime order p (hence, all isomorphic cyclic groups) together with a function e:G1​×G2​→GT​ such that for any a1​,a2​∈Z, g1​∈G1​, g2​∈G2​, we have e(a1​⋅g1​,a2​⋅g2​)=a1​a2​⋅e(g1​,g2​) e is called a \"pairing\" or a \"bilinear pairing\". What this lets us do is \"multiply in the scalar\" but only once. Fix a degree bound d on the polynomials we would like to be able to commit to. The KZG scheme, will let us commit to polynomials in Fp​[x]| |<------ b | OT Prot. | | |-------> x_b +----------+ Due to a result of Impagliazzo and Rudich in this paper , it is very unlikely that OT is possible without the use of public-key cryptography. However, OT can be efficiently extended. That is, starting with a small number of base OTs, one could create many more OTs with only symmetric primitives. The seminal work of IKNP03 presented a very efficient protocol for extending OTs, requiring only black-box use of symmetric primitives and λ base OTs, where λ is the security parameter. This doc focuses on the family of protocols inspired by IKNP.","breadcrumbs":"Two Party Computation » Oblivious Transfer » Oblivious Transfer","id":"66","title":"Oblivious Transfer"},"67":{"body":"We introduce the CO15 OT as the base OT. Notation . The protocol is described over an additive group (G,G,p,+) of prime order p generated by G. Denote H:(G×G)×G→{0,1}λ as a key-derivation function to extract a λ-bit key from group elements. Note that H here is different from the tweakable hash function in garbled circuit. In the CO15 protocol, the sender holds n 128-bit strings {(Xi0​,Xi1​)}i∈[n]​ and the receiver holds n-bit string b∈{0,1}n. The protocol is as follows. The sender samples y←Fp​, and computes S=y⋅G and T=y⋅S. The sender sends S to the receiver, who aborts if S∈/G. For i∈[n], the receiver samples xi​←Fp​, and computes: Ri​=bi​S+xi​G The receiver sends {Ri​}i∈[n]​ to the sender, who aborts if Ri​∈/G. For i∈[n], the sender computes ki0​=H(S,Ri​,yRi​), ki1​=H(S,Ri​,yRi​−T) and sends {(ci0​=ki0​⊕Xi0​,ci1​=ki1​⊕Xi1​)}i∈[n]​ to the receiver. For i∈[n], the receiver computes ki​=H(S,Ri​,xi​S), and outputs cibi​​⊕ki​ Correctness . The receiver always computes the hash of xi​S=(xi​y)G. The sender sends the hashes of yRi​ and yRi​−T. If bi​=0, ki0​ is the hash of yRi​=(yxi​)G, then the receive will get Xi0​. if bi​=1, ki1​ is the hash of yRi​−T=y(S+xi​G)−T=(yxi​)G, then the receiver will get Xi1​. Security . We refer the security analysis to CO15 .","breadcrumbs":"Two Party Computation » Oblivious Transfer » Base OT » Base OT","id":"67","title":"Base OT"},"68":{"body":"To meet the large-scale demand of OTs, OT extensions can be used. An OT extension protocol works by running a small number of base OTs that are used as a base for obtaining many OTs via the use of cheap symmetric cryptographic operations only. This is conceptually similar to hybrid encryption. In this subsection, we focus on the maliciously secure OT extension presented in KOS15 . To be compatible with the garbled circuit optimizations, Correlated OT (COT) is considered here. COT is a variant of OT, where the sender chooses a secret string Δ and the receiver has a choice bit b. After the protocol, then sender obtains two strings X0​=X and X1​=X⊕Δ and the receiver obtains Xb​, where X is a random string. Sender Receiver +----------+ Delta ------->| |<------ b | COT Prot.| x_0 = x, x_1 = x + Delta <-------| |-------> x_b +----------+ Notation . Let λ be the security parameter, we use λ=128 in zkOracles. Identify F2λ​ with the finite field F2λ​. Use \"⋅\" for multiplication in F2λ​ and \"⋆\" for the component-wise product in F2λ​. Given a matrix A, denote its rows by subindices ai​ and its columns by superindices ak. Use v[i] to denote the i-th entry of v. The KOS15 OT protocol to generate ℓ(≫λ) OTs is described as follows. Initialize The receiver samples λ pairs of λ-bit seeds {(k0i​,k1i​)}i=1λ​. The sender choose a uniformly random λ-bit string Δ. The two parties calls the base OT with inputs Δ and {(k0i​,k1i​)}i=1λ​. Note that the sender acts as the receiver in the base OT, and the receiver acts as the sender in the base OT. The sender obtains kΔi​i​ for i∈[λ]. Extend The receiver takes as input the choice bits x1​,...,xℓ​, defines ℓ′=ℓ+λ+s, where s is the statistic parameter, and we set s=40 in zkOracles. Let x=x1​∥x2​∥...∥xℓ​∥x′∈F2ℓ′​, with x′∈F2ℓ′−ℓ​ uniformly chosen. The receiver defines vectors x1​,...,xℓ′​ as xi​=x[i]⋅(1,...,1)∈F2λ​ for i∈[ℓ′]. The receiver expands k0i​ and k1i​ with a pseudo random generator (PRG), letting t0i​=PRG(k0i​)∈F2ℓ′​ and t1i​=PRG(k1i​)∈F2ℓ′​ for i∈[λ] The sender expands kΔi​i​ with the same PRG and gets tΔi​i​=PRG(kΔi​i​) for i∈[λ] The receiver computes ui=t0i​⊕t1i​⊕x∈F2ℓ′​ for i∈[λ] and sends them to the sender. The sender computes qi=Δi​⋅ui⊕tΔi​i​=t0i​⊕Δi​⋅x∈F2ℓ′​ for i∈[λ] Let qj​ be the j-th row of the ℓ′×λ matrix Q=[q1∣q2∣⋯∣qλ], and similarly let tj​ be the j-th row of T=[t01​∣t02​∣⋯∣t0λ​]. Note that qj​=tj​⊕(xj​⋆Δ)=tj​⊕(x[j]⋅Δ) for j∈[ℓ′] Correlation Check The sender and receiver run a πRAND​(F2λℓ′​) protocol to obtain random elements χ1​,...,χℓ′​. πRAND​(F2λℓ′​) will be described later. The receiver computes x=j=1∑ℓ′​x[j]⋅χj​ and t=j=1∑ℓ′​tj​⋅χj​ and sends them to the sender. The sender computes q=j=1∑ℓ′​qj​⋅χj​ and checks t=q⊕x⋅Δ. If the check fails, abort. Output The sender computes vj​=H(j,qj​)⊕H(j,qj​⊕Δ)⊕Δ for j∈[ℓ] and sends them to the receiver. H is a tweakable hash function as used in garbled circuit. The sender outputs (H(j,qj​), H(j,qj​)⊕Δ) For j∈[ℓ], the receiver outputs the following: If xj​=0, outputs H(j,tj​); If xj​=1, outputs H(j,tj​)⊕vj​ This ends the description of the KOS15 protocol. We note that a very recent work pointed out that there is a flaw in the security proof of the main lemma of KOS15 and provided a fix, which is less efficient. They also pointed out that this the original KOS15 protocol is still secure in practice. Therefore, we choose the original KOS15 protocol in zkOracles in this version. We begin to describe the πRAND​(F2λℓ​) protocol as follows. The sender and receiver locally choose uniformly 128-bit random seeds s0​ and s1​, respectively. The sender and receiver compute commitments of the seeds by choosing 128-bit random strings r0​,r1​ and generate c0​=SHA256(s0​∥r0​) and c1​=SHA256(s1​∥r1​), respectively. The sender sends c0​ to the receiver, and the receiver sends c1​ to the sender. On receiving the commitments, the sender and receiver open the commitments by sending (s0​,r0​) and (s1​,r1​) to each other, respectively. The sender checks c1​ by re-computing it with s1​,r1​, and the receiver checks c0​ by re-computing it with s0​,r0​. If checks fail, abort. The sender and receiver locally compute s=s0​⊕s1​ and generate random elements with a PRG as χ1​,...,χℓ​←PRG(s)","breadcrumbs":"Two Party Computation » Oblivious Transfer » OT Extension » OT Extension","id":"68","title":"OT Extension"},"69":{"body":"With garbled circuit and oblivious transfer, we are ready to describe the well-known Yao's method to construct secure two-party computation protocols for any polynomial-size function. Given any function F(x,y), where x is the private input of Alice, and y is the private input of Bob. Let x=x1​∥x2​∥⋯∥xℓ1​​ and y=y1​∥y2​∥⋯∥yℓ2​​, where xi​ and yi​ are bits. Let Alice be the garbler and Bob be the evaluator, Yao's protocol is described as follows. Alice and Bob run a COT protocol to generate ℓ2​ OTs, where Alice is the sender and Bob is the receiver. Alice obtains (Y1​,Y1​⊕Δ),...,(Yℓ2​​,Yℓ2​​⊕Δ), and Bob obtains (Y1​⊕y1​⋅Δ,...,Yℓ2​​⊕yℓ2​​⋅Δ) Alice chooses uniformly random labels X1​,X2​,...,Xℓ1​​. Alice uses (X1​,...,Xℓ1​​,Y1​,...,Yℓ2​​) and global Δ to generate the garbled circuit of F and sends GC(F) to Bob. Alice also sends the decoding information to Bob. Alice encodes her inputs and sends X1​⊕x1​⋅Δ,...,Xℓ1​​⊕xℓ1​​⋅Δ to Bob. With the encoded X labels and Y labels, Bob evaluates GC(F) to get the output labels, and then decodes these output labels with decoding information and gets the output bits.","breadcrumbs":"Two Party Computation » Full Protocol » Full Protocol","id":"69","title":"Full Protocol"},"7":{"body":"When cryptographers talk about \"groups\", they usually mean a \"computational group\", which is a group equipped with efficient algorithms for computing the group operation and inversion. This is different because a group in the mathematical sense need not have its operations be computable at all.","breadcrumbs":"Groups » More things to know","id":"7","title":"More things to know"},"70":{"body":"Many modern proof systems (and I think all that are in use) are constructed according to the following recipe. You start out with a class of computations. You devise a way to arithmetize those computations. That is, to express your computation as a statement about polynomials. More specifically, you describe what is often called an \"algebraic interactive oracle proof\" (AIOP) that encodes your computation. An AIOP is a protocol describing an interaction between a prover and a verifier, in which the prover sends the verifier some \"polynomial oracles\" (basically a black box function that given a point evaluates a polynomial at that point), the verifier sends the prover random challenges, and at the end, the verifier queries the prover's polynomials at points of its choosing and makes a decision as to whether it has been satisfied by the proof. An AIOP is an imagined interaction between parties. It is an abstract description of the protocol that will be \"compiled\" into a SNARK. There are several \"non-realistic\" aspects about it. One is that the prover sends the verifier black-box polynomials that the verifier can evaluate. These polynomials have degree comparable to the size of the computation being verified. If we implemented these \"polynomial oracles\" by having the prover really send the O(n) size polynomials (say by sending all their coefficients), then we would not have a zk-SNARK at all, since the verifier would have to read this linearly sized polynomial so we would lose succinctness, and the polynomials would not be black-box functions, so we may lose zero-knowledge. Instead, when we concretely instantiate the AIOP, we have the prover send constant-sized, hiding polynomial commitments . Then, in the phase of the AIOP where the verifier queries the polynomials, the prover sends an opening proof for the polynomial commitments which the verifier can check, thus simulating the activity of evaluating the prover's polynomials on your own. So this is the next step of making a SNARK: instantiating the AIOP with a polynomial commitment scheme of one's choosing. There are several choices here and these affect the properties of the SNARK you are constructing, as the SNARK will inherit efficiency and setup properties of the polynomial commitment scheme used. An AIOP describes an interactive protocol between the verifier and the prover. In reality, typically, we also want our proofs to be non-interactive. This is accomplished by what is called the Fiat--Shamir transformation . The basic idea is this: all that the verifier is doing is sampling random values to send to the prover. Instead, to generate a \"random\" value, the prover simulates the verifier by hashing its messages. The resulting hash is used as the \"random\" challenge. At this point we have a fully non-interactive proof. Let's review our steps. Start with a computation. Translate the computation into a statement about polynomials and design a corresponding AIOP. Compile the AIOP into an interactive protocol by having the prover send hiding polynomial commitments instead of polynomial oracles. Get rid of the verifier-interaction by replacing it with a hash function. I.e., apply the Fiat--Shamir transform.","breadcrumbs":"Proof Systems » Proof Systems Design Overview","id":"70","title":"Proof Systems Design Overview"},"71":{"body":"Finally we can discuss zk-SNARKs. In these notes right now I will not start with intuitive motivations for SNARKs because I can't think of them right now, but maybe later. Let f:A×W→Bool be a program (in some yet to be specified programming language). For a:A, zk-SNARKs let you produce proofs for statements of the form I know w:W such that f(a,w)=true in such a way that no information about w is revealed by the proof (this is the zero-knowledge or ZK part of them). In other words, zk-SNARKs allow you to prove execution of programs, where you are also allowed to hide a selected portion (i.e., the W portion) of the data involved in the computation. We could also phrase zk-SNARKs in terms of computing functions f:A×W→B, which seems more general, but actually this would be taken care of by the previous sense where we can only use boolean-valued functions, by proving against the function g:(A×B)×W→Bool defined by g((a,b),w):=b=?​f(a,w). SNARK stands for Succinct Non-interactive ARguments of Knowledge . If it also satisfies zero-knowledge, then it is called a zk-SNARK. Preprocessing SNARKs allow the verifier to precompute some encodings of the relation to run independently of the instance length. Definitions of these schemes owe the succinctness property to the fact that both the verifier and the proof length are (at most) polylogarithmic in the size of the computation (the witness length). Recent schemes such as Plonk, go beyond that definition and provide zkSNARKs with constant proof length. Unlike the verifier in preprocessing SNARKs, the prover can't be faster than linear, as it has to read the circuit at least.","breadcrumbs":"Proof Systems » zk-SNARKs » zk-SNARKs","id":"71","title":"zk-SNARKs"},"72":{"body":"We will define a notion of computation (a programming language basically) for which it is easy to construct zk-SNARKs.","breadcrumbs":"Proof Systems » zk-SNARKs » Expressing computations with \"iterating over trace tables\"","id":"72","title":"Expressing computations with \"iterating over trace tables\""},"73":{"body":"The proof system in Mina is a variant of PlonK . To understand PlonK, you can refer to our series of videos on the scheme. In this section we explain our variant, called kimchi . kimchi is not formally a zk-SNARK, as it is not succinct in the proof size. zk-SNARKs must have a log(n) proof size where n is the number of gates in the circuit. In kimchi, due to the Bootleproof polynomial commitment approach, our proofs are log(d) where d is the maximum degree of the committed polynomials (in the order of the number of rows). In practice, our proofs are in the order of dozens of kilobytes. Note that kimchi is a zk-SNARK in terms of verification complexity (log(n)) and the proving complexity is quasilinear (O(nlogn)) – recall that the prover cannot be faster than linear. Essentially what PlonK allows you to do is to, given a program with inputs and outputs, take a snapshot of its execution. Then, you can remove parts of the inputs, or outputs, or execution trace, and still prove to someone that the execution was performed correctly for the remaining inputs and outputs of the snapshot. There are a lot of steps involved in the transformation of \"I know an input to this program such that when executed with these other inputs it gives this output\" to \"here's a sequence of operations you can execute to convince yourself of that\". At the end of this chapter, you will understand that the protocol boils down to filling a number of tables (illustrated in the diagram below): tables that specify the circuit, and tables that describes an execution trace of the circuit (given some secret and public inputs). kimchi","breadcrumbs":"Overview » Overview","id":"73","title":"Overview"},"74":{"body":"Size: number of rows Columns: number of variables per rows Cell: a pair (row, column) Witness: the values assigned in all the cells Gate: polynomials that act on the variables in a row Selector vector: a vector of values 1 or 0 (not true for constant vector I think) that toggles gates and variables in a row Gadget: a series of contiguous rows with some specific gates set (via selector vectors)","breadcrumbs":"Overview » Glossary » Glossary","id":"74","title":"Glossary"},"75":{"body":"PlonK needs a domain to encode the circuit on (for each point we can encode a row/constraint/gate). We choose a domain H such that it is a multiplicative subgroup of the scalar field of our curve.","breadcrumbs":"Domain » Domain","id":"75","title":"Domain"},"76":{"body":"Furthermore, as FFT (used for interpolation, see the section on FFTs ) works best in domains of size 2k for some k, we choose H to be a subgroup of order 2k. Since the pasta curves both have scalar fields of prime orders p and q, their multiplicative subgroup is of order p−1 and q−1 respectively (without the zero element). The pasta curves were generated specifically so that both p−1 and q−1 are multiples of 232. We say that they have 2-adicity of 32. Looking at the definition of one of the pasta fields in Rust you can see that it is defined specifically for a trait related to FFTs: impl FftParameters for FqParameters { type BigInt = BigInteger; const TWO_ADICITY: u32 = 32; #[rustfmt::skip] const TWO_ADIC_ROOT_OF_UNITY: BigInteger = BigInteger([ 0x218077428c9942de, 0xcc49578921b60494, 0xac2e5d27b2efbee2, 0xb79fa897f2db056 ]); The 2-adicity of 32 means that there's a multiplicative subgroup of size 232 that exists in the field. The code above also defines a generator g for it, such that g232=1 and gi=1 for i∈[[1,232−1]] (so it's a primitive 232-th root of unity). Lagrange's theorem tells us that if we have a group of order n, then we'll have subgroups with orders dividing n. So in our case, we have subgroups with all the powers of 2, up to the 32-th power of 2. To find any of these groups, it is pretty straight forward as well. Notice that: let h=g2, then h231=g232=1 and so h generates a subgroup of order 31 let t=g22, then t230=g232=1 and so t generates a subgroup of order 30 and so on... In arkworks you can see how this is implemented: let size = n.next_power_of_two() as u64;\nlet log_size_of_group = ark_std::log2(usize::try_from(size).expect(\"too large\"));\nomega = Self::TWO_ADIC_ROOT_OF_UNITY;\nfor _ in log_size_of_group..Self::TWO_ADICITY { omega.square_in_place();\n} this allows you to easily find subgroups of different sizes of powers of 2, which is useful in zero-knowledge proof systems as FFT optimizations apply well on domains that are powers of 2. You can read more about that in the mina book .","breadcrumbs":"Domain » 2-adicity","id":"76","title":"2-adicity"},"77":{"body":"For each curve, we generate a domain H as the set H=1,ω,ω2,⋯,ωn−1. As we work in a multiplicative subgroup, the polynomial that vanishes in all of these points can be written and efficiently calculated as ZH​(x)=x∣H∣−1. This is because, by definition, ω∣H∣=1. If x∈H, then x=ωi for some i, and we have: x∣H∣=(ωi)∣H∣=(ω∣H∣)i=1i=1 This optimization is important, as without it calculating the vanishing polynomial would take a number of operations linear in ∣H∣ (which is not doable in a verifier circuit, for recursion). With the optimization, it takes us a logarithmic number of operation (using exponentiation by squaring ) to calculate the vanishing polynomial.","breadcrumbs":"Domain » Efficient computation of the vanishing polynomial","id":"77","title":"Efficient computation of the vanishing polynomial"},"78":{"body":"What's a lagrange base? Li​(x)=1 if x=gi, 0 otherwise.","breadcrumbs":"Lagrange Basis in Multiplicative Subgroups » Lagrange basis in multiplicative subgroups","id":"78","title":"Lagrange basis in multiplicative subgroups"},"79":{"body":"Arkworks has the formula to construct a lagrange base: Evaluate all Lagrange polynomials at τ to get the lagrange coefficients. Define the following as H: The coset we are in, with generator g and offset h m: The size of the coset H ZH​: The vanishing polynomial for H. ZH​(x)=∏i∈[m]​(x−h⋅gi)=xm−hm vi​: A sequence of values, where v0​=m∗hm−11​, and vi+1​=g⋅vi​ We then compute Li,H​(τ) as Li,H​(τ)=ZH​(τ)⋅τ−hgivi​​ However, if τ in H, both the numerator and denominator equal 0 when i corresponds to the value tau equals, and the coefficient is 0 everywhere else. We handle this case separately, and we can easily detect by checking if the vanishing poly is 0. following this, for h=1 we have: L0​(x)=m(x−1)ZH​(x)​ L1​(x)=m(x−g)ZH​(x)g​ L2​(x)=m(x−g2)ZH​(x)g2​ and so on","breadcrumbs":"Lagrange Basis in Multiplicative Subgroups » What's the formula?","id":"79","title":"What's the formula?"},"8":{"body":"Implement a type JacobianPoint and functions for computing the group operation Familiarize yourself with the types and traits in ark_ec. Specifically, todo Implement fn decompress(c: (F, bool)) -> (F, F)","breadcrumbs":"Groups » Exercises","id":"8","title":"Exercises"},"80":{"body":"https://en.wikipedia.org/wiki/Lagrange_polynomial#Barycentric_form","breadcrumbs":"Lagrange Basis in Multiplicative Subgroups » What's the logic here?","id":"80","title":"What's the logic here?"},"81":{"body":"So far we've talked about an interactive protocol between a prover and a verifier. The zero-knowledge proof was also in the honest verifier zero-knowedlge (HVZK) model, which is problematic. In practice, we want to remove the interaction and have the prover produce a proof by themselves, that anyone can verify.","breadcrumbs":"Non-Interactivity via Fiat-Shamir » Non-Interactivity via Fiat-Shamir","id":"81","title":"Non-Interactivity via Fiat-Shamir"},"82":{"body":"public-coin protocols are protocols were the messages of the verifier are simply random messages. This is important as our technique to transform an interactive protocol to a non-interactive protocol works on public-coin protocols.","breadcrumbs":"Non-Interactivity via Fiat-Shamir » Public-coin protocols","id":"82","title":"Public-coin protocols"},"83":{"body":"The whole idea is to replace the verifier by a random oracle, which in practice is a hash function. Note that by doing this, we remove potential leaks that can happen when the verifier acts dishonestly. Initially the Fiat-Shamir transformation was only applied to sigma protocols, named after the greek letter Σ due to its shape resembling the direction of messages (prover sends a commit to a verifier, verifier sends a challenge to a prover, prover sends the final proof to a verifier). A Z would have made more sense but here we are.","breadcrumbs":"Non-Interactivity via Fiat-Shamir » Fiat-Shamir trick","id":"83","title":"Fiat-Shamir trick"},"84":{"body":"As our protocol has more than three moves, where every even move is a challenge from the verifier, we need to generalize Fiat-Shamir. This is simple: every verifier move can be replaced by a hash of the transcript (every message sent and received so far) to obtain a challenge.","breadcrumbs":"Non-Interactivity via Fiat-Shamir » Generalization of Fiat-Shamir","id":"84","title":"Generalization of Fiat-Shamir"},"85":{"body":"While we use a hash function for that, a different construction called the duplex construction is particularly useful in such situations as they allow to continuously absorb the transcript and produce challenges, while automatically authenticating the fact that they produced a challenge. Merlin is a standardization of such a construction using the Strobe protocol framework (a framework to make use of a duplex construction). Note that the more recent Xoodyak (part of NIST's lightweight competition) could have been used for this as well. Note also that Mina uses none of these standards, instead it simply uses Poseidon (see section on poseidon).","breadcrumbs":"Non-Interactivity via Fiat-Shamir » In practice: a duplex construction as Merlin","id":"85","title":"In practice: a duplex construction as Merlin"},"86":{"body":"Plookup allows us to check if witness values belong to a look up table. This is usually useful for reducing the number of constraints needed for bit-wise operations. So in the rest of this document we'll use the XOR table as an example. 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For the Rust documentation, see here . Note that this book is a work in progress, not necessarily reflecting the current state of Mina. Authors: Izaak Meckler, Vanishree Rao, Matthew Ryan, Anaïs Querol, Joseph Spadavecchia, David Wong, Xiang Xie In memory of Vitaly Zelov.","breadcrumbs":"Introduction » Introduction","id":"0","title":"Introduction"},"1":{"body":"This document is intended primarily to communicate mathematical ideas to programmers with some experience of math. To that end, we will often be ambivalent about the difference between sets and types (in whatever programming language, but usually I am imagining some idealized form of Rust or OCaml). So for example we may write A is a type A is a set A is a type and these should all be interpreted the same way, assuming it makes sense in the context. 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So if we say \"let f:A→B\", we typically mean that A and B are types in some programming language which makes sense in the context (or maybe that they are sets, again depending on context) f is actually a program computing a function of the type A→B (again, in whatever sense of program makes sense in context)","breadcrumbs":"Terminology » Terminology","id":"1","title":"Terminology"},"10":{"body":"The purpose of this document is to give the reader mathematical, cryptographic, and programming context sufficient to become an effective practitioner of zero-knowledge proofs and ZK-SNARKs specifically.","breadcrumbs":"Fields » Intro","id":"10","title":"Intro"},"100":{"body":"Unlike the latest version of vanilla PlonK that implements the final check using a polynomial opening (via Maller's optimization), we implement it manually. (That is to say, Izaak implemented Maller's optimization for 5-wires.) But the check is not exactly f(ζ)=ZH​(ζ)t(ζ). This note describes how and why the implementation deviates a little.","breadcrumbs":"Final Check » Understanding the implementation of the f(ζ)=ZH​(ζ)t(ζ) check","id":"100","title":"Understanding the implementation of the f(ζ)=ZH​(ζ)t(ζ) check"},"101":{"body":"If you look at how the evaluation of f(z) is computed on the prover side (or the commitment of f is computed on the verifier side), you can see that f is missing two things: the public input part some terms in the permutation What is it missing in the permutation part? Let's look more closely. This is what we have: −1⋅⋅⋅⋅⋅⋅⋅⋅​z(ζω)⋅zkpm(ζ)⋅αPERM0(w[0](ζ)+β⋅σ[0](ζ)+γ)(w[1](ζ)+β⋅σ[1](ζ)+γ)(w[2](ζ)+β⋅σ[2](ζ)+γ)(w[3](ζ)+β⋅σ[3](ζ)+γ)(w[4](ζ)+β⋅σ[4](ζ)+γ)(w[5](ζ)+β⋅σ[5](ζ)+γ)β⋅σ[6](x)​​ In comparison, here is the list of the stuff we needed to have: the two sides of the coin: z(ζ)⋅⋅⋅⋅⋅⋅⋅⋅​zkpm(ζ)⋅αPERM0(w[0](ζ)+β⋅shift[0](ζ)+γ)(w[1](ζ)+β⋅shift[1](ζ)+γ)(w[2](ζ)+β⋅shift[2](ζ)+γ)(w[3](ζ)+β⋅shift[3](ζ)+γ)(w[4](ζ)+β⋅shift[4](ζ)+γ)(w[5](ζ)+β⋅shift[5](ζ)+γ)(w[6](ζ)+β⋅shift[6](ζ)+γ)​​ with shift[0]=1 and −1⋅⋅⋅⋅⋅⋅⋅⋅​z(ζω)⋅zkpm(ζ)⋅αPERM0(w[0](ζ)+β⋅σ[0](ζ)+γ)(w[1](ζ)+β⋅σ[1](ζ)+γ)(w[2](ζ)+β⋅σ[2](ζ)+γ)(w[3](ζ)+β⋅σ[3](ζ)+γ)(w[4](ζ)+β⋅σ[4](ζ)+γ)(w[5](ζ)+β⋅σ[5](ζ)+γ)(w[6](ζ)+β⋅σ[6](ζ)+γ)​​ the initialization: (z(ζ)−1)⋅L1​(ζ)⋅αPERM1 and the end of the accumulator: (z(ζ)−1)⋅Ln−k​(ζ)⋅αPERM2 You can read more about why it looks like that in this post . We can see clearly that the permutation stuff we have in f is clearly lacking. It's just the subtraction part (equation 2), and even that is missing some terms. It is missing exactly this: −1 ⋅⋅⋅⋅⋅⋅⋅⋅​z(ζω)⋅zkpm(ζ)⋅αPERM0(w[0](ζ)+β⋅σ[0](ζ)+γ)(w[1](ζ)+β⋅σ[1](ζ)+γ)(w[2](ζ)+β⋅σ[2](ζ)+γ)(w[3](ζ)+β⋅σ[3](ζ)+γ)(w[4](ζ)+β⋅σ[4](ζ)+γ)(w[5](ζ)+β⋅σ[5](ζ)+γ)(w[6](ζ)+γ)​​ So at the end, when we have to check for the identity f(ζ)=ZH​(ζ)t(ζ) we'll actually have to check something like this (I colored the missing parts on the left hand side of the equation): f(ζ)​+pub(ζ)+z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+βζ+γ)⋅(w[1](ζ)+β⋅shift[0](ζ)+γ)⋅(w[2](ζ)+β⋅shift[1](ζ)+γ)⋅(w[3](ζ)+β⋅shift[2](ζ)+γ)⋅(w[4](ζ)+β⋅shift[3](ζ)+γ)⋅(w[5](ζ)+β⋅shift[4](ζ)+γ)⋅(w[6](ζ)+β⋅shift[5](ζ)+γ)−z(ζω)⋅zkpm(ζ)⋅αPERM0⋅⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅⋅(w[6](ζ)+γ)+(z(ζ)−1)⋅L1​(ζ)⋅αPERM1+(z(ζ)−1)⋅Ln−k​(ζ)⋅αPERM2=t(ζ)(ζn−1)​​ This is not exactly what happens in the code. But if we move things around a bit, we end up with what's implemented. I highlight what changes in each steps. First, we just move things around: f(ζ)​+pub(ζ)+z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅shift[0]ζ+γ)⋅(w[1](ζ)+β⋅shift[1]ζ+γ)⋅(w[2](ζ)+β⋅shift[2]ζ+γ)⋅(w[3](ζ)+β⋅shift[3]ζ+γ)⋅(w[4](ζ)+β⋅shift[4]ζ+γ)⋅(w[5](ζ)+β⋅shift[5]ζ+γ)⋅(w[6](ζ)+β⋅shift[6]ζ+γ)−z(ζω)⋅zkpm(ζ)⋅αPERM0⋅⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅(w[6](ζ)+γ)−t(ζ)(ζn−1)=(1−z(ζ))L1​(ζ)αPERM1+(1−z(ζ))Ln−k​(ζ)αPERM2​​ here are the actual lagrange basis calculated with the formula here , oh and we actually use L0​ in the code, not L1​, so let's change that as well: f(ζ)++−−=​pub(ζ)z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅shift[0]ζ+γ)⋅(w[1](ζ)+β⋅shift[1]ζ+γ)⋅(w[2](ζ)+β⋅shift[2]ζ+γ)⋅(w[3](ζ)+β⋅shift[3]ζ+γ)⋅(w[4](ζ)+β⋅shift[4]ζ+γ)⋅(w[5](ζ)+β⋅shift[5]ζ+γ)⋅(w[6](ζ)+β⋅shift[6]ζ+γ)+z(ζω)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅(w[6](ζ)+γ)+t(ζ)(ζn−1)(1−z(ζ))[n(ζ−1)(ζn−1)​αPERM1+n(ζ−ωn−k)ωn−k(ζn−1)​αPERM2]​​ finally we extract some terms from the lagrange basis: ​[f(ζ)+pub(ζ)+z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅shift[0]ζ+γ)⋅(w[1](ζ)+β⋅shift[1]ζ+γ)⋅(w[2](ζ)+β⋅shift[2]ζ+γ)⋅(w[3](ζ)+β⋅shift[3]ζ+γ)⋅(w[4](ζ)+β⋅shift[4]ζ+γ)⋅(w[5](ζ)+β⋅shift[5]ζ+γ)⋅(w[6](ζ)+β⋅shift[6]ζ+γ)+−z(ζω)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅(w[6](ζ)+γ)−t(ζ)(ζn−1)]⋅(ζ−1)(ζ−ωn−k)=(1−z(ζ))[n(ζn−1)(ζ−ωn−k)​αPERM1+nωn−k(ζn−1)(ζ−1)​αPERM2]​​​ with αPERM0=α17,αPERM1=α18,αPERM2=α19 Why do we do things this way? Most likely to reduce Also, about the code: the division by n in the αPERM1 and the αPERM2 terms is missing (why?) the multiplication by wn−k in the αPERM2 term is missing (why?) As these terms are constants, and do not prevent the division by ZH​ to form t, t also omits them. In other word, they cancel one another.","breadcrumbs":"Final Check » What's f and what's missing in the final equation?","id":"101","title":"What's f and what's missing in the final equation?"},"102":{"body":"In verifier.rs you can see the following code (cleaned to remove anything not important) // compute the witness polynomials $w_0, \\cdots, w_14$ and the permutation polynomial $z$ in evaluation forms on different domains\nlet lagrange = index.cs.evaluate(&w, &z); // compute parts of the permutation stuff that is included in t\nlet (perm, bnd) = index.cs.perm_quot(&lagrange, &oracles, &z, &alpha[range::PERM])?; // divide contributions with vanishing polynomial\nlet (mut t, res) = (perm + some_other_stuff).divide_by_vanishing_poly() // add the other permutation stuff\nt += &bnd; // t is evaluated at zeta and sent... Notice that bnd is not divided by the vanishing polynomial . Also what's perm? Let's answer the latter TESTREMOVEME: perm(x)= −​aPERM0⋅zkpl(x)⋅[z(x)⋅(w[0](x)+γ+x⋅β⋅shift[0])⋅(w[1](x)+γ+x⋅β⋅shift[1])⋅…z(x⋅w)⋅(w[0](x)+γ+σ[0]⋅β) ⋅(w[1](x)+γ+σ[1]⋅β)⋅…]​​ and bnd is: bnd(x)=aPERM1⋅x−1z(x)−1​+aPERM2⋅x−sid[n−k]z(x)−1​ you can see that some terms are missing in bnd, specifically the numerator xn−1. Well, it would have been cancelled by the division by the vanishing polynomial, and this is the reason why we didn't divide that term by ZH​(x)=xn−1. Also, note that the same constant terms that were missing in f are missing in t.","breadcrumbs":"Final Check » What about t?","id":"102","title":"What about t?"},"103":{"body":"This document proposes a protocol change for kimchi .","breadcrumbs":"Maller's Optimization for Kimchi » Maller's optimization for Kimchi","id":"103","title":"Maller's optimization for Kimchi"},"104":{"body":"See the section on Maller's optimization for background.","breadcrumbs":"Maller's Optimization for Kimchi » What is Maller's optimization?","id":"104","title":"What is Maller's optimization?"},"105":{"body":"We want the verifier to form commitment to the following polynomial: L(x)=f(x)−ZH​(ζ)⋅t(x) They could do this like so: com(L)=com(f)−ZH​(ζ)⋅com(t) Since they already know f, they can produce com(f), the only thing they need is com(t). So the protocol looks like that: maller 15 1 In the rest of this document, we review the details and considerations needed to implement this change in kimchi .","breadcrumbs":"Maller's Optimization for Kimchi » Overview","id":"105","title":"Overview"},"106":{"body":"There's one challenge that prevents us from directly using this approach: com(t) is typically sent and received in several commitments (called chunks or segments throughout the codebase). As t is the largest polynomial, usually exceeding the size of the SRS (which is by default set to be the size of the domain).","breadcrumbs":"Maller's Optimization for Kimchi » How to deal with a chunked t?","id":"106","title":"How to deal with a chunked t?"},"107":{"body":"Thus, the verifier will have to produce split commitments of L and combine them with powers of ζn to verify an evaluation proof. Let's define L as: L=L0​+xnL1​+x2nL1​+⋯ where every Li​ is of degree n−1 at most. Then we have that com(L)=com(L0​)+com(xn⋅L1​)+com(x2n⋅L2​)+⋯ Which the verifier can't produce because of the powers of xn, but we can linearize it as we already know which x we're going to evaluate that polynomial with: com(L~)=com(L0​)+ζn⋅com(L1​)+ζ2n⋅com(L2​)+⋯","breadcrumbs":"Maller's Optimization for Kimchi » The verifier side","id":"107","title":"The verifier side"},"108":{"body":"This means that the prover will produce evaluation proofs on the following linearized polynomial: L~(x)=1⋅L0​(x)+ζn⋅L1​(x)+ζ2n⋅L2​(x)+⋯ which is the same as L(x) only if evaluated at ζ. As the previous section pointed out, we will need L (ζω) and L (ζω)=L(ζω).","breadcrumbs":"Maller's Optimization for Kimchi » The prover side","id":"108","title":"The prover side"},"109":{"body":"To create an evaluation proof, the prover also needs to produce the blinding factor rL​ associated with the chunked commitment of: L =f ​−(ζn−1)t~ To compute it, there are two rules to follow: when two commitment are added together, their associated blinding factors get added as well: com(a)+com(b)⟹ra​+rb​ when scaling a commitment, its blinding factor gets scalled too: n⋅com(a)⟹n⋅ra​ As such, if we know rf​ and rt​, we can compute: rL ​=rf ​​+(ζn−1)⋅rt~​ The prover knows the blinding factor of the commitment to t, as they committed to it. But what about f? They never commit to it really, and the verifier recreates it from scratch using: The commitments we sent them . In the linearization process, the verifier actually gets rid of most prover commitments, except for the commitment to the last sigma commitment Sσ6​. (TODO: link to the relevant part in the spec) As this commitment is public, it is not blinded. The public commitments . Think commitment to selector polynomials or the public input polynomial. These commitments are not blinded, and thus do not impact the calculation of the blinding factor. The evaluations we sent them . Instead of using commitments to the wires when recreating f, the verifier uses the (verified) evaluations of these in ζ. If we scale our commitment com(z) with any of these scalars, we will have to do the same with rz​. Thus, the blinding factor of L is (ζn−1)⋅rt ​.","breadcrumbs":"Maller's Optimization for Kimchi » Evaluation proof and blinding factors","id":"109","title":"Evaluation proof and blinding factors"},"11":{"body":"In this section we'll discuss the fundamental objects used in the construction of most ZK-SNARKs in use today. These objects are used extensively and without ceremony, so it's important to understand them very well. If you find you don't understand something at first, don't worry: practice is often the best teacher and in using these objects you will find they will become like the bicycle you've had for years: like an extension of yourself that you use without a thought.","breadcrumbs":"Fields » Some fundamental mathematical objects","id":"11","title":"Some fundamental mathematical objects"},"110":{"body":"The prover actually does not send a commitment to the full f polynomial. As described in the last check section . The verifier will have to compute the evaluation of L~(ζ) because it won't be zero. It should be equal to the following: ​f ​(ζ)−t (ζ)(ζn−1)=(ζ−1)(ζ−ωn−k)1−z(ζ)​[n(ζn−1)(ζ−ωn−k)​αPERM1+nωn−k(ζn−1)(ζ−1)​αPERM2]−pub(ζ)−z(ζ)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅shift[0]ζ+γ)⋅(w[1](ζ)+β⋅shift[1]ζ+γ)⋅(w[2](ζ)+β⋅shift[2]ζ+γ)⋅(w[3](ζ)+β⋅shift[3]ζ+γ)⋅(w[4](ζ)+β⋅shift[4]ζ+γ)⋅(w[5](ζ)+β⋅shift[5]ζ+γ)⋅(w[6](ζ)+β⋅shift[6]ζ+γ)+z(ζω)⋅zkpm(ζ)⋅αPERM0⋅(w[0](ζ)+β⋅σ[0](ζ)+γ)⋅(w[1](ζ)+β⋅σ[1](ζ)+γ)⋅(w[2](ζ)+β⋅σ[2](ζ)+γ)⋅(w[3](ζ)+β⋅σ[3](ζ)+γ)⋅(w[4](ζ)+β⋅σ[4](ζ)+γ)⋅(w[5](ζ)+β⋅σ[5](ζ)+γ)⋅(w[6](ζ)+γ)+​​ Because we use the inner product polynomial commitment , we also need: L (ζω)=f ​(ζω)−ZH​(ζ)⋅t~(ζω) Notice the ZH​(ζ). That evaluation must be sent as part of the proof as well.","breadcrumbs":"Maller's Optimization for Kimchi » The evaluation of L~","id":"110","title":"The evaluation of L~"},"111":{"body":"Now here's how we need to modify the current protocol: The evaluations f(ζ),f(ζω),t(ζ),t(ζω) (and their corresponding evaluation proofs) don't have to be part of the protocol anymore. The prover must still send the chunked commitments to t. The prover must create a linearized polynomial L by creating a linearized polynomial f ​ and a linearized polynomial t and computing: L =f ​+(ζn−1)⋅t While the verifier can compute the evaluation of L (ζ) by themselves, they don't know the evaluation of L (ζω), so the prover needs to send that. The verifier must recreate com(L ), the commitment to L , themselves so that they can verify the evaluation proofs of both L (ζ) and L (ζω). The evaluation of L~(ζω) must be absorbed in both sponges (Fq and Fr). maller 15 2 The proposal is implemented in #150 with the following details: the L~ polynomial is called ft. the evaluation of L~(ζ) is called ft_eval0. the evaluation L~(ζω) is called ft_eval1.","breadcrumbs":"Maller's Optimization for Kimchi » The actual protocol changes","id":"111","title":"The actual protocol changes"},"112":{"body":"In 2020, Plookup showed how to create lookup proofs. Proofs that some witness values are part of a lookup table . Two years later, an independent team published plonkup showing how to integrate Plookup into PlonK. This document specifies how we integrate Plookup in kimchi. It assumes that the reader understands the basics behind Plookup.","breadcrumbs":"Lookup Tables » Plookup in Kimchi","id":"112","title":"Plookup in Kimchi"},"113":{"body":"We integrate Plookup in kimchi with the following differences: we snake-ify the sorted table instead of wrapping it around (see later) we allow fixed-ahead-of-time linear combinations of columns of the queries we make we implemented different tables, like RangeCheck and XOR. we allow several lookups (or queries) to be performed within the same row zero-knowledgeness is added in a specific way (see later) The following document explains the protocol in more detail","breadcrumbs":"Lookup Tables » Overview","id":"113","title":"Overview"},"114":{"body":"As per the Plookup paper, the prover will have to compute three vectors: f, the (secret) query vector , containing the witness values that the prover wants to prove are part of the lookup table. t, the (public) lookup table . s, the (secret) concatenation of f and t, sorted by t (where elements are listed in the order they are listed in t). Essentially, Plookup proves that all the elements in f are indeed in the lookup table t if and only if the following multisets are equal: {(1+β)f,diff(t)} diff(sorted(f,t)) where diff is a new set derived by applying a \"randomized difference\" between every successive pairs of a vector, and (f,t) is the set union of f et t. More precisely, for a set S={s0​,s1​,⋯,sn​}, diff(S) is defined as the set {s0​+βs1​,s1​+βs2​,⋯,sn−1​+βsn​}. For example, with: f={5,4,1,5} t={1,4,5} we have: sorted(f,t)={1,1,4,4,5,5,5} {(1+β)f,diff(t)}={(1+β)5,(1+β)4,(1+β)1,(1+β)5,1+β4,4+β5} diff(sorted(f,t))={1+β1,1+β4,4+β4,4+β5,5+β5,5+β5} Note: This assumes that the lookup table is a single column. You will see in the next section how to address lookup tables with more than one column. The equality between the multisets can be proved with the permutation argument of PlonK, which would look like enforcing constraints on the following accumulator: init: acc0​=1 final: accn​=1 for every 0 RAM loading, disk -> RAM loading, and RAM r/w. This gives the maximum flexibility for circuit designers, and each primitive has uses for projects that we are pursuing. This design has been refined over several iterations to reduce the number of components exposed in the proof and the amount of computation that the verifier requires. The largest remaining parts represent the data itself and the method for combining them, where simplifying assumptions have been made to minimize their impact while still satisfying the design constraints.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » Why is this design the best in the space of possible designs?","id":"138","title":"Why is this design the best in the space of possible designs?"},"139":{"body":"Other designs building upon the lookup primitive reduce on some subset of this design, or on relaxing a design constraint that would remove some useful / required functionality.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What other designs have been considered and what is the rationale for not choosing them?","id":"139","title":"What other designs have been considered and what is the rationale for not choosing them?"},"14":{"body":"For a finite field Fp​ where p fits in n bits (i.e., p<2n) we have Addition, subtraction: O(n) Multiplication O(n2) Division O(n2) I believe, in practice significantly slower than multiplication. Actually, on a practical level, it's more accurate to model the complexity in terms of the number of limbs rather than the number of bits where a limb is 64 bits. Asymptotically it makes no difference but concretely it's better to think about the number of limbs for the most part. As a result you can see it's the smaller n is the better, especially with respect to multiplication, which dominates performance considerations for implementations of zk-SNARKs, since they are dominated by elliptic curve operations that consist of field operations. While still in development, Mina used to use a field of 753 bits or 12 limbs and now uses a field of 255 bits or 4 limbs. As a result, field multiplication became automatically sped up by a factor of 122/42=9, so you can see it's very useful to try to shrink the field size.","breadcrumbs":"Fields » Algorithmics of prime order fields","id":"14","title":"Algorithmics of prime order fields"},"140":{"body":"We are currently unable to expose certain useful functionality to users of Snapps / SnarkyJS, and are missing the ability to use that same functionality for important parts of the zkOracles ecosystem (such as parsing / scraping). We also are currently forced to hard-code any constraints that we need for particular uses in the verifier circuit, and users are unable to remove the parts they do not need and to replace them with other useful constraints that their circuits might benefit from. The Mina transaction model is unable to support a bytecode interpreter without this or similar extensions.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What is the impact of not doing this?","id":"140","title":"What is the impact of not doing this?"},"141":{"body":"(Elided)","breadcrumbs":"Lookup Tables » Extended Lookup Tables » Prior art","id":"141","title":"Prior art"},"142":{"body":"","breadcrumbs":"Lookup Tables » Extended Lookup Tables » Unresolved questions","id":"142","title":"Unresolved questions"},"143":{"body":"The maximum number of tables / entries of each kind that we want to support in the 'standard' protocol. Validation (or rejection) of the simplifying assumptions. Confirmation of the design constraints, or simplifications thereto. Necessity / utility of the permutation argument.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What parts of the design do you expect to resolve through the RFC process before this gets merged?","id":"143","title":"What parts of the design do you expect to resolve through the RFC process before this gets merged?"},"144":{"body":"None, this RFC has been developed in part from an initial design and iteration on implementations of parts of that design.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What parts of the design do you expect to resolve through the implementation of this feature before merge?","id":"144","title":"What parts of the design do you expect to resolve through the implementation of this feature before merge?"},"145":{"body":"Bytecode design and implementation details. Mina transaction interface for interacting with these features. Snarky / SnarkyJS interface for interacting with these features.","breadcrumbs":"Lookup Tables » Extended Lookup Tables » What related issues do you consider out of scope for this RFC that could be addressed in the future independently of the solution that comes out of this RFC?","id":"145","title":"What related issues do you consider out of scope for this RFC that could be addressed in the future independently of the solution that comes out of this RFC?"},"146":{"body":"This section explains how to design and add a custom constraint to our proof-systems library. PLONK is an AIOP. That is, it is a protocol in which the prover sends polynomials as messages and the verifier sends random challenges, and then evaluates the prover's polynomials and performs some final checks on the outputs. PLONK is very flexible. It can be customized with constraints specific to computations of interest. For example, in Mina, we use a PLONK configuration called kimchi that has custom constraints for poseidon hashing, doing elliptic curve operations, and more. A \"PLONK configuration\" specifies The set of types of constraints that you would like to be able to enforce. We will describe below how these types of constraints are specified. A number of \"eq-able\" columns W A number of \"advice\" columns A Under such configuration, a circuit is specified by A number of rows n A vector cs of constraint-types of length n. I.e., a vector that specifies, for each row, which types of constraints should be enforced on that row. A vector eqs : Vec<(Position, Position)> of equalities to enforce, where struct Position { row: usize, column: usize }. E.g., if the pair (Position { row: 0, col: 8 }, Position { row: 10, col: 2 }) is in eqs, then the circuit is saying the entries in those two positions should be equal, or in other words that they refer to the same value. This is where the distinction between \"eq-able\" and \"advice\" columns comes in. The column field of a position in the eqs array can only refer to one of the first W columns. Equalities cannot be enforced on entries in the A columns after that. Then, given such a circuit, PLONK lets you produce proofs for the statement I know W + A \"column vectors\" of field elements vs: [Vec; W + A] such that for each row index i < n, the constraint of type cs[i] holds on the values [vs[0][i], ..., vs[W+A - 1][i], vs[0][i+1], ..., vs[W+A - 1][i+1] and all the equalities in eqs hold. I.e., for (p1, p2) in eqs we have vs[p1.col][p1.row] == vs[p2.col][p2.row]. So, a constraint can check the values in two adjacent rows. Mathematically speaking, a constraint is a multivariate polynomial over the variables cCurr,i​,…,vCurr,W+A−1​,vNext,0​,…,vNext,W+A−1​. In other words, there is one variable corresponding to the value of each column in the \"current row\" and one variable correspond to the value of each column in the \"next row\". In Rust, vr,i​ is written E::cell(Column::Witness(i), r). So, for example, the variable vNext,3​ is written E::cell(Column::Witness(3), CurrOrNext::Next). let w = |i| v(Column::Witness(i)); Let's","breadcrumbs":"Custom Constraints » Specifying a constraint","id":"146","title":"Specifying a constraint"},"147":{"body":"The art in proof systems comes from knowing how to design a PLONK configuration to ensure maximal efficiency for the sorts of computations you are trying to prove. That is, how to choose the numbers of columns W and A, and how to define the set of constraint types. Let's describe the trade-offs involved here. The majority of the proving time for the PLONK prover is in committing to the W + A column polynomials, which have length equal to the number of rows n committing to the \"permutation accumulator polynomial, which has length n. committing to the quotient polynomial, which reduces to computing max(k, W) MSMs of size n, where k is the max degree of a constraint. performing the commitment opening proof, which is mostly dependent on the number of rows n. So all in all, the proving time is approximately equal to the time to perform W + A + 1 + max(k - 1, W) MSMs of size n, plus the cost of an opening proof for polynomials of degree n - 1. and maybe computing the combined constraint polynomial, which has degree k * n where k is the maximum degree of a constraint Increasing W and A increase proof size, and they potentially impact the prover-time as the prover must compute polynomial commitments to each column, and computing a polynomial commitment corresponds to doing one MSM (multi-scalar multiplication, also called a multi-exponentiation.) However, often increasing the number of columns allows you to decrease the number of rows required for a given computation. For example, if you can perform one Poseidon hash in 36 rows with 5 total columns, then you can also perform it in 12 (= 36 / 3) rows with 15 (= 5 * 3) total columns. Decreasing the number of rows (even while keeping the total number of table entries the same) is desirable because it reduces the cost of the polynomial commitment opening proof, which is dominated by a factor linear in the number of rows, and barely depends on the number of columns. Increasing the number of columns also increases verifier time, as the verifier must perform one scalar-multiplication and one hash per column. Proof length is also affected by a larger number of columns, as more polynomials need to be committed and sent along to the verifier. There is typically some interplay between these","breadcrumbs":"Custom Constraints » Defining a PLONK configuration","id":"147","title":"Defining a PLONK configuration"},"148":{"body":"The goal of this section is to draw the connections between the practical aspect of designing custom gates and the theory behind it. First, suppose we had the following layout of gates. This means that our circuit has four gates: a (single) generic gate, a (non-optimized) complete add, and then two different Example gates that we just make up for the example; ExamplePairs and ExampleTriples. row GateType:: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 Generic l r o 1 CompleteAdd x1 y1 x2 y2 x3 y3 inf same_x s inf_z x21_inv 2 ExamplePairs a e i o u y 3 ExampleTriples a b c d e f g h i j k l m n o The most important aspect to take into account when you are designing a new custom gate is writing the actual description of the new functionality. For this, we are using the Expression framework that allows us to write equations generically by representing each column of the table (wire) as a variable. Then we can write operations between them and define the gate constraints . From a mathematical perspective, these will have the shape of multivariate polynomials of as many variables as columns in the table (currently 15 for Kimchi) that is constrained to equal 0. In our example, we would define equations that represent the operations carried out by each of the gates. In the following equations we use the variable names that we wrote in the cells above, and we don't imply that the same name in a different row correspond to the same value (we are not imposing equalities between gates yet): Generic: l ⋅cl​+ r ⋅cr​+ o ⋅co​+ l ⋅ r ⋅cm​+cc​=0 CompleteAdd: x21_inv ⋅( x2 − x1 )−(1− same_x)=0 same_x ⋅( x2 − x1 )=0 same_x ⋅(2⋅ s ⋅ y1 −3⋅ x12)+(1− same_x )⋅((x2 − x1 )⋅ s − y2 + y1 )=0 x1 + x2 + x3 − s2=0 s ⋅( x1 − x3 )− y1 − y3=0 ( y2 − y1 )⋅( same_x − inf )=0 ( y2 − y1 )⋅ inf_z − inf =0 ExamplePairs: a − e =0 i − o =0 u − y =0 ExampleTriples: a ⋅ b ⋅ c =0 d ⋅ e ⋅ f =0 g ⋅ h ⋅ i =0 j ⋅ k ⋅ l =0 m ⋅ n ⋅ o =0 Nonetheless, when we write these constraints in proof-systems, we use a shared framework that helps us use the columns in each row as placeholders. For example, the i-th column of the current row Curr is represented as: E::::cell(Column::Witness(i), CurrOrNext::Curr) where the E type is a shorthand for Expr. More succinctly, we refer to the above simply as: witness_curr(i) For readability, let's just refer to them here as w0..w14. Then, the above constraints would look like the following: Generic: w0 ⋅cl​+ w1 ⋅cr​+ w2 ⋅co​+ w0 ⋅ w1 ⋅cm​+cc​=0 CompleteAdd: w10 ⋅( w2 − w0 )−(1− w7)=0 w7 ⋅( w2 − w0 )=0 w7 ⋅(2⋅ w8 ⋅ w1 −3⋅ w02)+(1− w7 )⋅((w2 − w0 )⋅ w8 − w3 + w1 )=0 w0 + w2 + w4 − w82=0 w8 ⋅( w0 − w4 )− w1 − w4=0 ( w3 − w1 )⋅( w7 − w6 )=0 ( w3 − w1 )⋅ w9 − w6 =0 ExamplePairs: w0 − w1 =0 w2 − w3 =0 w4 − w5 =0 ExampleTriples: w0 ⋅ w1 ⋅ w2 =0 w3 ⋅ w4 ⋅ w5 =0 w6 ⋅ w7 ⋅ w8 =0 w9 ⋅ w10 ⋅ w11 =0 w12 ⋅ w13 ⋅ w14 =0 Looking at the list of equations, each of them look like polynomials in the variables w0..w14, constrained to zero. But so far, the type of equations we had studied were univariate (i.e. f(X) rather than f(X0​,...,X14​)). But this will not be a problem, since these multivariate equations will soon become univariate with a simple trick: interpolation . Right now, we are describing the gate constraints row-wise (i.e. horizontally ). But as we said before, our domain will be determined by the number of rows; we will transform these equations to a vertical and univariate behaviour. The strategy to follow will be obtaining witness polynomials that evaluate to the cells of the execution trace. First , note that so far we didn't mention any values for the w0..w14 terms, but instead they are placeholders for what is yet to come . And what is this you may ask: the witness of the relation . Put it another way, the witness will be the instantiation of the cells of the table that will make all of the above constraints hold. And where do we get these values from? Well, the prover will know or will compute them . Second , we count the number of rows that we ended up with: here it's four (conveniently, a power of two). What we want to do is to create as many witness polynomials as the number of columns in the execution trace. These polynomials, that we can call w0​(X),...,w14​(X), must be design to evaluate to the cells of the trace. Formally we define each wj​(X) as: G=:wj​(gi)=trace[i,j] Meaning that wi​(x) returns the i-th column of the x-th row of the execution trace. And how can we create such polynomials? We will use interpolation, which can be efficiently computed over power-of-two-sized groups, as it is the case in the example. Recalling Lagrangian terms , they act like linearly independent selectors over a set, in particular: Let G be a multiplicative group formed by the powers of a generator g. Given any element x in the group, the i-th Lagrangian term evaluates to 1 iff x=gi and 0 iff x=gi. ∀ x∈G:LgiG​(x)=1⟺x=θ and LgiG​(x)=0⟺x=θ Given that our group has size 4, the number of rows (ignoring ZK for now), G will be the set {1,g,g2,g3}. This means that we can build such 15 witness polynomials as: ∀j∈[0..15):wj​(X)=i=0∑3​trace[i,j]⋅LgiG​(X) Then, the above constraints will become the following equations using our 15 univariate witness polynomials: Generic: w0​(X)⋅cl​+w1​(X)⋅cr​+w2​(X)⋅co​+w0​(X)⋅w1​(X)⋅cm​+cc​=0 CompleteAdd: w10​(X)⋅(w2​(X)−w0​(X))−(1−w7​(X))=0 w7​(X)⋅(w2​(X)−w0​(X))=0 w7​(X)⋅(2⋅w8​(X)⋅w1​(X)−3⋅w0​(X)2)+(1−w7​(X))⋅((w2​(X)−w0​(X))⋅w8​(X)−w3​(X)+w1​(X))=0 w0​(X)+w2​(X)+w4​(X)−w8​(X)2=0 w8​(X)⋅(w0​(X)−w4​(X))−w1​(X)−w4​(X)=0 (w3​(X)−w1​(X))⋅(w7​(X)−w6​(X))=0 (w3​(X)−w1​(X))⋅w9​(X)−w6​(X)=0 ExamplePairs: w0​(X)−w1​(X)=0 w2​(X)−w3​(X)=0 w4​(X)−w5​(X)=0 ExampleTriples: w0​(X)⋅w1​(X)⋅w2​(X)=0 w3​(X)⋅w4​(X)⋅w5​(X)=0 w6​(X)⋅w7​(X)⋅w8​(X)=0 w9​(X)⋅w10​(X)⋅w11​(X)=0 w12​(X)⋅w13​(X)⋅w14​(X)=0 These are the 16 constraints that will need to be satisfied by a correct witness for the relation. We already know how to check many equations at once (combining them with powers of alpha), and how to check that the above holds for the full domain G (by providing the quotient polynomial of the huge equation after dividing by the vanishing polynomial on G). All there's left to do is making a correspondence between the constraints that need to be satisfied, and the particular row in the execution trace to which they apply. This implies, that before even aggregating the 16 constraints into a single one, we need to impose this restriction. In proof-systems, we represent this restriction with the GateType selector. In particular, we would multiply each of the constraints above by the Expr terms: Index(GateType::YourType), which will be transformed (again) into mutually exclusive selector polynomials. This means, that each row can only have one of these polynomials being nonzero (equal to 1), whereas the evaluation of all of the other selector polynomials for the given row should be 0. These polynomials are known by the verifier, who could check that this is indeed the case. But what will these polynomials look like? I am sure the above paragraph is familiar for you already. Once again, we will be able to build them using interpolation with our 4 Lagrangian terms for G. In our example: generic(X) will equal 1 when X=g0 and 0 otherwise; add(X) will equal 1 when X=g1 and 0 otherwise; pairs(X) will equal 1 when X=g2 and 0 otherwise; and triples(X) will equal 1 when X=g3 and 0 otherwise. These polynomials are, in fact, a description of the circuit. Note that we managed to make constraints for different gates already independent from each other (thanks to the mutually exclusivity). Nonetheless, granting independence within the gate is still needed. Here's where we will be using the powers of alpha: within a gate the powers cannot be repeated, but they can be reused among gates. Then, we will need 7 different powers of alpha for this stage of the protocol, as this is the largest number of different constraints within the same gate. Putting all of the above together, this means that our LONG and single constraint will look like: gates(X)= generic(X)⋅ α0⋅(w0​(X)⋅cl​+w1​(X)⋅cr​+w2​(X)⋅co​+w0​(X)⋅w1​(X)⋅cm​+cc​) +add(X)⋅( α0⋅(w10​(X)⋅(w2​(X)−w0​(X))−(1−w7​(X))) +α1⋅(w7​(X)⋅(w2​(X)−w0​(X))) +α2⋅(w7​(X)⋅(2⋅w8​(X)⋅w1​(X)−3⋅w0​(X)2)+(1−w7​(X))⋅((w2​(X)−w0​(X))⋅w8​(X)−w3​(X)+w1​(X))) +α3⋅(w0​(X)+w2​(X)+w4​(X)−w8​(X)2) +α4⋅(w8​(X)⋅(w0​(X)−w4​(X))−w1​(X)−w4​(X)) +α5⋅((w3​(X)−w1​(X))⋅(w7​(X)−w6​(X))) +α6⋅((w3​(X)−w1​(X))⋅w9​(X)−w6​(X))) +pairs(X)⋅( α0⋅(w0​(X)−w1​(X)) +α1⋅(w2​(X)−w3​(X)) +α2⋅(w4​(X)−w5​(X))) +triples(X)⋅( α0⋅(w0​(X)⋅w1​(X)⋅w2​(X)) +α1⋅(w3​(X)⋅w4​(X)⋅w5​(X)) +α2⋅(w6​(X)⋅w7​(X)⋅w8​(X)) +α3⋅(w9​(X)⋅w10​(X)⋅w11​(X)) +α4⋅(w12​(X)⋅w13​(X)⋅w14​(X))) Finally, providing q(X)=gates(X)/vG​(X) and performing the check on a random X∈F \\ G would give the verifier the assurance of the following: The prover knows a polynomial gates(X) that equals zero on any x∈{1,g,g2,g3}. Nonetheless, it would still remain to verify that gates(X) actually corresponds to the encoding of the actual constraints. Meaning, checking that this polynomial encodes the column witnesses, and the agreed circuit. So instead of providing just gates(X) (actually, a commitment to it), the prover can send commitments to each of th 15 witness polynomials, so that the verifier can reconstruct the huge constraint using the encodings of the circuit (which is known ahead).","breadcrumbs":"Custom Gates » Custom gates","id":"148","title":"Custom gates"},"149":{"body":"This document is meant to explain the foreign field addition gate in Kimchi.","breadcrumbs":"Custom Gates » Foreign Field Addition » Foreign Field Addition RFC","id":"149","title":"Foreign Field Addition RFC"},"15":{"body":"The next object to know about is polynomials. A polynomial is a syntactic object that also defines a function. Specifically, let R be a field (or more generally it could even be a ring, which is like a field but without necessarily having division, like the integers Z). And pick a variable name like x. Then R[x] (pronounced, \"R adjoin x\" or \"polynomials in x over R\" or \"polynomials in x with R coefficients\") is the set of syntactic expressions of the form a0​+a1​x+a2​x2+⋯+ad​xd where each ai​∈R and d is any natural number. If we wanted to be very formal about it, we could define R[x] to be the set of lists of elements of R (the lists of coefficients), and the above is just a suggestive way of writing that list. An important fact about polynomials over R is that they can be interpreted as functions R→R. In other words, there is a function eval:R[x]→(R→R) defined by eval(a0​+a1​x+⋯+ad​xd)=(y:R)↦a0​+a1​y+⋯+ad​yd where x is just a variable name without a value assigned to it and eval maps it to a field element y and evaluates the function as the inner product between the list of coefficients and the powers of y. It is important to remember that polynomials are different than the functions that they compute. Polynomials are really just lists of coefficients. Over some fields, there are polynomials p1 and p2 such that eval(p1)=eval(p2), but p1=p2. For example, consider in F2​[x] the polynomials x and x2. These map to the same function F2​→F2​ (meaning for x∈R=F2​={0,1} it holds that x=x2), but are distinct polynomials. Some definitions and notation If f is a polynomial in R[x] and a∈R, we write f(a) for eval(f)(a) If f is a polynomial, the degree of f is the largest d for which the coefficient of xd is non-zero in f. For example, the degree of x3+2x10 is 10. We will use the notation R[x]n. In order to deal with this, we will divide foreign field elements into limbs that fit in our native field. We want to be compatible with the foreign field multiplication gate, and thus the parameters we will be using are the following: 3 limbs of 88 bits each (for a total of 264 bits) So f<2264, concretely: The modulus f of the foreign field Ff​ is 256 bit The modulus of our the native field Fn​ is 255 bit In other words, using 3 limbs of 88 bits each allows us to represent any foreign field element in the range [0,2264) for foreign field addition, but only up to 2259 for foreign field multiplication. Thus, with the current configuration of our limbs, our foreign field must be smaller than 2259 (because 2264⋅2255>22592+2259, more on this in Foreign Field Multiplication or the original FFmul RFC .","breadcrumbs":"Custom Gates » Foreign Field Addition » Overview","id":"150","title":"Overview"},"151":{"body":"Let's take a closer look at what we have, if we split our variables in limbs (using little endian) bits 0..............87|88...........175|176...........263 a = (-------a0-------|-------a1-------|-------a2-------)\n+/-\nb = (-------b0-------|-------b1-------|-------b2-------)\n=\nr = (-------r0-------|-------r1-------|-------r2-------) mod(f) We will perform the addition in 3 steps, one per limb. Now, when we split long additions in this way, we must be careful with carry bits between limbs. Also, even if a and b are foreign field elements, it could be the case that a+b is already larger than the modulus (in this case a+b could be at most 2f−2). But it could also be the case that the subtraction produces an underflow because a > >\n= c_0 c_1 c_2\nq = -1 | 0 | 1\n·\nf = (-------f0-------|-------f1-------|-------f2-------)\n+\nr = (-------r0-------|-------r1-------|-------r2-------) Our witness computation is currently using the BigUint library, which takes care of all of the intermediate carry limbs itself so we do not have to address all casuistries ourselves (such as propagating the low carry flag to the high limb in case the middle limb is all zeros).","breadcrumbs":"Custom Gates » Foreign Field Addition » Splitting the addition","id":"151","title":"Splitting the addition"},"152":{"body":"Last but not least, we should perform some range checks to make sure that the result r is contained in Ff​. This is important because there could be other values of the result which still fit in <2264 but are larger than f, and we must make sure that the final result is the minimum one (that we will be referring to as rmin​ in the following). Ideally, we would like to reuse some gates that we already have. In particular, we can perform range checks for 0≤X<23ℓ=23⋅88. But we want to check that 0≤rmin​ >\n= k_0 k_1\nf\n+\nu = (-------u0-------|-------u1-------|-------u2-------) Following the steps above, and representing this equation in limb form, we have: u0​u1​u2​​=r0​+0−f0​−k0​⋅288=r1​+0−f1​−k1​⋅288+k0​=r2​+288−f2​+k1​​ But now we also have to check that 0≤r. But this is implicitly covered by r being a field element of at most 264 bits (range check). When we have a chain of additions ai​+bi​=ri​ with ai​=ri−1​, we could apply the field membership check naively to every intermediate ri​, however it is sufficient to apply it only once at the end of the computations for rn​, and keep intermediate ri​∈[0,2264), in a \"raw\" form. Generally speaking, treating intermediate values lazily helps to save a lot of computation in many different FF addition and multiplication scenarios.","breadcrumbs":"Custom Gates » Foreign Field Addition » Upper bound check for foreign field membership","id":"152","title":"Upper bound check for foreign field membership"},"153":{"body":"Mathematically speaking, a subtraction within a field is no more than an addition over that field. Negative elements are not different from \"positive\" elements in finite fields (or in any modular arithmetic). Our witness computation code computes negative sums by adding the modulus to the result. To give a general example, the element −e within a field Fm​ of order m and e > > c_0 c_1 0 Inspired by the halving approach in foreign field multiplication, an optimized version of the above gate results in a reduction by 2 in the number of constraints and by 1 in the number of witness cells needed. The main idea is to condense the claims about the low and middle limbs in one single larger limb of 176 bits, which fit in our native field. This way, we can get rid of the low carry flag, its corresponding carry check, and the three result checks become just two. bits 0..............87|88...........175|176...........263\nr = (-------r0------- -------r1-------|-------r2-------)\n=\na = (-------a0------- -------a1-------|-------a2-------)\n+\ns = 1 | -1\n·\nb = (-------b0------- -------b1-------|-------b2-------)\n-\nq = -1 | 0 | 1\n·\nf = (-------f0------- -------f1-------|-------f2-------) > > c 0 These are the new equations: rbot​rtop​​=(a0​+288⋅a1​)+s⋅(b0​+288⋅b1​)−q⋅(f0​+288⋅f1​)−c⋅2176=a2​+s⋅b2​−q⋅f2​+c​ with rtop​=r2​ and c=c1​.","breadcrumbs":"Custom Gates » Foreign Field Addition » Optimization","id":"154","title":"Optimization"},"155":{"body":"The full foreign field addition/subtraction gadget will be composed of: 1 public input row containing the value 1; n rows with ForeignFieldAdd gate type: for the actual n chained additions or subtractions; 1 ForeignFieldAdd row for the bound addition; 1 Zero row for the final bound addition. 1 RangeCheck gadget for the first left input of the chain a1​:=a; Then, n of the following set of rows: 1 RangeCheck gadget for the i-th right input bi​; 1 RangeCheck gadget for the i-th result which will correspond to the (i+1)-th left input of the chain ri​=ai+1​. 1 final RangeCheck gadget for the bound check u. A total of 20 rows with 15 columns in Kimchi for 1 addition. All ranges below are inclusive. Row(s) Gate type(s) Witness 0 PublicInput 1 1..n ForeignFieldAdd +/- n+1 ForeignFieldAdd bound n+2 Zero bound n+3..n+6 multi-range-check left n+7+8i..n+10+8i multi-range-check right n+11+8i..n+14+8i multi-range-check r 9n+7..9n+10 multi-range-check u This mechanism can chain foreign field additions together. Initially, there are n foreign field addition gates, followed by a foreign field addition gate for the bound addition (whose current row corresponds to the next row of the last foreign field addition gate), and an auxiliary Zero row that holds the upper bound. At the end, an initial left input range check is performed, which is followed by a n pairs of range check gates for the right input and intermediate result (which become the left input for the next iteration). After the chained inputs checks, a final range check on the bound takes place. For example, chaining the following set of 3 instructions would result in a full gadget with 37 rows: add(add(add(a,b),c),d) Row(s) Gate type(s) Witness 0 PublicInput 1 1 ForeignFieldAdd a+b 2 ForeignFieldAdd (a+b)+c 3 ForeignFieldAdd ((a+b)+c)+d 4 ForeignFieldAdd bound 5 Zero bound 6..9 multi-range-check a 10..13 multi-range-check b 14..17 multi-range-check a+b 18..21 multi-range-check c 22..25 multi-range-check a+b+c 26..29 multi-range-check d 30..33 multi-range-check a+b+c+d 34..37 multi-range-check bound Nonetheless, such an exhaustive set of checks are not necessary for completeness nor soundness. In particular, only the very final range check for the bound is required. Thus, a shorter gadget that is equally valid and takes (8⋅n+4) fewer rows could be possible if we can assume that the inputs of each addition are correct foreign field elements. It would follow the next layout (with inclusive ranges): Row(s) Gate type(s) Witness 0 public input row for soundness of bound overflow flag 1 1..n ForeignFieldAdd n+1 ForeignFieldAdd n+2 Zero n+3..n+6 multi-range-check for bound u Otherwise, we would need range checks for each new input of the chain, but none for intermediate results; implying 4⋅n fewer rows. Row(s) Gate type(s) Witness 0 public input row for soundness of bound overflow flag 1 1..n ForeignFieldAdd ai​+bi​ n+1 ForeignFieldAdd n+2 Zero n+3..n+6 multi-range-check for first left input a1​ n+7+4i..n+10+4i multi-range-check for i-th right input bi​ 5n+7..5n+10 multi-range-check for bound u For more details see the Bound Addition section in Foreign Field Multiplication or the original Foreign Field Multiplication RFC .","breadcrumbs":"Custom Gates » Foreign Field Addition » Gadget","id":"155","title":"Gadget"},"156":{"body":"For the full mode of tests of this gate, we need to perform 4 range checks to assert that the limbs of a,b,r,u have a correct size, meaning they fit in 288 (and thus, range-checking a,b,r,u for 2264). Because each of these elements is split into 3 limbs, we will have to use 3 copy constraints between the RangeCheck gates and the ForeignFieldAdd rows (per element). That amounts to 12 copy constraints. Recalling that Kimchi only allows for the first 7 columns of each row to host a copy constraint, we necessarily have to use 2 rows for the actual addition gate. The layout of these two rows is the following: Curr Next ... Final Column ForeignFieldAdd ForeignFieldAdd Zero 0 a0​ (copy) r0​ (copy) u0​ (copy) 1 a1​ (copy) r1​ (copy) u1​ (copy) 2 a2​ (copy) r2​ (copy) u2​ (copy) 3 b0​ (copy) 4 b1​ (copy) 5 b2​ (copy) 6 q 7 c 8 9 10 11 12 13 14","breadcrumbs":"Custom Gates » Foreign Field Addition » Layout","id":"156","title":"Layout"},"157":{"body":"So far, we have pointed out the following sets of constraints: Main addition 0=a0​+b0​−r0​−q⋅f0​−288⋅c0​ 0=a1​+b1​−r1​−q⋅f1​+c0​−288⋅c1​ 0=a2​+b2​−r2​−q⋅f2​+c1​ Carry checks 0=c0​⋅(c0​+1)⋅(c0​−1) 0=c1​⋅(c0​+1)⋅(c1​−1) Sign checks TODO: decide if we really want to keep this check or leave it implicit as it is a coefficient value 0=(s+1)⋅(s−1) Overflow checks 0=q⋅(q−s)","breadcrumbs":"Custom Gates » Foreign Field Addition » Constraints","id":"157","title":"Constraints"},"158":{"body":"When we use this gate as part of a larger chained gadget, we should optimize the number of range check rows to avoid redundant checks. In particular, if the result of an addition becomes one input of another addition, there is no need to check twice that the limbs of that term have the right length. The sign is now part of the coefficients of the gate. This will allow us to remove the sign check constraint and release one witness cell element. But more importantly, it brings soundness to the gate as it is now possible to wire the 1 public input to the overflow witness of the final bound check of every addition chain.","breadcrumbs":"Custom Gates » Foreign Field Addition » Optimizations","id":"158","title":"Optimizations"},"159":{"body":"This document is an original RFC explaining how we constrain foreign field multiplication (i.e. non-native field multiplication) in the Kimchi proof system. For a more recent RFC, see FFmul RFC . Changelog Author(s) Date Details Joseph Spadavecchia and Anais Querol October 2022 Design of foreign field multiplication in Kimchi","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Foreign Field Multiplication","id":"159","title":"Foreign Field Multiplication"},"16":{"body":"An important fact for zk-SNARKs about polynomials is that we can use them to encode arrays of field elements. In other words, there is a way of converting between polynomials of degree d and arrays of field elements of length d+1. This is important for a few reasons It will allow us to translate statements about arrays of field elements into statements about polynomials, which will let us construct zk-SNARKs. It will allow us to perform certain operations on polynomials more efficiently. So let's get to defining and proving this connection between arrays of field elements. The first thing to understand is that we won't be talking about arrays directly in this section. Instead, we'll be talking about functions A→F where A is a finite subset of F. The idea is, if A has size n, and we put an ordering on A, then we can think of a function A→F as the same thing as an immutable array of length n, since such a function can be thought of as returning the value of an array at the input position. With this understanding in hand, we can start describing the \"fundamental theorem of polynomials\". If A⊆F has size d+1, this theorem will define an isomorphism between functions A→F and F[x]≤d​, the set of polynomials of degree at most d. Now let's start defining this isomorphism. One direction of it is very easy to define. It is none other than the evaluation map, restricted to A: evalA​evalA​​:F[x]≤d​→(A→F)(c0​+⋯+cd​xd)=a↦i≤d∑​ci​ai​ We would now like to construct an inverse to this map. What would that mean? It would be a function that takes as input a function φ:A→F (remember, basically an array of length ∣A∣), and returns a polynomial f which agrees with φ on the set A. In other words, we want to construct a polynomial that interpolates between the points (a,φ(a)) for a∈A. Our strategy for constructing this polynomial will be straightforward. For each a∈A, we will construct a polynomial fa​ that is equal to φ(a) when evaluated at a, and equal to 0 when evaluated anywhere else in the set A. Then, our final polynomial will be f:=∑a∈A​fa​. Then, when f is evaluated at a0​∈A, only the fa0​​ term will be non-zero (and equal to φ(a0​) as desired), all the other terms will disappear. Constructing the interpolation map requires a lemma. Lemma. (construction of vanishing polynomials) Let S⊆F. Then there is a polynomial vS​∈F[x] of degree ∣S∣ such that vS​ evaluates to 0 on S, and is non-zero off of S. vS​ is called the vanishing polynomial of S. Proof. Let vS​=∏s∈S​(x−s). Clearly vS​ has degree ∣S∣. If t∈S, then vS​(t)=0 since x−t is one of the terms in the product defining vS​, so when we evaluate at t, t−t=0 is one of the terms. If t∈/S, then all the terms in the product are non-zero, and thus vS​(t) is non-zero as well. □ Now we can define the inverse map. Define interpA​interpA​​:(A→F)→F[x]≤d​(f)=a∈A∑​vA∖{a}​(a)f(a)​⋅vA∖{a}​​ Since each vA∖{a}​ has degree d, this polynomial has degree d. Now we have, for any b∈A, evalA​(interpA​(f))(b)​=a∈A∑​vA∖{a}​(a)f(a)​⋅vA∖{a}​(b)=vA∖{b}​(b)f(b)​⋅vA∖{b}​(b)+a=b∑​vA∖{a}​(a)f(a)​⋅vA∖{a}​(b)=f(b)+a=b∑​vA∖{a}​(a)f(a)​⋅0=f(b)​ Thus evalA​∘interpA​ is the identity. So we have successfully devised a way of interpolating a set of d+1 points with a polynomial of degree d. What remains to be seen is that these two maps are inverse in the other direction. That is, that for any polynomial f∈F[x], we have interpA​(evalA​(f))=f This says that if we interpolate the function that has the values of f on A, we get back f itself. This essentially says that there is only one way to interpolate a set of d+1 points with a degree d polynomial. So let's prove it by proving that statement. Lemma: polynomials that agree on enough points are equal. Let F be a field and suppose f,g∈F[x] have degree at most d. Let A⊆F have size d+1. Suppose for all a∈A, f(a)=g(a). Then f=g as polynomials. Proof. Define h:=f−g. Our strategy will be to show that h is the zero polynomial. This would then imply that f=g as polynomials. Note that h vanishes on all of A since f and g are equal on A. To that end, let a∈A. Then we can apply the polynomial division algorithm to divide h by x−a and obtain polynomials qa​,ra​ such that h=qa​(x−a)+ra​ and ra​ has degree less than 1. I.e., ra​ is a constant in F. Now, 0=h(a)=qa​(a)⋅0+ra​=ra​ so ra​=0 and thus h=qa​(x−a). Note that qa​ is 0 on all elements b∈A with b=a since h(b)=qa​(b)(b−a)=0, but b−a=0. Thus, if we iterate the above, enumerating the elements of A as a0​,…,ad​, we find that h​=(x−a0​)qa0​​=(x−a0​)(x−a1​)qa1​​=…=(x−a0​)(x−a1​)…(x−ad​)qad​​​ Now, if qad​​ is not the zero polynomial, then h will have degree at least d+1 since it would have as a factor the d+1 linear terms (x−ai​). But since f,g both have degree at most d and h=f−g, h has degree at most d as well. Thus, qad​​=0, which means h=0 as well. This gives us the desired conclusion, that f=g as polynomials. □ Now we can easily show that interpolating the evaluations of a polynomial yields that polynomial itself. Let f∈F[x]≤d​. Then interpA​(evalA​(f)) is a polynomial of degree at most d that agrees with f on A, a set of size d+1. Thus, by the lemma, they are equal as polynomials. So indeed interpA​(evalA​(f))=f for all f∈F[x]. So far we have proven that interpA​ and evalA​ give an isomorphism of sets (i.e., a bijection) between the sets A→F and F[x]≤d​. But we can take this theorem a bit further. The set of functions A→F can be given the structure of a ring, where addition and multiplication happen pointwise. I.e., for f,g:A→F we define f+g:=a↦f(a)+g(a) and f⋅g:=a↦f(a)⋅g(a). Then we can strengthen our theorem to say Fundamental theorem of polynomials (final version) Let d∈N and let A⊆F with ∣A∣=d+1. With evalA​:F[x]≤d​→(A→F)interpA​:(A→F)→F[x]≤d​ defined as above, these two functions define an isomorphism of rings. That is is, they are mutually inverse and each one respects addition, subtraction and multiplication. The fundamental theorem of polynomials is very important when it comes to computing operations on polynomials. As we will see in the next section , the theorem will help us to compute the product of degree n polynomials in time O(nlogn), whereas the naive algorithm takes time O(n2). To put this in perspective, if n=216, n2 is 4096 times larger than nlogn and the gap only gets bigger as n grows.","breadcrumbs":"Polynomials » Fundamental theorem of polynomials","id":"16","title":"Fundamental theorem of polynomials"},"160":{"body":"This gate constrains a⋅b=cmodf where a,b,c∈Ff​, a foreign field with modulus f, using the native field Fn​ with prime modulus n.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Overview","id":"160","title":"Overview"},"161":{"body":"In foreign field multiplication the foreign field modulus f could be bigger or smaller than the native field modulus n. When the foreign field modulus is bigger, then we need to emulate foreign field multiplication by splitting the foreign field elements up into limbs that fit into the native field element size. When the foreign modulus is smaller everything can be constrained either in the native field or split up into limbs. Since our projected use cases are when the foreign field modulus is bigger (more on this below) we optimize our design around this scenario. For this case, not only must we split foreign field elements up into limbs that fit within the native field, but we must also operate in a space bigger than the foreign field. This is because we check the multiplication of two foreign field elements by constraining its quotient and remainder. That is, renaming c to r, we must constrain a⋅b=q⋅f+r, where the maximum size of q and r is f−1 and so we have a⋅b​≤q(f−1)​​⋅f+r(f−1)​​≤f2−1.​ Thus, the maximum size of the multiplication is quadratic in the size of foreign field. Naïve approach Naïvely, this implies that we have elements of size f2−1 that must split them up into limbs of size at most n−1. For example, if the foreign field modulus is 256 and the native field modulus is 255 bits, then we'd need log2​((2256)2−1)≈512 bits and, thus, require 512/255≈3 native limbs. However, each limb cannot consume all 255 bits of the native field element because we need space to perform arithmetic on the limbs themselves while constraining the foreign field multiplication. Therefore, we need to choose a limb size that leaves space for performing these computations. Later in this document (see the section entitled \"Choosing the limb configuration\") we determine the optimal number of limbs that reduces the number of rows and gates required to constrain foreign field multiplication. This results in ℓ=88 bits as our optimal limb size. In the section about intermediate products we place some upperbounds on the number of bits required when constraining foreign field multiplication with limbs of size ℓ thereby proving that the computations can fit within the native field size. Observe that by combining the naïve approach above with a limb size of 88 bits, we would require 512/88≈6 limbs for representing foreign field elements. Each limb is stored in a witness cell (a native field element). However, since each limb is necessarily smaller than the native field element size, it must be copied to the range-check gate to constrain its value. Since Kimchi only supports 7 copyable witness cells per row, this means that only one foreign field element can be stored per row. This means a single foreign field multiplication would consume at least 4 rows (just for the operands, quotient and remainder). This is not ideal because we want to limit the number of rows for improved performance. Chinese remainder theorem Fortunately, we can do much better than this, however, by leveraging the chinese remainder theorem (CRT for short) as we will now show. The idea is to reduce the number of bits required by constraining our multiplication modulo two coprime moduli: 2t and n. By constraining the multiplication with both moduli the CRT guarantees that the constraints hold with the bigger modulus 2t⋅n. The advantage of this approach is that constraining with the native modulus n is very fast, allowing us to speed up our non-native computations. This practically reduces the costs to constraining with limbs over 2t, where t is something much smaller than 512. For this to work, we must select a value for t that is big enough. That is, we select t such that 2t⋅n>f2−1. As described later on in this document, by doing so we reduce the number of bits required for our use cases to 264. With 88 bit limbs this means that each foreign field element only consumes 3 witness elements, and that means the foreign field multiplication gate now only consumes 2 rows. The section entitled \"Choosing t\" describes this in more detail. Overall approach Bringing it all together, our approach is to verify that a⋅b=q⋅f+r​​ over Z+. In order to do this efficiently we use the CRT, which means that the equation holds mod M=2t⋅n. For the equation to hold over the integers we must also check that each side of the equation is less than 2t⋅n. The overall steps are therefore Apply CRT to equation (1) Check validity with binary modulus mod2t Check validity with native modulus modn Check each side of equation (1) is less than M a⋅b<2t⋅n q⋅f+r<2t⋅n This then implies that a⋅b=cmodf. where c=r. That is, a multiplied with b is equal to c where a,b,c∈Ff​. There is a lot more to each of these steps. That is the subject of the rest of this document. Other strategies Within our overall approach, aside from the CRT, we also use a number of other strategies to reduce the number and degree of constraints. Avoiding borrows and carries Intermediate product elimination Combining multiplications The rest of this document describes each part in detail.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Approach","id":"161","title":"Approach"},"162":{"body":"This section describes important parameters that we require and how they are computed. Native field modulus n Foreign field modulus f Binary modulus 2t CRT modulus 2t⋅n Limb size in bits ℓ Choosing t Under the hood, we constrain a⋅b=q⋅f+rmod2t⋅n. Since this is a multiplication in the foreign field f, the maximum size of q and r are f−1, so this means a⋅b​≤(f−1)⋅f+(f−1)≤f2−1.​ Therefore, we need the modulus 2t⋅n such that 2t⋅n>f2−1, which is the same as saying, given f and n, we must select t such that 2t⋅nt​≥f2≥2log2​(f)−log2​(n).​ Thus, we have an lower bound on t. Instead of dynamically selecting t for every n and f combination, we fix a t that will work for the different selections of n and f relevant to our use cases. To guide us, from above we know that tmin​=2log2​(f)−log2​(n) and we know the field moduli for our immediate use cases. vesta = 2^254 + 45560315531506369815346746415080538113 (255 bits)\npallas = 2^254 + 45560315531419706090280762371685220353 (255 bits)\nsecp256k1 = 2^256 - 2^32 - 2^9 - 2^8 - 2^7 - 2^6 - 2^4 - 1 (256 bits) So we can create a table n f tmin​ vesta secp256k1 258 pallas secp256k1 258 vesta pallas 255 pallas vesta 255 and know that to cover our use cases we need t≥258. Next, given our native modulus n and t, we can compute the maximum foreign field modulus supported . Actually, we compute the maximum supported bit length of the foreign field modulus f=2m. 2t⋅nt+log2​(n)​≥f2≥(2m)2=22m>log2​(22m)=2m​ So we get mmax​=2t+log2​(n)​. With t=258,n=255 we have mmax​​=2258+255​=256.5,​ which is not enough space to handle anything larger than 256 bit moduli. Instead, we will use t=264, giving mmax​=259 bits. The above formula is useful for checking the maximum number of bits supported of the foreign field modulus, but it is not useful for computing the maximum foreign field modulus itself (because 2mmax​ is too coarse). For these checks, we can compute our maximum foreign field modulus more precisely with maxmod​=⌊2t⋅n​⌋. The max prime foreign field modulus satisfying the above inequality for both Pallas and Vesta is 926336713898529563388567880069503262826888842373627227613104999999999999999607. Choosing the limb configuration Choosing the right limb size and the right number of limbs is a careful balance between the number of constraints (i.e. the polynomial degree) and the witness length (i.e. the number of rows). Because one limiting factor that we have in Kimchi is the 12-bit maximum for range check lookups, we could be tempted to introduce 12-bit limbs. However, this would mean having many more limbs, which would consume more witness elements and require significantly more rows. It would also increase the polynomial degree by increasing the number of constraints required for the intermediate products (more on this later). We need to find a balance between the number of limbs and the size of the limbs. The limb configuration is dependent on the value of t and our maximum foreign modulus (as described in the previous section). The larger the maximum foreign modulus, the more witness rows we will require to constrain the computation. In particular, each limb needs to be constrained by the range check gate and, thus, must be in a copyable (i.e. permuteable) witness cell. We have at most 7 copyable cells per row and gates can operate on at most 2 rows, meaning that we have an upperbound of at most 14 limbs per gate (or 7 limbs per row). As stated above, we want the foreign field modulus to fit in as few rows as possible and we need to constrain operands a,b, the quotient q and remainder r. Each of these will require cells for each limb. Thus, the number of cells required for these is cells=4⋅limbs It is highly advantageous for performance to constrain foreign field multiplication with the minimal number of gates. This not only helps limit the number of rows, but also to keep the gate selector polynomial small. Because of all of this, we aim to constrain foreign field multiplication with a single gate (spanning at most 2 rows). As mentioned above, we have a maximum of 14 permuteable cells per gate, so we can compute the maximum number of limbs that fit within a single gate like this. limbsmax​​=⌊cells/4⌋=⌊14/4⌋=3​ Thus, the maximum number of limbs possible in a single gate configuration is 3. Using limbsmax​=3 and t=264 that covers our use cases (see the previous section), we are finally able to derive our limbs size ℓ​=limbsmax​t​=264/3=88​ Therefore, our limb configuration is: Limb size ℓ=88 bits Number of limbs is 3","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Parameter selection","id":"162","title":"Parameter selection"},"163":{"body":"When we constrain a⋅b−q⋅f=rmod2t we want to be as efficient as possible. Observe that the expansion of a⋅b−q⋅f into limbs would also have subtractions between limbs, requiring our constraints to account for borrowing. Dealing with this would create undesired complexity and increase the degree of our constraints. In order to avoid the possibility of subtractions we instead use a⋅b+q⋅f′ where f′​=−fmod2t=2t−f​ The negated modulus f′ becomes part of our gate coefficients and is not constrained because it is publicly auditable. Using the substitution of the negated modulus, we now must constrain a⋅b+q⋅f′=rmod2t. Observe that f′<2t since f<2t and that f′>f when f<2t−1.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Avoiding borrows","id":"163","title":"Avoiding borrows"},"164":{"body":"This section explains how we expand our constraints into limbs and then eliminate a number of extra terms. We must constrain a⋅b+q⋅f′=rmod2t on the limbs, rather than as a whole. As described above, each foreign field element x is split into three 88-bit limbs: x0​,x1​,x2​, where x0​ contains the least significant bits and x2​ contains the most significant bits and so on. Expanding the right-hand side into limbs we have ​(a0​+a1​⋅2ℓ+a2​⋅22ℓ)⋅(b0​+b1​⋅2ℓ+b2​⋅22ℓ)+(q0​+q1​⋅2ℓ+q2​⋅22ℓ)⋅(f0′​+f1′​⋅2ℓ+f2′​⋅22ℓ)= a0​⋅b0​+a0​⋅b1​⋅2ℓ+a0​⋅b2​⋅22ℓ +a1​⋅b0​⋅2ℓ+a1​⋅b1​⋅22ℓ+a1​⋅b2​⋅23ℓ +a2​⋅b0​⋅22ℓ+a2​⋅b1​⋅23ℓ+a2​⋅b2​⋅24ℓ+ q0​⋅f0′​+q0​⋅f1′​⋅2ℓ+q0​⋅f2′​⋅22ℓ +q1​⋅f0′​⋅2ℓ+q1​⋅f1′​⋅22ℓ+q1​⋅f2′​⋅23ℓ +q2​⋅f0′​⋅22ℓ+q2​⋅f1′​⋅23ℓ+q2​⋅f2′​⋅24ℓ=a0​⋅b0​+q0​⋅f0′​+2ℓ⋅(a0​⋅b1​+a1​⋅b0​+q0​⋅f1′​+q1​⋅f0′​)+22ℓ⋅(a0​⋅b2​+a2​⋅b0​+q0​⋅f2′​+q2​⋅f0′​+a1​⋅b1​+q1​⋅f1′​)+23ℓ⋅(a1​⋅b2​+a2​⋅b1​+q1​⋅f2′​+q2​⋅f1′​)+24ℓ⋅(a2​⋅b2​+q2​⋅f2′​)​ Since t=3ℓ, the terms scaled by 23ℓ and 24ℓ are a multiple of the binary modulus and, thus, congruent to zero mod2t. They can be eliminated and we don't need to compute them. So we are left with 3 intermediate products that we call p0​,p1​,p2​: Term Scale Product p0​ 1 a0​b0​+q0​f0′​ p1​ 2ℓ a0​b1​+a1​b0​+q0​f1′​+q1​f0′​ p2​ 22ℓ a0​b2​+a2​b0​+q0​f2′​+q2​f0′​+a1​b1​+q1​f1′​ So far, we have introduced these checked computations to our constraints Computation of p0​,p1​,p2​","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Intermediate products","id":"164","title":"Intermediate products"},"165":{"body":"Let's call p:=ab+qf′mod2t. Remember that our goal is to constrain that p−r=0mod2t (recall that any more significant bits than the 264th are ignored in mod2t). Decomposing that claim into limbs, that means 22ℓ(p2​−r2​)+2ℓ(p1​−r1​)+p0​−r0​=0mod2t.​​ We face two challenges Since p0​,p1​,p2​ are at least 2ℓ bits each, the right side of the equation above does not fit in Fn​ The subtraction of the remainder's limbs r0​ and r1​ could require borrowing For the moment, let's not worry about the possibility of borrows and instead focus on the first problem.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Constraining mod2t","id":"165","title":"Constraining mod2t"},"166":{"body":"The first problem is that our native field is too small to constrain 22ℓ(p2​−r2​)+2ℓ(p1​−r1​)+p0​−r0​=0mod2t. We could split this up by multiplying a⋅b and q⋅f′ separately and create constraints that carefully track borrows and carries between limbs. However, a more efficient approach is combined the whole computation together and accumulate all the carries and borrows in order to reduce their number. The trick is to assume a space large enough to hold the computation, view the outcome in binary and then split it up into chunks that fit in the native modulus. To this end, it helps to know how many bits these intermediate products require. On the left side of the equation, p0​ is at most 2ℓ+1 bits. We can compute this by substituting the maximum possible binary values (all bits set to 1) into p0​=a0​b0​+q0​f0′​ like this maxbits(p0​)​=log2​(a0​(2ℓ−1)​​b0​(2ℓ−1)​​+q0​(2ℓ−1)​​f0′​(2ℓ−1)​​)=log2​(2(22ℓ−2ℓ+1+1))=log2​(22ℓ+1−2ℓ+2+2).​ So p0​ fits in 2ℓ+1 bits. Similarly, p1​ needs at most 2ℓ+2 bits and p2​ is at most 2ℓ+3 bits. Term p0​ p1​ p2​ Bits 2ℓ+1 2ℓ+2 2ℓ+3 The diagram below shows the right hand side of the zero-sum equality from equation (2). That is, the value p−r. Let's look at how the different bits of p0​,p1​,p2​,r0​,r1​ and r2​ impact it. 0 L 2L 3L 4L\n|-------------|-------------|-------------|-------------|-------------| :\n|--------------p0-----------:-| 2L + 1 : |-------------:-p1------------| 2L + 2 p10➚ : p11➚ |----------------p2-------------| 2L + 3 :\n|-----r0------| : : |-----r1------| : |-----r2------|\n\\__________________________/ \\______________________________/ ≈ h0 ≈ h1 Within our native field modulus we can fit up to 2ℓ+δfk quickly overflows our CRT modulus 2t⋅n. For example, assuming our maximum foreign modulus of f=2259 and either of Kimchi's native moduli (i.e. Pallas or Vesta), fk>2t⋅n for k>2. That is, an overflow is possible for a chain of greater than 1 foreign field multiplication. Thus, we must check a,b,q and r are less than f for each multiplication. Fortunately, in many situations the input operands may already be checked either as inputs or as outputs of previous operations, so they may not be required for each multiplication operation. Thus, the q′ and r′ checks (or equivalently q and r) are our main focus because they must be done for every multiplication. Compute bound q′=q+f′ with addition gate (2 rows) Compute bound r′=r+f′ with addition gate (2 rows) Range check q′<2t (4 rows) Range check r′<2t (4 rows) This costs 12 rows per multiplication. In a subsequent section, we will reduce it to 8 rows.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Cost of bound checks","id":"174","title":"Cost of bound checks"},"175":{"body":"Due to the limited number of permutable cells per gate, we do not have enough cells for copy constraining q′ and r′ (or q and r) to their respective range check gadgets. To address this issue, we must decompose q′ into 2 limbs instead of 3, like so q′=q01′​+22ℓ⋅q2′​ and q01′​=q0′​+2ℓ⋅q1′​ Thus, q′ is decomposed into two limbs q01′​ (at most 2ℓ bits) and q2′​ (at most ℓ bits). Note that q′ must be range checked by a multi-range-check gadget. To do this the multi-range-check gadget must Store a copy of the limbs q0′​,q1′​ and q2′​ in its witness Range check that they are ℓ bit each Constrain that q01′​=q0′​+2ℓ⋅q1′​ (this is done with a special mode of the multi-range-check gadget) Have copy constraints for q01′​ and q2′​ as outputs of the ForeignFieldMul gate and inputs to the multi-range-check gadget Note that since the foreign field multiplication gate computes q′ from q which is already in the witness and q01′​ and q2′​ have copy constraints to a multi-range-check gadget that fully constrains their decomposition from q′, then the ForeignFieldMul gate does not need to store an additional copy of q0′​ and q1′​.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 2-limb decomposition","id":"175","title":"2-limb decomposition"},"176":{"body":"Since the q2ℓ where 2ℓ−1 is the maximum possible size of x1​ (before it overflows) and ϕ0​ is the overflow bit from the addition of the least significant limbs x0​ and f0′​. This means >2ℓ−ϕ0​−f1′​ We cannot allow x to overflow the foreign field, so we also have >x1​<(f−x0​)/22ℓ> Thus, >2ℓ−ϕ0​−f1′​<(f−x0​)/22ℓ=f/22ℓ−x0​/22ℓ> Since x0​/22ℓ=ϕ0​ we have >2ℓ−ϕ0​−f1′​ so >2ℓ−f1′​ Notice that f/22ℓ=f1​. Now we have >2ℓ−f1′​⟺>f1′​>2ℓ−f1​> However, this is a contradiction with the definition of our negated foreign field modulus limb f1′​=2ℓ−f1​. ■ We have proven that qcarry2′​ is always zero, so that allows use to simplify our constraints. We now have q0​∈[0,2ℓ) multi-range-check q1​∈[0,2ℓ) multi-range-check q01′​=q0′​+2ℓ⋅q1′​ multi-range-check qcarry01′​∈[0,2) ForeignFieldMul 22ℓ⋅qcarry01′​=s01​−q01′​ ForeignFieldMul q2​∈[0,2ℓ) multi-range-check q2′​=s2​+w01​ ForeignFieldMul In other words, we have eliminated constraint (7) and removed qcarry2′​ from the witness. Since we already needed to range-check q or q′, the total number of new constraints added is 4: 3 added to ForeignFieldMul and 1 added to multi-range-check gadget for constraining the decomposition of q01′​. This saves 2 rows per multiplication.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » An optimization","id":"176","title":"An optimization"},"177":{"body":"Due to the limited number of cells accessible to gates, we are not able to chain multiplications into multiplications. We can chain foreign field additions into foreign field multiplications, but currently do not support chaining multiplications into additions (though there is a way to do it).","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Chaining multiplications","id":"177","title":"Chaining multiplications"},"178":{"body":"Now we collect all of the checks that are required to constrain foreign field multiplication","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Constraining the computation","id":"178","title":"Constraining the computation"},"179":{"body":"If the a operand has not been constrained to [0,f) by any previous foreign field operations, then we constrain it like this Compute bound a′=a+f′ with addition gate (2 rows) ForeignFieldAdd Range check a′∈[0,2t) (4 rows) multi-range-check","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 1. Range constrain a","id":"179","title":"1. Range constrain a"},"18":{"body":"There are three common representations for polynomials used in computer implementations. Dense coefficient form. A degree d polynomial is represented as a vector of length d+1 of all the coefficients. Entry i of the vector corresponds to the coefficient ai​. This corresponds to the DensePolynomial type in arkworks. Sometimes this is also described as writing the polynomial \"in the monomial basis\", because it amounts to writing the polynomial as a linear combination of the monomials xi. Sparse coefficient form. If a polynomial does not have very many non-zero coefficients, the above representation is wasteful. In sparse coefficient form, you represent a polynomial as a vector (or potentially a hash-map) of pairs (usize, F) where F is the type of coefficients. The polynomial corresponding to the list [(i_0, b_0), ..., (i_n, b_n)] is b0​xi0​+⋯+bn​xin​ Evaluation form. We fix an index set A⊆F, with A={a0​,…,ad​}, and represent a polynomial f∈F[x]≤d​ as the vector [f(a_0), ..., f(a_d)]. By the fundamental theorem of polynomials, this is a valid way of representing the polynomials, since the coefficients form can be obtained by using the interpA​ function. The evaluation form is very important. The reason is that multiplying two polynomials in the evaluation form takes time O(n). You just multiply the two vectors entry-wise. By contrast, the coefficient forms naively require time O(n2) to multiply. Now, there is a trick. For certain sets A⊆F, we can efficiently translate between the dense coefficients form and the evaluation form. That is, for certain A, the functions interpA​ and evalA​ can be computed more efficiently than O(n2).","breadcrumbs":"Polynomials » Computer representation","id":"18","title":"Computer representation"},"180":{"body":"If the b operand has not been constrained to [0,f) by any previous foreign field operations, then we constrain it like this Compute bound b′=b+f′ with addition gate (2 rows) ForeignFieldAdd Range check b′∈[0,2t) (4 rows) multi-range-check","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 2. Range constrain b","id":"180","title":"2. Range constrain b"},"181":{"body":"The quotient q is constrained to [0,f) for each multiplication as part of the multiplication gate Compute bound q′=q+f′ with ForeignFieldMul constraints Range check q′∈[0,2t) (4 rows) multi-range-check Range check q≥0 ForeignFieldMul (implicit by storing q in witness)","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 3. Range constrain q","id":"181","title":"3. Range constrain q"},"182":{"body":"The remainder r is constrained to [0,f) for each multiplication using an external addition gate. Compute bound r′=r+f′ with addition gate (2 rows) ForeignFieldAdd Range check r′∈[0,2t) (4 rows) multi-range-check","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 4. Range constrain r","id":"182","title":"4. Range constrain r"},"183":{"body":"Compute and constrain the intermediate products p0​,p1​ and p2​ as: p0​=a0​⋅b0​+q0​⋅f0′​ ForeignFieldMul p1​=a0​⋅b1​+a1​⋅b0​+q0​⋅f1′​+q1​⋅f0′​ ForeignFieldMul p2​=a0​⋅b2​+a2​⋅b0​+a1​⋅b1​+q0​⋅f2′​+q2​⋅f0′​+q1​⋅f1′​ ForeignFieldMul where each of them is about 2ℓ-length elements.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 5. Compute intermediate products","id":"183","title":"5. Compute intermediate products"},"184":{"body":"Compute and constrain the native modulus values, which are used to check the constraints modulo n in order to apply the CRT an​=22ℓ⋅a2​+2ℓ⋅a1​+a0​modn bn​=22ℓ⋅b2​+2ℓ⋅b1​+b0​modn qn​=22ℓ⋅q2​+2ℓ⋅q1​+q0​modn rn​=22ℓ⋅r2​+2ℓ⋅r1​+r0​modn fn​=22ℓ⋅f2​+2ℓ⋅f1​+f0​modn","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 6. Native modulus checked computations","id":"184","title":"6. Native modulus checked computations"},"185":{"body":"Check that p1​=2ℓ⋅p11​+p10​: p1​=2ℓ⋅p11​+p10​ ForeignFieldMul Range check p10​∈[0,2ℓ) multi-range-check Range check p11​∈[0,2ℓ+2) p11​=p111​⋅2ℓ+p110​ ForeignFieldMul Range check p110​∈[0,2ℓ) multi-range-check Range check p111​∈[0,22) with a degree-4 constraint ForeignFieldMul","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 7. Decompose middle intermediate product","id":"185","title":"7. Decompose middle intermediate product"},"186":{"body":"Now we have to constrain the zero sum (p0​−r0​)+288(p1​−r1​)+2176(p2​−r2​)=0mod2t We constrain the first and the second halves as v0​⋅22ℓ=p0​+2ℓ⋅p10​−r0​−2ℓ⋅r1​ ForeignFieldMul v1​⋅2ℓ=(p111​⋅2ℓ+p110​)+p2​−r2​+v0​ ForeignFieldMul And some more range checks Check that v0​∈[0,22) with a degree-4 constraint ForeignFieldMul Check that v1​∈[0,2ℓ+3) Check v1​=v11​⋅288+v10​ ForeignFieldMul Check v11​∈[0,23] ForeignFieldMul Check v10​<2ℓ with range constraint multi-range-check To check that v11​∈[0,23) (i.e. that v11​ is at most 3 bits long) we first range-check v11​∈[0,212) with a 12-bit plookup. This means there can be no higher bits set beyond the 12-bits of v11​. Next, we scale v11​ by 29 in order to move the highest 12−3=9 bits beyond the 12th bit. Finally, we perform a 12-bit plookup on the resulting value. That is, we have Check v11​∈[0,212) with a 12-bit plookup (to prevent any overflow) Check scaledv11​​=29⋅v11​ Check scaledv11​​ is a 12-bit value with a 12-bit plookup Kimchi's plookup implementation is extremely flexible and supports optional scaling of the lookup target value as part of the lookup operation. Thus, we do not require two witness elements, two lookup columns, nor the scaledv11​​=29⋅v11​ custom constraint. Instead we can just store v11​ in the witness and define this column as a \"joint lookup\" comprised of one 12-bit plookup on the original cell value and another 12-bit plookup on the cell value scaled by 29, thus, yielding a 3-bit check. This eliminates one plookup column and reduces the total number of constraints.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 8. Zero sum for multiplication","id":"186","title":"8. Zero sum for multiplication"},"187":{"body":"Using the checked native modulus computations we constrain that an​⋅bn​−qn​⋅fn​−rn​=0modn.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 9. Native modulus constraint","id":"187","title":"9. Native modulus constraint"},"188":{"body":"Compute and constrain the intermediate sums s01​ and s2​ as: s01​=q01​+f01′​ s2​=q2​+f2′​ q01​=q0​+2ℓ⋅q1​ f01′​=f0′​+2ℓ⋅f1′​","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 10. Compute intermediate sums","id":"188","title":"10. Compute intermediate sums"},"189":{"body":"Check that q01′​=q0′​+2ℓ⋅q1′​. Done by (3) above with the multi-range-check on q′ q01′​=q0′​+2ℓ⋅q1′​ Range check q0′​∈[0,2ℓ) Range check q1′​∈[0,2ℓ)","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 11. Decompose the lower quotient bound","id":"189","title":"11. Decompose the lower quotient bound"},"19":{"body":"The algorithm that allows us to multiply polynomials in O(nlogn) is called the Cooley-Tukey fast Fourier transform , or FFT for short. It was discovered by Gauss 160 years earlier, but then separately rediscovered and publicized by Cooley-Tukey. The heart of Cooley-Tukey FFT is actually about converting between coefficient and evaluation representations, rather than the multiplication itself. Given polynomials p and q in dense coefficient representation, it works like this. Convert p and q from coefficient to evaluation form in O(nlogn) using Cooley-Tukey FFT Compute r=p∗q in evaluation form by multiplying their points pairwise in O(n) Convert r back to coefficient form in O(nlogn) using the inverse Cooley-Tukey FFT The key observation is that we can choose any n distinct evaluation points to represent any degree n−1 polynomial. The Cooley-Tukey FFT works by selecting points that yield an efficient FFT algorithm. These points are fixed and work for any polynomials of a given degree. The next section describes the Cooley-Tukey FFT in detail.","breadcrumbs":"Polynomials » Multiplying Polynomials » Multiplying polynomials","id":"19","title":"Multiplying polynomials"},"190":{"body":"We constrain that s01​−q01′​+22ℓ⋅(s2​−q2′​)=0mod2t by constraining the two halves 22ℓ⋅qcarry01′​=s01​−q01′​ 2ℓ⋅qcarry2′​=s2​+qcarry01′​−q2′​ We also need a couple of small range checks Check that qcarry01′​ is boolean ForeignFieldMul Check that q2′​ is boolean ForeignFieldMul","breadcrumbs":"Custom Gates » Foreign Field Multiplication » 12. Zero sum for quotient bound addition","id":"190","title":"12. Zero sum for quotient bound addition"},"191":{"body":"Based on the constraints above, we need the following 12 values copied from the range check gates. a0, a1, a2, b0, b1, b2, q0, q1, q2, r0, r1, r2 Since we need 12 copied values for the constraints, they must span 2 rows. The q>s01​>>>​=q01​+f01′​=q0​+2ℓ⋅q1​+f0′​+2ℓ⋅f1′​=q0​+f0′​+2ℓ⋅(q1′​+f1′​)=q0′​+2ℓ⋅q1′​>​> where q0′​=q0​+f0′​ and q1′​=q1​+f1′​ can be done with checked computations. Next, for applying the CRT we compute an​=22ℓ⋅a2​+2ℓ⋅a1​+a0​modn bn​=22ℓ⋅b2​+2ℓ⋅b1​+b0​modn qn​=22ℓ⋅q2​+2ℓ⋅q1​+q0​modn rn​=22ℓ⋅r2​+2ℓ⋅r1​+r0​modn fn​=22ℓ⋅f2​+2ℓ⋅f1​+f0​modn","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Checked computations","id":"192","title":"Checked computations"},"193":{"body":"In total we require the following checks p111​∈[0,22) v10​∈[0,2ℓ) multi-range-check p110​∈[0,2ℓ) multi-range-check p10​∈[0,2ℓ) multi-range-check p11​=2ℓ⋅p111​+p110​ p1​=2ℓ⋅p11​+p10​ v0​∈[0,22) 22ℓ⋅v0​=p0​+2ℓ⋅p10​−r0​−2ℓ⋅r1​ v11​∈[0,23) v1​=2ℓ⋅v11​+v10​ 2ℓ⋅v1​=v0​+p11​+p2​−r2​ an​⋅bn​−qn​⋅fn​=rn​ q0′​∈[0,2ℓ) multi-range-check q1′​∈[0,2ℓ) multi-range-check q2′​∈[0,2ℓ) multi-range-check q01′​=q0′​+2ℓ⋅q1′​ multi-range-check qcarry01′​∈[0,2) 22ℓ⋅qcarry01′​=s01​−q01′​ q2′​=s2​+qcarry01′​","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Checks","id":"193","title":"Checks"},"194":{"body":"These checks can be condensed into the minimal number of constraints as follows. First, we have the range-check corresponding to (1) as a degree-4 constraint C1: p111​⋅(p111​−1)⋅(p111​−2)⋅(p111​−3) Checks (2) - (4) are all handled by a single multi-range-check gadget. C2: multi-range-check v10​,p10​,p110​ Next (5) and (6) can be combined into C3: 2ℓ⋅(2ℓ⋅p111​+p110​)+p10​=p1​ Now we have the range-check corresponding to (7) as another degree-4 constraint C4: v0​⋅(v0​−1)⋅(v0​−2)⋅(v0​−3) Next we have check (8) C5: 22ℓ⋅v0​=p0​+2ℓ⋅p10​−r0​−2ℓ⋅r1​ Up next, check (9) is a 3-bit range check C6: Plookup v11​ (12-bit plookup scaled by 29) Now checks (10) and (11) can be combined into C7: 2ℓ⋅(v11​⋅2ℓ+v10​)=p2​+p11​+v0​−r2​ Next, for our use of the CRT, we must constrain that a⋅b=q⋅f+rmodn. Thus, check (12) is C8: an​⋅bn​−qn​⋅fn​=rn​ Next we must constrain the quotient bound addition. Checks (13) - (16) are all combined into multi-range-check gadget C9: multi-range-check q0′​,q1′​,q2′​ and q01′​=q0′​+2ℓ⋅q1′​. Check (17) is a carry bit boolean check C10: qcarry01′​⋅(qcarry01′​−1) Next, check (18) is C11: 22ℓ⋅qcarry10′​=s01​−q01′​ Finally, check (19) is C12: q2′​=s2​+qcarry01′​ The Zero gate has no constraints and is just used to hold values required by the ForeignFieldMul gate.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » Constraints","id":"194","title":"Constraints"},"195":{"body":"The following checks must be done with other gates to assure the soundness of the foreign field multiplication Range check input multi-range-check a multi-range-check b Range check witness data multi-range-check q′ and check q01′​=q0′​+2ℓq1′​ multi-range-check ℓ-bit limbs: p10​,p110​,p111​ Range check output multi-range-check either r or r′ Compute and constrain r′ the bound on r ForeignFieldAdd r+2t=1⋅f+r′ Copy constraints must connect the above witness cells to their respective input cells within the corresponding external check gates witnesses.","breadcrumbs":"Custom Gates » Foreign Field Multiplication » External checks","id":"195","title":"External checks"},"196":{"body":"The Keccak gadget is comprised of 3 circuit gates (Xor16, Rot64, and Zero) Keccak works with 64-bit words. The state is represented using 5×5 matrix of 64 bit words. Each compression step of Keccak consists of 24 rounds. Let us denote the state matrix with A (indexing elements as A[x,y]), from which we derive further states as follows in each round. Each round then consists of the following 5 steps: C[x]D[x]E[x,y]B[y,2x+3y]F[x,y]Fp[0,0]​=A[x,0]⊕A[x,1]⊕A[x,2]⊕A[x,3]⊕A[x,4]=C[x−1]⊕ROT(C[x+1],1)=A[x,y]⊕D[x]=ROT(E[x,y],ρ[x,y])=B[x,y]⊕((NOT B[x+1,y])AND B[x+2,y])=F[0,0]⊕RC​​ for 0≤x,y≤4 and ρ[x,y] is the rotation offset defined for Keccak. The values are in the table below extracted from the Keccak reference https://keccak.team/files/Keccak-reference-3.0.pdf x = 3 x = 4 x = 0 x = 1 x = 2 y = 2 155 231 3 10 171 y = 1 55 276 36 300 6 y = 0 28 91 0 1 190 y = 4 120 78 210 66 253 y = 3 21 136 105 45 15","breadcrumbs":"Custom Gates » Keccak » Keccak Gate","id":"196","title":"Keccak Gate"},"197":{"body":"The atomic operations are XOR, ROT, NOT, AND. In the sections below, we will describe the gates for these operations. Below are some common approaches followed in their design. To fit within 15 wires, we first decompose each word into its lower and upper 32-bit components. A gate for an atomic operation works with those 32-bit components at a time. Before we describe the specific gate design approaches, below are some constraints in the Kimchi framework that dictated those approaches. only 4 lookups per row only first 7 columns are available to the permutation polynomial","breadcrumbs":"Custom Gates » Keccak » Design Approach:","id":"197","title":"Design Approach:"},"198":{"body":"It is clear from the definition of the rotation gate that its constraints are complete (meaning that honest instances always satisfy the constraints). It is left to be proven that the proposal is sound. In this section, we will give a proof that as soon as we perform the range checks on the excess and shifted parts of the input, only one possible assignment satisfies the constraints. This means that there is no dishonest instance that can make the constraints pass. We will also give an example where one could find wrong rotation witnesses that would satisfy the constraints if we did not check the range.","breadcrumbs":"Custom Gates » Keccak » Rot64","id":"198","title":"Rot64"},"199":{"body":"First of all, we will illustrate the necessity of range-checks with a simple example. For the sake of readability, we will use some toy field lengths. In particular, let us assume that our words have 4 bits, meaning all of the elements between 0x0 and 0xF. Next, we will be using the native field F32​. As we will later see, this choice of field lengths is not enough to perform any 4-bit rotation, since the operations in the constraints would overflow the native field. Nonetheless, it will be sufficient for our example where we will only rotate by 1 bit. Assume we want to rotate the word 0b1101 (meaning 13) by 1 bit to the left. This gives us the rotated word 0b1011 (meaning 11). The excess part of the word is 0b1, whereas the shifted part corresponds to 0b1010. We recall the constraints for the rotation gate: word⋅2rotrotated​=excess⋅2len+shifted=excess+shifted​ Applied to our example, this results in the following equations: 13⋅211​=excess⋅16+shifted=excess+shifted​ We can easily check that the proposed values of the shifted 0b1010=10 and the excess 0b1=1 values satisfy the above constraint because 26=1⋅16+10 and 11=1+10. Now, the question is: can we find another value for excess and shifted, such that their addition results in an incorrect rotated word? The answer to this question is yes, due to diophantine equations . We basically want to find x,y such that 26=x⋅16+y( mod 32). The solution to this equation is: ∀k∈[0…31]:xy​=k ∧=26−16⋅k​ We chose these word and field lengths to better understand the behaviour of the solution. Here, we can see two \"classes\" of evaluations. If we choose an even k, then y will have the following shape: 26−16⋅(2⋅n)⟺26−32n≡32​26 Meaning, if x=2n then y=26. If on the other hand, we chose an odd k, then y will have the following shape instead: 26−16⋅(2⋅n+1)⟺26−32n−16≡32​26−16=10 Meaning, if x=2n+1 then y=10. Thus, possible solutions to the diophantine equation are: xy​=0,1,2,3,4,5…=26,10,26,10,26,10…​ Note that our valid witness is part of that set of solutions, meaning x=1 and y=10. Of course, we can also find another dishonest instantiation such as x=0 and y=26. Perhaps one could think that we do not need to worry about this case, because the resulting rotation word would be 0+26=26, and if we later use that result as an input to a subsequent gate such as XOR, the value 26 would not fit and at some point the constraint system would complain. Nonetheless, we still have other solutions to worry about, such as (x=3,y=10) or (x=5,y=10), since they would result in a rotated word that would fit in the word length of 4 bits, yet would be incorrect (not equal to 11). All of the above incorrect solutions differ in one thing: they have different bit lengths. This means that we need to range check the values for the excess and shifted witnesses to make sure they have the correct length.","breadcrumbs":"Custom Gates » Keccak » Necessity of range checks","id":"199","title":"Necessity of range checks"},"2":{"body":"Groups are an algebraic structure which are used in most widely deployed signature schemes are used in most widely deployed zk-SNARKs generalize the notion of a collection of symmetries of an object generalize certain aspects of numbers I will not touch on the last two too much. First let's do a quick definition and then some examples. Definition. A group is a set G with a binary operation ∗:G×G→G, an identity e∈G for that operation, and an inverse map ()−1:G→G such that ∗ is associative: x∗(y∗z)=(x∗y)∗z for all x,y,z. e is an identity for ∗. I.e., e∗x=x∗e=x for all x. ()−1 is an inverse map for ∗ with identity e. I.e., x∗x−1=x−1∗x=e for all x. So basically, an invertible binary operation. Definition in hand, we can see some examples: The integers Z with +, 0 as identity, and negation as inverse. For any natural number n, the integers mod n. This can be thought of as the set of numbers {0,…,n−1} with the operation being + followed by taking remainder mod n. It can also be thought of as the group of rotations of an n-gon. For any set X, the set of invertible functions X→X (a.k.a permutations) with function composition as the operation, the identity function as the identity, and function inverse as the inverse operation. The set of translations of a plane R×R with composition of translations (since the composition of translations is a translation). The identity is translating by (0,0) and inverse is reversing the translation. This is equivalent to the group R×R with coordinate-wise addition as the operation and coordinate-wise negation as the inverse. The set of rotations of a sphere with composition as the operation, doing nothing as the identity, and performing the rotation in reverse as the inverse map. For any field F, the set F×:=F∖{0}, with field multiplication as the operation, 1 as the identity, and x↦1/x as the inverse map. This is called the group of units of F. Sometimes, instead of x∗y we simply write xy when ∗ is clear from context.","breadcrumbs":"Groups » Groups","id":"2","title":"Groups"},"20":{"body":"This section describes how the Cooley-Tukey fast Fourier transform works. As we learned in the previous section, the key is to select evaluation points that yield an efficient FFT algorithm. Specifically, say we have ω∈F such that ωn=1, and ωr=1 for any 00, then we will split f into two polynomials, recursively call FFT on them, and reconstruct the result from the recursive calls. To that end, define f0​ to be the polynomial whose coefficients are all the even-index coefficients of f and f1​ the polynomial whose coefficients are all the odd-index coefficients of f. In terms of the array representation, this just means splitting out every other entry into two arrays. So that can be done in time O(n). Write f=∑i<2k​ci​xi, so that f0​=∑i<2k−1​c2i​xi and f1​=∑i<2k−1​c2i+1​xi. Then f(x)​=i<2k∑​ci​xi=i<2k−1∑​c2i​x2i+i<2k−1∑​c2i+1​x2i+1=i<2k−1∑​c2i​(x2)i+i<2k−1∑​c2i+1​x⋅(x2)i=i<2k−1∑​c2i​(x2)i+xi<2k−1∑​c2i+1​(x2)i=f0​(x2)+xf1​(x2)​ Now, notice that if ω is a 2kth root of unity, then ω2 is a 2k−1th root of unity. Thus we can recurse with FFT(k−1,ω2,f0​) and similarly for f1​. Let [e0,0​,…,e0,2k−1−1​][e1,0​,…,e1,2k−1−1​]​=FFT(k−1,ω2,f0​)=FFT(k−1,ω2,f1​)​ By assumption ei,j​=fi​((ω2)j). So, for any j we have f(ωj)​=f0​((ω2)j)+ωjf1​((ω2)j)​ Now, since j may be larger than 2k−1−1, we need to reduce it mod 2k−1, relying on the fact that if τ is an nth root of unity then τj=τjmodn since τn=1. Thus, (ω2)j=(ω2)jmod2k−1 and so we have f(ωj)​=f0​((ω2)jmod2k−1)+ωjf1​((ω2)jmod2k−1)=e0,jmod2k−1​+ωje1,jmod2k−1​​ We can compute the array W=[1,ω,…,ω2k−1] in time O(n) (since each entry is the previous entry times ω). Then we can compute each entry of the output in O(1) as f(ωj)​=e0,jmod2k−1​+W[j]⋅e1,jmod2k−1​​ There are n such entries, so this takes time O(n). This concludes the recursive definition of the algorithm FFT(k,ω,f). Algorithm: computing evalAk​​ Input f=[c0​,…,c2k−1​] the coefficients of polynomial f(x)=∑i<2k​ci​xi Compute W←[1,ω,ω2,...,ω2k−1] FFT(k,ω,f)→[f(1),f(ω),f(ω2)…,f(ω2k−1)] if k==0 return f else Compute f0​=[c0​,c2​,...,c2k−2​] the even coefficients of f, corresponding to f0​(x)=∑i<2k−1​c2i​xi Compute f1​=[c1​,c3​,...,c2k−1​] the odd coefficients of f, corresponding to f1​(x)=∑i<2k−1​c2i+1​xi e0​←FFT(k−1,ω2,f0​) e1​←FFT(k−1,ω2,f1​) for j∈[0,2k−1] Fj​←e0,jmod2k−1​+W[j]⋅e1,jmod2k−1​ return F Now let's analyze the time complexity. Let T(n) be the complexity on an instance of size n (that is, for n=2k). Looking back at what we have done, we have done O(n) for computing f0​ and f1​ two recursive calls, each of size n/2 O(n) for computing the powers of ω O(n) for combining the results of the recursive calls In total, this is O(n)+2T(n/2). Solving this recurrence yields T(n)=O(n)⋅logn=O(nlogn). Basically, there are logn recursions before we hit the base case, and each step takes time O(n). □ Now, in practice there are ways to describe this algorithm non-recursively that have better concrete performance, but that's out of scope for this document. Read the code if you are interested. Using the FFT algorithm to compute interpAk​​ So far we have a fast way to compute evalAk​​(f) all at once, where Ak​ is the set of powers of a 2kth root of unity ω. For convenience let n=2k. Now we want to go the other way and compute a polynomial given an array of evaluations. Specifically, n evaluations [f(x0​),f(x1​),...,f(xn−1​)] uniquely define a degree n−1 polynomial. This can be written as a system of n equations f(x0​)f(x1​)⋮f(xn−1​)​=c0​+c1​x0​+…+cn−1​x0n−1​=c0​+c1​x1​+…+cn−1​x1n−1​=c0​+c1​xn−1​+…+cn−1​xn−1n−1​,​ which can be rewritten as a matrix vector product. ​f(x0​)f(x1​)⋮f(xn−1​)​​=​11⋮1​x0​x1​⋮xn−1​​⋯⋯⋱⋯​x0n−1​x1n−1​⋮xn−1n−1​​​×​c0​c1​⋮cn−1​​​ This n×n matrix is a Vandermonde matrix and it just so happens that square Vandermonde matrices are invertible, iff the xi​ are unique. Since we purposely selected our xi​ to be the powers of ω, a primitive n-th root of unity, by definition xi​=ωi are unique. Therefore, to compute the polynomial given the corresponding array of evaluations (i.e. interpolation) we can solve for the polynomial's coefficients using the inverse of the matrix. ​c0​c1​⋮cn−1​​​=​11⋮1​1ω⋮ωn−1​⋯⋯⋱⋯​1n−1ωn−1⋮ω(n−1)(n−1)​​−1×​f(1)f(ω)⋮f(ωn−1)​​ All we need now is the inverse of this matrix, which is slightly complicated to compute. I'm going to skip it for now, but if you have the details please make a pull request. Substituting in the inverse matrix we obtain the equation for interpolation. ​c0​c1​⋮cn−1​​​=n1​​11⋮1​1ω−1⋮ω−(n−1)​⋯⋯⋱⋯​1n−1ω−(n−1)⋮ω−(n−1)(n−1)​​×​f(1)f(ω)⋮f(ωn−1)​​ Observe that this equation is nearly identical to the original equation for evaluation, except with the following substitution. ωi⇒n1​ω−1i Consequently and perhaps surprisingly, we can reuse the FFT algorithm evalAk​​ in order to compute the inverse-- interpAk​​. So, suppose we have an array [a0​,…,an−1​] of field elements (which you can think of as a function Ak​→F) and we want to compute the coefficients of a polynomial f with f(ωi)=ai​. To this end, define a polynomial g by g=∑j264, then b′ cannot be smaller than 264 as it was assumed. CONTRADICTION. b=b′ and a=a′: then 264(a−a′)≡n​0 and this can only happen if a′=a+kn. But since n>2r, then a′ cannot be smaller than 2r as it was assumed. CONTRADICTION. a=a′ and b=b′: then we have something like 264α+β≡n​0. This means β≡n​−264α=k⋅n−264α for any k. According to the assumption, both 0≤a<2r and 0≤a′<2r. This means, the difference α:=(a−a′) lies anywhere in between the following interval: 1−2r≤α≤2r−1 We plug in this interval to the above equation to obtain the following interval for β: k⋅n−264(1−2r)≤β≤k⋅n−264(2r−1) We look at this interval from both sides of the inequality: β≥kn−264+264+r and β≤kn+264−264+r and we wonder if kn−264+264+r≤kn+264−264+r is at all possible. We rewrite as follows: 264+r−2642⋅264+r264+r​≤264−264+r≤2⋅264≤264​ But this can only happen if r≤0, which is impossible since we assume 0p. We have a proof system (Kimchi) over Fp​ and Fq​, where commitments to public inputs are: Pp​=⟨e,H⟩∈Ep​(Fq​)Pq​=⟨e,G⟩∈Eq​(Fp​) Respectively. See Pasta Curves for more details. When referring to the Fq​-side we mean the proof system for circuit over the field Fq​.","breadcrumbs":"Deferred Computation » Deferred Computation","id":"217","title":"Deferred Computation"},"218":{"body":"In pickles-rs we have the notion of \"passing\" a variable (including the transcript) from one side to the other. e.g. when a field element α∈Fp​ needs to be used as a scalar on Fq​. This document explains what goes on \"under the hood\". Let us start by understanding why: Let v∈Fq​ be a scalar which we want to use to do both: Field arithmetic in Fq​ Scalar operations on Eq​(Fp​) In order to do so efficiently, we need to split these operations across two circuits (and therefore proofs) because: Emulating Fq​ arithmetic in Fp​ is very expensive, e.g. computing v⋅w requires O(log(q)2) multiplications over Fp​: 100's of gates for a single multiplication. Since Eq​(Fp​)⊆Fp​×Fp​ we cannot compute [v]⋅H∈Eq​(Fp​) over Fq​ efficiently, because, like before, emulating Fp​ arithmetic in Fq​ is very expensive...","breadcrumbs":"Deferred Computation » Public Inputs / Why Passing","id":"218","title":"Public Inputs / Why Passing"},"219":{"body":"The solution is to \"pass\" a value v between the two proofs, in other words to have two values v ∈Fp​ and v∈Fq​ which are equal as integers i.e. lift(v)=lift(v )∈Z: they represent \"the same number\". A naive first attempt would be to simply add v~∈Fp​ to the witness on the Fp​-side, however this has two problems: Insufficient Field Size: pq; always the case for any cycle of curves, p is only ≈q​ smaller than q, by Hasse. Now \"v \" is \"represented\" by the two values h ,l~∈Fp​. Note that no decomposition is nessary if the \"original value\" v was in Fp​, since Fq​ is big enough to hold the lift of any element in Fp​. Problem 2: Compute Commitment to the Public Input of other side To solve the binding issue we will add l,h to the public inputs on the Fp​-side, for simplicity we will describe the case where l,h are the only public inputs in the Fp​-side, which means that the commitment Pp​∈Ep​(Fq​) to the public inputs on the Fp​ side is: Pp​=[h]⋅Gh​+[l]⋅Gl​∈Ep​(Fq​) At this point it is important to note that Ep​ is defined over Fq​! Which means that we can compute Pp​∈Ep​(Fq​) efficiently on the Fq​-side! Therefore to enforce the binding, we: Add a sub-circuit which checks: Pp​=[h]⋅Gh​+[l]⋅Gl​∈Ep​(Fq​),∣h∣=⌊logp⌋∣l∣=1 Add Pp​=(x,y)∈Fq​×Fq​ to the public input on the Fq​-side.","breadcrumbs":"Deferred Computation » Solution","id":"219","title":"Solution"},"22":{"body":"Implement types DensePolynomial and Evaluations that wrap a Vec and implement the FFT algorithms described above for converting between them Familiarize yourself with the types and functions provided by ark_poly","breadcrumbs":"Polynomials » Fast Fourier Transform » Exercises","id":"22","title":"Exercises"},"220":{"body":"At this point the statement of the proof in Fq​-side is: the Fq​-proof is sound, condition on providing an opening of Pp​ that satisifies the Fp​-relation. At this point you can stop and verify the proof (in the case of a \"step proof\" you would), by recomputing Pp​ outside the circuit while checking the Fp​-relation manually \"in the clear\". However, when recursing (e.g. wrapping in a \"wrap proof\") we need to \"ingest\" this public input Pp​; after all, to avoid blowup in the proof size everything (proofs/accumulators/public inputs etc.) must eventually become part of the witness and every computation covered by a circuit... To this end, the wrap proof is a proof for the Fp​-relation with the public input Pp​ which additionally verifies the Fq​-proof. The next \"step\" proof then verifies this wrap proof which means that Pp​ then becomes part of the witness!","breadcrumbs":"Deferred Computation » We recurse onwards...","id":"220","title":"We recurse onwards..."},"221":{"body":"We can arbitrarily choose which side should compute the public input of the other, in pickles we let \"wrap\" compute the commitment to the public input.","breadcrumbs":"Deferred Computation » In Pickles","id":"221","title":"In Pickles"},"222":{"body":"Enforces that the public input of the proof verified on the Fr side is equal to the Fp input computed on Fr side.","breadcrumbs":"Deferred Computation » Enforcing Equality","id":"222","title":"Enforcing Equality"},"223":{"body":"This section contains a series of diagrams giving an overview of implementation of pickles, closely following the structures and abstractions in the code. However, they are very technical and are primarily intended for developer use. The primary source of all the images in this section is pickles_structure.drawio file, and if one edits it one must re-generate the .svg renders by doing \"export -> selection\". The legend of the diagrams is quite straightforward: The black boxes are data structures that have names and labels following the implementation. MFNStep/MFNWrap is an abbreviation from MessagesForNextStep and MessagesForNextWrap that is used for brevity. Most other datatypes are exactly the same as in the codebase. The blue boxes are computations. Sometimes, when the computation is trivial or only vaguely indicated, it is denoted as a text sign directly on an arrow. Arrows are blue by default and denote moving a piece of data from one place to another with no (or very little) change. Light blue arrows are denoting witness query that is implemented through the handler mechanism. The \"chicken foot\" connector means that this arrow accesses just one field in an array: such an arrow could connect e.g. a input field of type old_a: A in a structure Vec<(A,B)> to an output new_a: A, which just means that we are inside a for loop and this computation is done for all the elemnts in the vector/array. Colour of the field is sometimes introduced and denotes how many steps ago was this piece of data created. The absense of the colour means either that (1) the data structure contains different subfields of different origin, or that (2) it was not coloured but it could be. The colours are assigned according to the following convention:","breadcrumbs":"Technical Diagrams » Pickles Technical Diagrams","id":"223","title":"Pickles Technical Diagrams"},"224":{"body":"The following is the diagram that explains the Wrap computation. The left half of it corresponds to the wrap.ml file and the general logic of proof creation, while the right part of it is wrap_main.ml/wrap_verifier.ml and explains in-circuit computation. This diagram explains the deferred_values computation which is in the heart of wrap.ml represented as a blue box in the middle of the left pane of the main Wrap diagram. Deferred values for Wrap are computed as follows:","breadcrumbs":"Technical Diagrams » Wrap Computatiton and Deferred Values","id":"224","title":"Wrap Computatiton and Deferred Values"},"225":{"body":"The following is the diagram that explains the Step computation, similarly to Wrap. The left half of it corresponds to the general logic in step.ml, and the right part of it is step_main.ml and explains in-circuit computation. We provide no deferred_values computation diagram for Step, but it is very conceptually similar to the one already presented for Wrap.","breadcrumbs":"Technical Diagrams » Step Computation","id":"225","title":"Step Computation"},"226":{"body":"Poseidon is a hash function that can efficiently run in a zk circuit. (See poseidon-hash.info ) It is based on the sponge function , with a state composed of field elements and a permutation based on field element operation (addition and exponentiation). The permutation contains an S-box (exponentiation of a group element), adding constants to the state, and matrix multiplication of the state (multiplications and additions) with an MDS matrix . Since a field element is around 255-bit, a single field element is enough as the capacity of the sponge to provide around 116-bit security. Note We might want to piggy back on the zcash poseidon spec at some point (perhaps by making this an extension of the zcash poseidon spec).","breadcrumbs":"Poseidon hash » Poseidon hash function","id":"226","title":"Poseidon hash function"},"227":{"body":"We define a base sponge, and a scalar sponge. Both must be instantiated when verifying a proof (this is due to recursion-support baked in Kimchi). External users of kimchi (or pickles) are most likely to interact with a wrap proof (see the pickles specification ). As such, the sponges they need to instantiate are most likely to be instantiated with: Poseidon-Fp for base sponge Poseidon-Fq for the scalar sponge","breadcrumbs":"Poseidon hash » APIs","id":"227","title":"APIs"},"228":{"body":"new(params) -> BaseSponge. Creates a new base sponge. BaseSponge.absorb(field_element). Absorbs a field element by calling the underlying sponge absorb function. BaseSponge.absorb_point(point). Absorbs an elliptic curve point. If the point is the point at infinity, absorb two zeros. Otherwise, absorb the x and y coordinates with two calls to absorb. BaseSponge.absorb_scalar(field_element_of_scalar_field). Absorbs a scalar. If the scalar field is smaller than the base field (e.g. Fp is smaller than Fq), then the scalar is casted to a field element of the base field and absorbed via absorb. Otherwise, the value is split between its least significant bit and the rest. Then both values are casted to field elements of the base field and absorbed via absorb (the high bits first, then the low bit). BaseSponge.digest() -> field_element. The squeeze function of the underlying sponge function is called and the first field element is returned. BaseSponge.digest_scalar() -> field_element_of_scalar_field. BaseSponge.challenge // TODO: specify. BaseSponge.challenge_fq // TODO: specify.","breadcrumbs":"Poseidon hash » Base Sponge","id":"228","title":"Base Sponge"},"229":{"body":"new(params) -> ScalarSponge ScalarSponge.absorb(scalar_field_element) ScalarSponge.digest() -> scalar_field_element ScalarSponge.challenge // TODO: specify","breadcrumbs":"Poseidon hash » Scalar Sponge","id":"229","title":"Scalar Sponge"},"23":{"body":"A \"commitment scheme\" is a cryptographic scheme that lets us provide a \"commitment\" to a given piece of data, in such a way that we can later \"open\" that commitment. I recommend checking out section 2.4.1 of David's book Real-World Cryptography. There are two properties we typically expect of our commitment schemes: Hiding. A commitment to a piece of data should not reveal anything about that data. Binding. There should be no ambiguity about what data is committed to. That is, it should not be possible for the committer to open the commitment to a piece of data other than the one they originally committed to. There are various kinds of commitments that allow opening in different ways, revealing only part of the data, or some function of the data. Sometimes it is even useful to elide the hiding property entirely--- so-called non-hiding commitments.","breadcrumbs":"Commitments » Commitments","id":"23","title":"Commitments"},"230":{"body":"Note that every operation is done in the field of the sponge. In this section we define the high-level algorithm behind the permutation and the sponge. The permutation is never used directly by users, it is used only by the sponge function.","breadcrumbs":"Poseidon hash » Algorithms","id":"230","title":"Algorithms"},"231":{"body":"In pseudo-code: def sbox(field_element): # modular exponentiation return field_element^7 # apply MDS matrix\ndef apply_mds(state): n = [0, 0, 0] n[0] = state[0] * mds[0][0] + state[1] * mds[0][1] + state[2] * mds[0][2] n[1] = state[0] * mds[1][0] + state[1] * mds[1][1] + state[2] * mds[1][2] n[2] = state[0] * mds[2][0] + state[1] * mds[2][1] + state[2] * mds[2][2] return n # a round in the permutation\ndef apply_round(round, state): # sbox state[0] = sbox(state[0]) state[1] = sbox(state[1]) state[2] = sbox(state[2]) # apply MDS matrix state = apply_mds(state) # add round constant state[0] += round_constants[round][0] state[1] += round_constants[round][1] state[2] += round_constants[round][2] # the permutation\ndef permutation(state): round_offset = 0 if ARK_INITIAL: constant = round_constants[0] state[0] += constant[0] state[1] += constant[1] state[2] += constant[2] round_offset = 1 for round in range(round_offset, ROUNDS + round_offset): apply_round(round, state)","breadcrumbs":"Poseidon hash » Permutation","id":"231","title":"Permutation"},"232":{"body":"In pseudo-code: def new(): return { \"state\": [0] * RATE, # `RATE` field elements set to 0 \"mode\": \"absorbing\", \"offset\": 0, } def absorb(sponge, field_element): # if we're changing mode, reset the offset if sponge.mode == \"squeezing\": sponge.mode = \"absorbing\" sponge.offset = 0 # we reached the end of the rate, permute elif sponge.offset == RATE: sponge.state = permutation(sponge.state) sponge.offset = 0 # absorb by adding to the state sponge.state[sponge.offset] += field_element sponge.offset += 1 def squeeze(sponge): # permute when changing mode or when we already squeezed everything if sponge.mode == \"absorbing\" or sponge.offset == RATE: sponge.mode = \"squeezing\" sponge.state = permutation(sponge.state) sponge.offset = 0 result = sponge.state[sponge.offset] sponge.offset += 1 return result","breadcrumbs":"Poseidon hash » Sponge","id":"232","title":"Sponge"},"233":{"body":"We instantiate two versions of Poseidon, one for the field Fp, and one for the field Fq (see the pasta specification ). Both share the following sponge configuration: capacity 1 rate: 2 and the following permutation configuration: number of full rounds: 55 sbox: 7 ARK_INITIAL: false","breadcrumbs":"Poseidon hash » Instantiations","id":"233","title":"Instantiations"},"234":{"body":"You can find the MDS matrix and round constants we use in /poseidon/src/pasta/fp_kimchi.rs .","breadcrumbs":"Poseidon hash » Poseidon-Fp","id":"234","title":"Poseidon-Fp"},"235":{"body":"You can find the MDS matrix and round constants we use in /poseidon/src/pasta/fp_kimchi.rs .","breadcrumbs":"Poseidon hash » Poseidon-Fq","id":"235","title":"Poseidon-Fq"},"236":{"body":"We have test vectors contained in /poseidon/tests/test_vectors/kimchi.json .","breadcrumbs":"Poseidon hash » Test vectors","id":"236","title":"Test vectors"},"237":{"body":"our ocaml implementation: https://github.com/minaprotocol/mina/blob/develop/src/lib/random_oracle/random_oracle.mli relies on random_oracle_input: https://github.com/minaprotocol/mina/blob/develop/src/lib/random_oracle_input/random_oracle_input.ml is instantiated with two types of fields: https://github.com/minaprotocol/mina/blob/develop/src/nonconsensus/snark_params/snark_params_nonconsensus.ml","breadcrumbs":"Poseidon hash » Pointers to the OCaml code","id":"237","title":"Pointers to the OCaml code"},"238":{"body":"Our polynomial commitment scheme is a bootleproof-type of commitment scheme, which is a mix of: The polynomial commitment scheme described in appendix A.1 of the PCD paper . The zero-knowledge opening described in section 3.1 of the HALO paper .","breadcrumbs":"Polynomial Commitment » Kimchi Polynomial Commitment","id":"238","title":"Kimchi Polynomial Commitment"},"239":{"body":"The two curves pallas and vesta (pa(llas ve)sta) created by the Zcash team. Each curve's scalar field is the other curve's base field, which is practical for recursion (see Pickles). supporting evidence mina parameters: https://github.com/o1-labs/proof-systems/tree/master/curves/src/pasta arkworks ark-pallas , pallas and vesta Note that in general Fq refers to the base field (in which the curve is defined over), while Fr refers to the scalar field (defined by the order of the curve). But in our code, because of the cycles: Fp refers to the base field of Pallas, and the scalar field of Vesta Fq refers to the base field of Vesta, and the scalar field of Pallas Note that ∣Fq​∣>∣Fp​∣. In pickles: Vesta is also referred to as the step curve, or the tick curve. Pallas is also referred to as the wrap curve, or the tock curve.","breadcrumbs":"Pasta Curves » Pasta Curves","id":"239","title":"Pasta Curves"},"24":{"body":"The simplest kind of a commitment is one in which opening reveals all of the underlying data. Let's give a simple construction of such a scheme for committing to field elements. Suppose we have Fp​ a prime order field, with p being rather large (say on the order of 2256). A hash function h:List(Fp​)→Fp​. Then we define commit(x,r)=h([x,r])open(c)=(x,r)verify(c,x,r)=true/false The argument r of the commit function is data that must only be known to the committer (until the commitment is opened). When a committer wants to commit to a field element x, they sample a random \"blinder\" r∈Fp​ and hash it together with x to form the commitment. To open the commitment c, they simply provide the committed value together with the blinder. Alternatively, if the verifier already knows x, they can just provide r, i.e. open(x,c)=r. Finally, given the commitment and the opening, we can verify whether the input was the value originally committed to using the verify function. If the hash function is collision-resistant, then this function is binding (because there's no way the committer could find another preimage of h([x,r])). If the hash function is one-way, then this commitment is also hiding (assuming x is revealed only as part of the opening). Commitments are often used in protocols between provers and verifiers. The following illustration provides an example with a prover named Peggy and a verifier named Victor. Peggyinputblinder​ }→commit​ commitment→open​ ​ ⟶⋮ ⟶⋮ ​Victorcommitment input,blinder commitmentinputblinder​ }→verify​→true/false​ ​ Here Peggy commits to an input using a blinder, obtains the commitment and sends it to Victor. The interlocutors continue their protocol, but eventually to convince Victor of her claims, Peggy must send the opening proof to her earlier commitment. Victor verifies the opening (i.e. the input and blinder) against the commitment. If the verification fails, then Victor knows that Peggy was trying to trick him, otherwise Victor has sufficient assurances that Peggy was telling the truth.","breadcrumbs":"Commitments » Simple commitments","id":"24","title":"Simple commitments"},"240":{"body":"curve equation: y2=x3+5 base field: 28948022309329048855892746252171976963363056481941560715954676764349967630337 scalar field: 28948022309329048855892746252171976963363056481941647379679742748393362948097 mina generator: (1,12418654782883325593414442427049395787963493412651469444558597405572177144507) arkworks generator: (−1,2) endo: endo_q = 2D33357CB532458ED3552A23A8554E5005270D29D19FC7D27B7FD22F0201B547\nendo_r = 397E65A7D7C1AD71AEE24B27E308F0A61259527EC1D4752E619D1840AF55F1B1 You can use sage to test this: Fp = GF(28948022309329048855892746252171976963363056481941560715954676764349967630337)\nPallas = EllipticCurve(Fp, [0, 5])\nPallas.count_points()","breadcrumbs":"Pasta Curves » Pallas","id":"240","title":"Pallas"},"241":{"body":"curve equation: y2=x3+5 base field: 28948022309329048855892746252171976963363056481941647379679742748393362948097 scalar field: 28948022309329048855892746252171976963363056481941560715954676764349967630337 mina generator: (1,11426906929455361843568202299992114520848200991084027513389447476559454104162) arkworks generator: (−1,2) endo: endo_q = 06819A58283E528E511DB4D81CF70F5A0FED467D47C033AF2AA9D2E050AA0E4F\nendo_r = 12CCCA834ACDBA712CAAD5DC57AAB1B01D1F8BD237AD31491DAD5EBDFDFE4AB9 You can use sage to test this: Fq = GF(28948022309329048855892746252171976963363056481941647379679742748393362948097)\nVesta = EllipticCurve(Fq, [0, 5])\nVesta.count_points()","breadcrumbs":"Pasta Curves » Vesta","id":"241","title":"Vesta"},"242":{"body":"This document specifies kimchi , a zero-knowledge proof system that's a variant of PLONK. This document does not specify how circuits are created or executed, but only how to convert a circuit and its execution into a proof. Table of content: Overview Tables used to describe a circuit Tables produced during proof creation Dependencies Polynomial Commitments Poseidon hash function Pasta Constraints Linearization Permutation Lookup The Lookup Tables The Lookup Selectors Producing the sorted table as the prover Gates Double Generic Gate Poseidon Elliptic Curve Addition Endo Scalar Endo Scalar Multiplication Scalar Multiplication Range Check Foreign Field Addition Foreign Field Multiplication Rotation Xor Gadgets Not And Setup Common Index Lookup Index Prover Index Verifier Index Proof Construction & Verification Proof Structure Proof Creation Proof Verification Fiat-Shamir argument Partial verification Batch verification of proofs Optimizations Security Considerations","breadcrumbs":"Kimchi » Kimchi","id":"242","title":"Kimchi"},"243":{"body":"There are three main algorithms to kimchi: Setup : takes a circuit and produces a prover index, and a verifier index. Proof creation : takes the prover index, and the execution trace of the circuit to produce a proof. Proof verification : takes the verifier index and a proof to verify. As part of these algorithms, a number of tables are created (and then converted into polynomials) to create a proof.","breadcrumbs":"Kimchi » Overview","id":"243","title":"Overview"},"244":{"body":"The following tables are created to describe the circuit: Gates . A circuit is described by a series of gates, that we list in a table. The columns of the tables list the gates, while the rows are the length of the circuit. For each row, only a single gate can take a value 1 while all other gates take the value 0. row Generic Poseidon CompleteAdd VarBaseMul EndoMul EndoMulScalar 0 1 0 0 0 0 0 1 0 1 0 0 0 0 Coefficients . The coefficient table has 15 columns, and is used to tweak the gates. Currently, only the Generic and the Poseidon gates use it (refer to their own sections to see how). All other gates set their values to 0. row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 / / / / / / / / / / / / / / / Wiring (or Permutation, or sigmas) . For gates to take the outputs of other gates as inputs, we use a wiring table to wire registers together. To learn about registers, see the next section. It is defined at every row, but only for the first 7 registers. Each cell specifies a (row, column) tuple that it should be wired to. Cells that are not connected to another cell are wired to themselves. Note that if three or more registers are wired together, they must form a cycle. For example, if register (0, 4) is wired to both registers (80, 6) and (90, 0) then you would have the following table: row 0 1 2 3 4 5 6 0 0,0 0,1 0,2 0,3 80,6 0,5 0,6 ... 80 80,0 80,1 80,2 80,3 80,4 80,5 90,0 ... 90 0,4 90,1 90,2 90,3 90,4 90,5 90,6 The lookup feature is currently optional, as it can add some overhead to the protocol. In the case where you would want to use lookups, the following tables would be needed: Lookup Tables . The different lookup tables that are used in the circuit. For example, the XOR lookup table: l r o 1 0 1 0 1 1 1 1 0 0 0 0 Lookup selectors . A lookup selector is used to perform a number of queries in different lookup tables. Any gate can advertise its use of a lookup selector (so a lookup selector can be associated to several gates), and on which rows they want to use them (current and/or next). In cases where a gate need to use lookups in its current row only, and is the only one performing a specific combination of queries, then its gate selector can be used in place of a lookup selector. As with gates, lookup selectors (including gates used as lookup selectors) are mutually exclusives (only one can be used on a given row). For example, suppose we have two lookup selectors: row ChaChaQuery ChaChaFinalQuery 0 0 0 1 1 0 Where each applies 4 queries. A query is a table describing which lookup table it queries, and the linear combination of the witness to use in the query. For example, the following table describes a query into the XOR table made out of linear combinations of registers (checking that r0​⊕r2​=2⋅r1​): table_id l r o XOR 1, r0 1, r2 2, r1","breadcrumbs":"Kimchi » Tables used to describe a circuit","id":"244","title":"Tables used to describe a circuit"},"245":{"body":"The following tables are created by the prover at runtime: Registers (or Witness) . Registers are also defined at every row, and are split into two types: the IO registers from 0 to 6 usually contain input or output of the gates (note that a gate can output a value on the next row as well). I/O registers can be wired to each other (they'll be forced to have the same value), no matter what row they're on (for example, the register at row:0, col:4 can be wired to the register at row:80, col:6). The rest of the registers, 7 through 14, are called advice registers as they can store values that useful only for the row's active gate. Think of them as intermediary or temporary values needed in the computation when the prover executes a circuit. row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 / / / / / / / / / / / / / / / Wiring (Permutation) trace . You can think of the permutation trace as an extra register that is used to enforce the wiring specified in the wiring table. It is a single column that applies on all the rows as well, which the prover computes as part of a proof. row pt 0 / Queries trace . These are the actual values made by queries, calculated by the prover at runtime, and used to construct the proof. Table trace . Represents the concatenation of all the lookup tables, combined into a single column at runtime by both the prover and the verifier. Sorted trace . Represents the processed (see the lookup section) concatenation of the queries trace and the table trace. It is produced at runtime by the prover. The sorted trace is long enough that it is split in several columns. Lookup (aggregation, or permutation) trace . This is a one column table that is similar to the wiring (permutation) trace we talked above. It is produced at runtime by the prover.","breadcrumbs":"Kimchi » Tables produced during proof creation","id":"245","title":"Tables produced during proof creation"},"246":{"body":"To specify kimchi, we rely on a number of primitives that are specified outside of this specification. In this section we list these specifications, as well as the interfaces we make use of in this specification.","breadcrumbs":"Kimchi » Dependencies","id":"246","title":"Dependencies"},"247":{"body":"Refer to the specification on polynomial commitments . We make use of the following functions from that specification: PolyCom.non_hiding_commit(poly) -> PolyCom::NonHidingCommitment PolyCom.commit(poly) -> PolyCom::HidingCommitment PolyCom.evaluation_proof(poly, commitment, point) -> EvaluationProof PolyCom.verify(commitment, point, evaluation, evaluation_proof) -> bool","breadcrumbs":"Kimchi » Polynomial Commitments","id":"247","title":"Polynomial Commitments"},"248":{"body":"Refer to the specification on Poseidon . We make use of the following functions from that specification: Poseidon.init(params) -> FqSponge Poseidon.update(field_elem) Poseidon.finalize() -> FieldElem specify the following functions on top: Poseidon.produce_challenge() (TODO: uses the endomorphism) Poseidon.to_fr_sponge() -> state_of_fq_sponge_before_eval, FrSponge With the current parameters: S-Box alpha: 7 Width: 3 Rate: 2 Full rounds: 55 Round constants: fp_kimchi , fq_kimchi MDS matrix: fp_kimchi , fq_kimchi","breadcrumbs":"Kimchi » Poseidon hash function","id":"248","title":"Poseidon hash function"},"249":{"body":"Kimchi is made to work on cycles of curves, so the protocol switch between two fields Fq and Fr, where Fq represents the base field and Fr represents the scalar field. See the Pasta curves specification .","breadcrumbs":"Kimchi » Pasta","id":"249","title":"Pasta"},"25":{"body":"Instead of a cryptographic hash function, we can use elliptic curve scalar multiplication to construct a commitment scheme. Here scalar multiplication is used like a one-way function based on the hardness assumption of the elliptic curve discrete logarithm problem (ECDLP). Suppose we have Fp​ a prime order field, with p being large (e.g. something like 2256). Publicly agreed generator point G over an elliptic curve E(Fp​) Another publicly agreed curve point H for which no one knows the discrete logarithm commit(x,r)=xG+rHopen(c)=(x,r) where x∈Fp​ is the value being committed to, r∈Fp​ is a random blinding factor and the commitment c=commit(x,r) is a curve point. These commitments are algebraic (i.e. they do not use a boolean-based cryptographic hash function) and have homomorphic properties: you can add commitments together to form another commitment of the added committed values. For example, if you have commitments A and B, you can perform: A+B​=commit(xa​,ra​)+commit(xb​,rb​)=xa​G+ra​H+xb​G+rb​H=(xa​+xb​)G+(ra​+rb​)H=commit(xa​+xb​,ra​+rb​)​ In other words, the sum of commitments A and B is equal to the commitment of the sum of the two committed values xa​ and xb​ and blinders ra​ and rb​. This is possible because in such a scheme scaling is commutative with adding scalars. As a cryptographic primitive, the ability to find a public curve point H for which no one knows the discrete logarithm may, at first, seem rather mind-blowing and powerful. Actually, it's as easy as it is awesome to find such a point--- simply perform rejection sampling by cryptographically hashing G (or, respectively, the hash output), using the output as the x-coordinate of a candidate point on E and checking whether it's valid. The first valid curve point obtained is H and by the hardness assumption of the ECDLP, no one knows it. Since approximately half of the hash outputs will be valid curve points on E, sampling will terminate very quickly. Indeed, as we will see later, this process can be used to sample many public curve points G1​,…,Gn​ for which the discrete logarithms are unknown; the so-called hash to curve algorithm.","breadcrumbs":"Commitments » Algebraic and homomorphic commitments","id":"25","title":"Algebraic and homomorphic commitments"},"250":{"body":"Kimchi enforces the correct execution of a circuit by creating a number of constraints and combining them together. In this section, we describe all the constraints that make up the main polynomial f once combined. We define the following functions: combine_constraints(range_alpha, constraints), which takes a range of contiguous powers of alpha and a number of constraints. It returns the sum of all the constraints, where each constraint has been multiplied by a power of alpha. In other words it returns: i∑​αi⋅constrainti​ The different ranges of alpha are described as follows: gates . Offset starts at 0 and 21 powers of α are used Permutation . Offset starts at 21 and 3 powers of α are used Note As gates are mutually exclusive (a single gate is used on each row), we can reuse the same range of powers of alpha across all the gates.","breadcrumbs":"Kimchi » Constraints","id":"250","title":"Constraints"},"251":{"body":"TODO","breadcrumbs":"Kimchi » Linearization","id":"251","title":"Linearization"},"252":{"body":"The permutation constraints are the following 4 constraints: The two sides of the coin (with shift0​=1): ​z(x)⋅zkpm(x)⋅αPERM0⋅(w0​(x)+β⋅shift0​x+γ)⋅(w1​(x)+β⋅shift1​x+γ)⋅(w2​(x)+β⋅shift2​x+γ)⋅(w3​(x)+β⋅shift3​x+γ)⋅(w4​(x)+β⋅shift4​x+γ)⋅(w5​(x)+β⋅shift5​x+γ)⋅(w6​(x)+β⋅shift6​x+γ)​​ and ​−1⋅z(xω)⋅zkpm(x)⋅αPERM0⋅(w0​(x)+β⋅σ0​(x)+γ)⋅(w1​(x)+β⋅σ1​(x)+γ)⋅(w2​(x)+β⋅σ2​(x)+γ)⋅(w3​(x)+β⋅σ3​(x)+γ)⋅(w4​(x)+β⋅σ4​(x)+γ)⋅(w5​(x)+β⋅σ5​(x)+γ)⋅(w6​(x)+β⋅σ6​(x)+γ)⋅​​ the initialization of the accumulator: (z(x)−1)L1​(x)αPERM1 and the accumulator's final value: (z(x)−1)Ln−k​(x)αPERM2 You can read more about why it looks like that in this post . The quotient contribution of the permutation is split into two parts perm and bnd. They will be used by the prover. perm(x)=​aPERM0⋅zkpl(x)⋅[z(x)⋅(w0​(x)+γ+x⋅β⋅shift0​)⋅(w1​(x)+γ+x⋅β⋅shift1​)⋅(w2​(x)+γ+x⋅β⋅shift2​)⋅(w3​(x)+γ+x⋅β⋅shift3​)⋅(w4​(x)+γ+x⋅β⋅shift4​)⋅(w5​(x)+γ+x⋅β⋅shift5​)⋅(w6​(x)+γ+x⋅β⋅shift6​)⋅−z(x⋅w)⋅(w0​(x)+γ+σ0​⋅β)⋅(w1​(x)+γ+σ1​⋅β)⋅(w2​(x)+γ+σ2​⋅β)⋅(w3​(x)+γ+σ3​⋅β)⋅(w4​(x)+γ+σ4​⋅β)⋅(w5​(x)+γ+σ5​⋅β)⋅(w6​(x)+γ+σ6​⋅β)⋅]​​ and bnd: bnd(x)=aPERM1⋅x−1z(x)−1​+aPERM2⋅x−sid[n−k]z(x)−1​ The linearization: scalar⋅σ6​(x) where scalar is computed as: z(ζω)βαPERM0zkpl(ζ)⋅(γ+βσ0​(ζ)+w0​(ζ))⋅(γ+βσ1​(ζ)+w1​(ζ))⋅(γ+βσ2​(ζ)+w2​(ζ))⋅(γ+βσ3​(ζ)+w3​(ζ))⋅(γ+βσ4​(ζ)+w4​(ζ))⋅(γ+βσ5​(ζ)+w5​(ζ))⋅​​ To compute the permutation aggregation polynomial, the prover interpolates the polynomial that has the following evaluations. The first evaluation represents the initial value of the accumulator: z(g0)=1 For i=0,⋅,n−4, where n is the size of the domain, evaluations are computed as: z(gi+1)=z1​/z2​ with z1​=​ (w0​(gi+sid(gi)⋅beta⋅shift0​+γ)⋅ (w1​(gi)+sid(gi)⋅beta⋅shift1​+γ)⋅ (w2​(gi)+sid(gi)⋅beta⋅shift2​+γ)⋅ (w3​(gi)+sid(gi)⋅beta⋅shift3​+γ)⋅ (w4​(gi)+sid(gi)⋅beta⋅shift4​+γ)⋅ (w5​(gi)+sid(gi)⋅beta⋅shift5​+γ)⋅ (w6​(gi)+sid(gi)⋅beta⋅shift6​+γ)​​ and z2​=​ (w0​(gi)+σ0​⋅beta+γ)⋅ (w1​(gi)+σ1​⋅beta+γ)⋅ (w2​(gi)+σ2​⋅beta+γ)⋅ (w3​(gi)+σ3​⋅beta+γ)⋅ (w4​(gi)+σ4​⋅beta+γ)⋅ (w5​(gi)+σ5​⋅beta+γ)⋅ (w6​(gi)+σ6​⋅beta+γ)​​ We randomize the evaluations at n - zk_rows + 1 and n - zk_rows + 2 in order to add zero-knowledge to the protocol. For a valid witness, we then have have z(gn−zkr​ows)=1.","breadcrumbs":"Kimchi » Permutation","id":"252","title":"Permutation"},"253":{"body":"Lookups in kimchi allows you to check if a single value, or a series of values, are part of a table. The first case is useful to check for checking if a value belongs to a range (from 0 to 1,000, for example), whereas the second case is useful to check truth tables (for example, checking that three values can be found in the rows of an XOR table) or write and read from a memory vector (where one column is an index, and the other is the value stored at that index). Note Similarly to the generic gate, each values taking part in a lookup can be scaled with a fixed field element. The lookup functionality is an opt-in feature of kimchi that can be used by custom gates. From the user's perspective, not using any gates that make use of lookups means that the feature will be disabled and there will be no overhead to the protocol. Please refer to the lookup RFC for an overview of the lookup feature. In this section, we describe the tables kimchi supports, as well as the different lookup selectors (and their associated queries) The Lookup Tables Kimchi currently supports two lookup tables: /// The table ID associated with the XOR lookup table.\npub const XOR_TABLE_ID: i32 = 0; /// The range check table ID.\npub const RANGE_CHECK_TABLE_ID: i32 = 1; XOR The lookup table for 4-bit xor. Note that it is constructed so that (0, 0, 0) is the last position in the table. This is because tables are extended to the full size of a column (essentially) by padding them with their final value. And, having the value (0, 0, 0) here means that when we commit to this table and use the dummy value in the lookup_sorted columns, those entries that have the dummy value of 0=0+j∗0+j2∗0 will translate into a scalar multiplication by 0, which is free. 12-bit Check The range check table is a single-column table containing the numbers from 0 to 2^12 (excluded). This is used to check that the value fits in 12 bits. Runtime tables Another type of lookup tables has been suggested in the Extended Lookup Tables . The Lookup Selectors XorSelector . Performs 4 queries to the XOR lookup table. l r o - l r o - l r o - l r o 1, r3 1, r7 1, r11 - 1, r4 1, r8 1, r12 - 1, r5 1, r9 1, r13 - 1, r6 1, r10 1, r14 Producing the sorted table as the prover Because of our ZK-rows, we can't do the trick in the plookup paper of wrapping around to enforce consistency between the sorted lookup columns. Instead, we arrange the LookupSorted table into columns in a snake-shape. Like so, _ _\n| | | | |\n| | | | |\n|_| |_| | or, imagining the full sorted array is [ s0, ..., s8 ], like s0 s4 s4 s8\ns1 s3 s5 s7\ns2 s2 s6 s6 So the direction (\"increasing\" or \"decreasing\" (relative to LookupTable) is if i % 2 = 0 { Increasing } else { Decreasing } Then, for each i < max_lookups_per_row, if i % 2 = 0, we enforce that the last element of LookupSorted(i) = last element of LookupSorted(i + 1), and if i % 2 = 1, we enforce that the first element of LookupSorted(i) = first element of LookupSorted(i + 1).","breadcrumbs":"Kimchi » Lookup","id":"253","title":"Lookup"},"254":{"body":"A circuit is described as a series of gates. In this section we describe the different gates currently supported by kimchi, the constraints associated to them, and the way the register table, coefficient table, and permutation can be used in conjunction. TODO: for each gate describe how to create it? Double Generic Gate The double generic gate contains two generic gates. A generic gate is simply the 2-fan in gate specified in the vanilla PLONK protocol that allows us to do operations like: addition of two registers (into an output register) or multiplication of two registers equality of a register with a constant More generally, the generic gate controls the coefficients ci​ in the equation: c0​⋅l+c1​⋅r+c2​⋅o+c3​⋅(l×r)+c4​ The layout of the gate is the following: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 l1 r1 o1 l2 r2 o2 where l1, r1, and o1 (resp. l2, r2, o2) are the left, right, and output registers of the first (resp. second) generic gate. The selectors are stored in the coefficient table as: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 l1 r1 o1 m1 c1 l2 r2 o2 m2 c2 with m1 (resp. m2) the mul selector for the first (resp. second) gate, and c1 (resp. c2) the constant selector for the first (resp. second) gate. The constraints: w0​⋅c0​+w1​⋅c1​+w2​⋅c2​+w0​⋅w1​⋅c3​+c4​ w3​⋅c5​+w4​⋅c6​+w5​⋅c7​+w3​w4​c8​+c9​ where the ci​ are the coefficients. Poseidon The poseidon gate encodes 5 rounds of the poseidon permutation. A state is represents by 3 field elements. For example, the first state is represented by (s0, s0, s0), and the next state, after permutation, is represented by (s1, s1, s1). Below is how we store each state in the register table: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 s0 s0 s0 s4 s4 s4 s1 s1 s1 s2 s2 s2 s3 s3 s3 s5 s5 s5 The last state is stored on the next row. This last state is either used: with another Poseidon gate on that next row, representing the next 5 rounds. or with a Zero gate, and a permutation to use the output elsewhere in the circuit. or with another gate expecting an input of 3 field elements in its first registers. Note As some of the poseidon hash variants might not use 5k rounds (for some k), the result of the 4-th round is stored directly after the initial state. This makes that state accessible to the permutation. We define Mr,c​ as the MDS matrix at row r and column c. We define the S-box operation as wS for S the SPONGE_BOX constant. We store the 15 round constants ri​ required for the 5 rounds (3 per round) in the coefficient table: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 The initial state, stored in the first three registers, are not constrained. The following 4 states (of 3 field elements), including 1 in the next row, are constrained to represent the 5 rounds of permutation. Each of the associated 15 registers is associated to a constraint, calculated as: first round: w6​−(r0​+(M0,0​w0S​+M0,1​w1S​+M0,2​w2S​)) w7​−(r1​+(M1,0​w0S​+M1,1​w1S​+M1,2​w2S​)) w8​−(r2​+(M2,0​w0S​+M2,1​w1S​+M2,2​w2S​)) second round: w9​−(r3​+(M0,0​w6S​+M0,1​w7S​+M0,2​w8S​)) w10​−(r4​+(M1,0​w6S​+M1,1​w7S​+M1,2​w8S​)) w11​−(r5​+(M2,0​w6S​+M2,1​w7S​+M2,2​w8S​)) third round: w12​−(r6​+(M0,0​w9S​+M0,1​w10S​+M0,2​w11S​)) w13​−(r7​+(M1,0​w9S​+M1,1​w10S​+M1,2​w11S​)) w14​−(r8​+(M2,0​w9S​+M2,1​w10S​+M2,2​w11S​)) fourth round: w3​−(r9​+(M0,0​w12S​+M0,1​w13S​+M0,2​w14S​)) w4​−(r10​+(M1,0​w12S​+M1,1​w13S​+M1,2​w14S​)) w5​−(r11​+(M2,0​w12S​+M2,1​w13S​+M2,2​w14S​)) fifth round: w0,next​−(r12​+(M0,0​w3S​+M0,1​w4S​+M0,2​w5S​)) w1,next​−(r13​+(M1,0​w3S​+M1,1​w4S​+M1,2​w5S​)) w2,next​−(r14​+(M2,0​w3S​+M2,1​w4S​+M2,2​w5S​)) where wi,next​ is the polynomial wi​(ωx) which points to the next row. Elliptic Curve Addition The layout is 0 1 2 3 4 5 6 7 8 9 10 x1 y1 x2 y2 x3 y3 inf same_x s inf_z x21_inv where (x1, y1), (x2, y2) are the inputs and (x3, y3) the output. inf is a boolean that is true iff the result (x3, y3) is the point at infinity. The rest of the values are inaccessible from the permutation argument, but same_x is a boolean that is true iff x1 == x2. The following constraints are generated: constraint 1: x0​=w2​−w0​ (w10​⋅x0​−F(1)−w7​) constraint 2: x0​=w2​−w0​ w7​⋅x0​ constraint 3: x0​=w2​−w0​ x1​=w3​−w1​ x2​=w0​⋅w0​ w7​⋅(2⋅w8​⋅w1​−2⋅x2​−x2​)+(F(1)−w7​)⋅(x0​⋅w8​−x1​) constraint 4: w0​+w2​+w4​−w8​⋅w8​ constraint 5: w8​⋅(w0​−w4​)−w1​−w5​ constraint 6: x1​=w3​−w1​ x1​⋅(w7​−w6​) constraint 7: x1​=w3​−w1​ x1​⋅w9​−w6​ Endo Scalar We give constraints for the endomul scalar computation. Each row corresponds to 8 iterations of the inner loop in \"Algorithm 2\" on page 29 of the Halo paper . The state of the algorithm that's updated across iterations of the loop is (a, b). It's clear from that description of the algorithm that an iteration of the loop can be written as (a, b, i) -> ( 2 * a + c_func(r_{2 * i}, r_{2 * i + 1}), 2 * b + d_func(r_{2 * i}, r_{2 * i + 1}) ) for some functions c_func and d_func. If one works out what these functions are on every input (thinking of a two-bit input as a number in {0,1,2,3}), one finds they are given by c_func(x), defined by c_func(0) = 0 c_func(1) = 0 c_func(2) = -1 c_func(3) = 1 d_func(x), defined by d_func(0) = -1 d_func(1) = 1 d_func(2) = 0 d_func(3) = 0 One can then interpolate to find polynomials that implement these functions on {0,1,2,3}. You can use sage , as R = PolynomialRing(QQ, 'x')\nc_func = R.lagrange_polynomial([(0, 0), (1, 0), (2, -1), (3, 1)])\nd_func = R.lagrange_polynomial([(0, -1), (1, 1), (2, 0), (3, 0)]) Then, c_func is given by 2/3 * x^3 - 5/2 * x^2 + 11/6 * x and d_func is given by 2/3 * x^3 - 7/2 * x^2 + 29/6 * x - 1 <=> c_func + (-x^2 + 3x - 1) We lay it out the witness as 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Type n0 n8 a0 b0 a8 b8 x0 x1 x2 x3 x4 x5 x6 x7 ENDO where each xi is a two-bit \"crumb\". We also use a polynomial to check that each xi is indeed in {0,1,2,3}, which can be done by checking that each xi​ is a root of the polyunomial below: crumb(x)\n= x (x - 1) (x - 2) (x - 3)\n= x^4 - 6*x^3 + 11*x^2 - 6*x\n= x *(x^3 - 6*x^2 + 11*x - 6) Each iteration performs the following computations Update n: ni+1​=2⋅ni​+xi​ Update a: ai+1​=2⋅ai​+ci​ Update b: bi+1​=2⋅bi​+di​ Then, after the 8 iterations, we compute expected values of the above operations as: expected_n8 := 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * (2 * n0 + x0) + x1 ) + x2 ) + x3 ) + x4 ) + x5 ) + x6 ) + x7 expected_a8 := 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * (2 * a0 + c0) + c1 ) + c2 ) + c3 ) + c4 ) + c5 ) + c6 ) + c7 expected_b8 := 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * (2 * b0 + d0) + d1 ) + d2 ) + d3 ) + d4 ) + d5 ) + d6 ) + d7 Putting together all of the above, these are the 11 constraints for this gate Checking values after the 8 iterations: Constrain n: 0 = expected_n8 - n8 Constrain a: 0 = expected_a8 - a8 Constrain b: 0 = expected_b8 - b8 Checking the crumbs, meaning each x is indeed in the range {0,1,2,3}: Constrain x0​: 0 = x0 * ( x0^3 - 6 * x0^2 + 11 * x0 - 6 ) Constrain x1​: 0 = x1 * ( x1^3 - 6 * x1^2 + 11 * x1 - 6 ) Constrain x2​: 0 = x2 * ( x2^3 - 6 * x2^2 + 11 * x2 - 6 ) Constrain x3​: 0 = x3 * ( x3^3 - 6 * x3^2 + 11 * x3 - 6 ) Constrain x4​: 0 = x4 * ( x4^3 - 6 * x4^2 + 11 * x4 - 6 ) Constrain x5​: 0 = x5 * ( x5^3 - 6 * x5^2 + 11 * x5 - 6 ) Constrain x6​: 0 = x6 * ( x6^3 - 6 * x6^2 + 11 * x6 - 6 ) Constrain x7​: 0 = x7 * ( x7^3 - 6 * x7^2 + 11 * x7 - 6 ) Endo Scalar Multiplication We implement custom gate constraints for short Weierstrass curve endomorphism optimised variable base scalar multiplication. Given a finite field Fq​ of order q, if the order is not a multiple of 2 nor 3, then an elliptic curve over Fq​ in short Weierstrass form is represented by the set of points (x,y) that satisfy the following equation with a,b∈Fq​ and 4a3+27b2=Fq​​0: E(Fq​):y2=x3+ax+b If P=(xp​,yp​) and T=(xt​,yt​) are two points in the curve E(Fq​), the goal of this operation is to perform the operation 2P±T efficiently as (P±T)+P. S = (P + (b ? T : −T)) + P The same algorithm can be used to perform other scalar multiplications, meaning it is not restricted to the case 2⋅P, but it can be used for any arbitrary k⋅P. This is done by decomposing the scalar k into its binary representation. Moreover, for every step, there will be a one-bit constraint meant to differentiate between addition and subtraction for the operation (P±T)+P: In particular, the constraints of this gate take care of 4 bits of the scalar within a single EVBSM row. When the scalar is longer (which will usually be the case), multiple EVBSM rows will be concatenated. Row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Type i xT yT Ø Ø xP yP n xR yR s1 s3 b1 b2 b3 b4 EVBSM i+1 = = xS yS n' xR' yR' s1' s3' b1' b2' b3' b4' EVBSM The layout of this gate (and the next row) allows for this chained behavior where the output point of the current row S gets accumulated as one of the inputs of the following row, becoming P in the next constraints. Similarly, the scalar is decomposed into binary form and n (n′ respectively) will store the current accumulated value and the next one for the check. For readability, we define the following variables for the constraints: endo := EndoCoefficient xq1 :=(1+(endo−1)⋅b1​)⋅xt​ xq2 :=(1+(endo−1)⋅b3​)⋅xt​ yq1 :=(2⋅b2​−1)⋅yt​ yq2 :=(2⋅b4​−1)⋅yt​ These are the 11 constraints that correspond to each EVBSM gate, which take care of 4 bits of the scalar within a single EVBSM row: First block: (xq1 - xp) * s1 = yq1 - yp (2 * xp – s1^2 + xq1) * ((xp – xr) * s1 + yr + yp) = (xp – xr) * 2 * yp (yr + yp)^2 = (xp – xr)^2 * (s1^2 – xq1 + xr) Second block: (xq2 - xr) * s3 = yq2 - yr (2*xr – s3^2 + xq2) * ((xr – xs) * s3 + ys + yr) = (xr – xs) * 2 * yr (ys + yr)^2 = (xr – xs)^2 * (s3^2 – xq2 + xs) Booleanity checks: Bit flag b1​: 0 = b1 * (b1 - 1) Bit flag b2​: 0 = b2 * (b2 - 1) Bit flag b3​: 0 = b3 * (b3 - 1) Bit flag b4​: 0 = b4 * (b4 - 1) Binary decomposition: Accumulated scalar: n_next = 16 * n + 8 * b1 + 4 * b2 + 2 * b3 + b4 The constraints above are derived from the following EC Affine arithmetic equations: (1) => (xq1​​−xp​)⋅s1​=yq1​​−yp​ (2&3) => (xp​–xr​)⋅s2​=yr​+yp​ (2) => (2⋅xp​+xq1​​–s12​)⋅(s1​+s2​)=2⋅yp​ <=> (2⋅xp​–s12​+xq1​​)⋅((xp​–xr​)⋅s1​+yr​+yp​)=(xp​–xr​)⋅2⋅yp​ (3) => s12​−s22​=xq1​​−xr​ <=> (yr​+yp​)2=(xp​–xr​)2⋅(s12​–xq1​​+xr​) (4) => (xq2​​−xr​)⋅s3​=yq2​​−yr​ (5&6) => (xr​–xs​)⋅s4​=ys​+yr​ (5) => (2⋅xr​+xq2​​–s32​)⋅(s3​+s4​)=2⋅yr​ <=> (2⋅xr​–s32​+xq2​​)⋅((xr​–xs​)⋅s3​+ys​+yr​)=(xr​–xs​)⋅2⋅yr​ (6) => s32​–s42​=xq2​​−xs​ <=> (ys​+yr​)2=(xr​–xs​)2⋅(s32​–xq2​​+xs​) Defining s2​ and s4​ as s2​:=2∗xP​+xT​−s12​2⋅yP​​−s1​ s4​:=2∗xR​+xT​−s32​2⋅yR​​−s3​ Gives the following equations when substituting the values of s2​ and s4​: (xq1 - xp) * s1 = (2 * b1 - 1) * yt - yp (2 * xp – s1^2 + xq1) * ((xp – xr) * s1 + yr + yp) = (xp – xr) * 2 * yp (yr + yp)^2 = (xp – xr)^2 * (s1^2 – xq1 + xr) (xq2 - xr) * s3 = (2 * b2 - 1) * yt - yr (2 * xr – s3^2 + xq2) * ((xr – xs) * s3 + ys + yr) = (xr – xs) * 2 * yr (ys + yr)^2 = (xr – xs)^2 * (s3^2 – xq2 + xs) Scalar Multiplication We implement custom Plonk constraints for short Weierstrass curve variable base scalar multiplication. Given a finite field Fq​ of order q, if the order is not a multiple of 2 nor 3, then an elliptic curve over Fq​ in short Weierstrass form is represented by the set of points (x,y) that satisfy the following equation with a,b∈Fq​ and 4a3+27b2=Fq​​0: E(Fq​):y2=x3+ax+b If P=(xp​,yp​) and Q=(xq​,yq​) are two points in the curve E(Fq​), the algorithm we represent here computes the operation 2P+Q (point doubling and point addition) as (P+Q)+Q. Info Point Q=(xq​,yq​) has nothing to do with the order q of the field Fq​. The original algorithm that is being used can be found in the Section 3.1 of https://arxiv.org/pdf/math/0208038.pdf , which can perform the above operation using 1 multiplication, 2 squarings and 2 divisions (one more squaring) if P=Q), thanks to the fact that computing the Y-coordinate of the intermediate addition is not required. This is more efficient to the standard algorithm that requires 1 more multiplication, 3 squarings in total and 2 divisions. Moreover, this algorithm can be applied not only to the operation 2P+Q, but any other scalar multiplication kP. This can be done by expressing the scalar k in biwise form and performing a double-and-add approach. Nonetheless, this requires conditionals to differentiate 2P from 2P+Q. For that reason, we will implement the following pseudocode from https://github.com/zcash/zcash/issues/3924 (where instead, they give a variant of the above efficient algorithm for Montgomery curves b⋅y2=x3+a⋅x2+x). Acc := [2]T\nfor i = n-1 ... 0: Q := (k_{i + 1} == 1) ? T : -T Acc := Acc + (Q + Acc)\nreturn (k_0 == 0) ? Acc - P : Acc The layout of the witness requires 2 rows. The i-th row will be a VBSM gate whereas the next row will be a ZERO gate. Row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Type i xT yT x0 y0 n n' x1 y1 x2 y2 x3 y3 x4 y4 VBSM i+1 x5 y5 b0 b1 b2 b3 b4 s0 s1 s2 s3 s4 ZERO The gate constraints take care of 5 bits of the scalar multiplication. Each single bit consists of 4 constraints. There is one additional constraint imposed on the final number. Thus, the VarBaseMul gate argument requires 21 constraints. For every bit, there will be one constraint meant to differentiate between addition and subtraction for the operation (P±T)+P: S = (P + (b ? T : −T)) + P We follow these criteria: If the bit is positive, the sign should be a subtraction If the bit is negative, the sign should be an addition Then, paraphrasing the above, we will represent this behavior as: S = (P - (2 * b - 1) * T ) + P Let us call Input the point with coordinates (xI, yI) and Target is the point being added with coordinates (xT, yT). Then Output will be the point with coordinates (xO, yO) resulting from O = ( I ± T ) + I Info Do not confuse our Output point (xO, yO) with the point at infinity that is normally represented as O. In each step of the algorithm, we consider the following elliptic curves affine arithmetic equations: s1​:=xi​−xt​yi​−(2⋅b−1)⋅yt​​ s2​:=2∗xi​+xt​−s12​2⋅yi​​−s1​ xo​:=xt​+s22​−s12​ yo​:=s2​⋅(xi​−xo​)−yi​ For readability, we define the following 3 variables in such a way that s2​ can be expressed as u / t: rx :=s12​−xi​−xt​ t :=xi​− rx ⟺2⋅xi​−s12​+xt​ u :=2⋅yi​− t ⋅s1​⟺2⋅yi​−s1​⋅(2⋅xi​−s12​+xt​) Next, for each bit in the algorithm, we create the following 4 constraints that derive from the above: Booleanity check on the bit b: 0 = b * b - b Constrain s1​: (xI - xT) * s1 = yI – (2b - 1) * yT Constrain Output X-coordinate xo​ and s2​: 0 = u^2 - t^2 * (xO - xT + s1^2) Constrain Output Y-coordinate yo​ and s2​: 0 = (yO + yI) * t - (xI - xO) * u When applied to the 5 bits, the value of the Target point (xT, yT) is maintained, whereas the values for the Input and Output points form the chain: [(x0, y0) -> (x1, y1) -> (x2, y2) -> (x3, y3) -> (x4, y4) -> (x5, y5)] Similarly, 5 different s0..s4 are required, just like the 5 bits b0..b4. Finally, the additional constraint makes sure that the scalar is being correctly expressed into its binary form (using the double-and-add decomposition) as: n′=25⋅n+24⋅b0​+23⋅b1​+22⋅b2​+21⋅b3​+b4​ This equation is translated as the constraint: Binary decomposition: 0 = n' - (b4 + 2 * (b3 + 2 * (b2 + 2 * (b1 + 2 * (b0 + 2*n))))) Range Check The multi range check gadget is comprised of three circuit gates (RangeCheck0, RangeCheck1 and Zero) and can perform range checks on three values (v0​,v1​ and v2​) of up to 88 bits each. Values can be copied as inputs to the multi range check gadget in two ways: (Standard mode) With 3 copies, by copying v0​,v1​ and v2​ to the first cells of the first 3 rows of the gadget. In this mode the first gate coefficient is set to 0. (Compact mode) With 2 copies, by copying v2​ to the first cell of the first row and copying v10​=v0​+2ℓ⋅v1​ to the 2nd cell of row 2. In this mode the first gate coefficient is set to 1. The RangeCheck0 gate can also be used on its own to perform 64-bit range checks by constraining witness cells 1-2 to zero. Byte-order: Each cell value is in little-endian byte order Limbs are mapped to columns in big-endian order (i.e. the lowest columns contain the highest bits) We also have the highest bits covered by copy constraints and plookups, so that we can copy the highest two constraints to zero and get a 64-bit lookup, which are envisioned to be a common case The values are decomposed into limbs as follows: L is a 12-bit lookup (or copy) limb, C is a 2-bit \"crumb\" limb (we call half a nybble a crumb). <----6----> <------8------> v0 = L L L L L L C C C C C C C C v1 = L L L L L L C C C C C C C C <2> <--4--> <---------------18----------------> v2 = C C L L L L C C C C C C C C C C C C C C C C C C Witness structure: Row Contents 0 v0​ 1 v1​ 2 v2​ 3 v0​,v1​,v2​ The first 2 rows contain v0​ and v1​ and their respective decompositions into 12-bit and 2-bit limbs The 3rd row contains v2​ and part of its decomposition: four 12-bit limbs and the 1st 10 crumbs The final row contains v0​'s and v1​'s 5th and 6th 12-bit limbs as well as the remaining 10 crumbs of v2​ Note Because we are constrained to 4 lookups per row, we are forced to postpone some lookups of v0 and v1 to the final row. Constraints: For efficiency, the limbs are constrained differently according to their type: 12-bit limbs are constrained with plookups 2-bit crumbs are constrained with degree-4 constraints x(x−1)(x−2)(x−3) Layout: This is how the three 88-bit inputs v0​,v1​ and v2​ are laid out and constrained. vipj is the jth 12-bit limb of value vi​ vicj is the jth 2-bit crumb limb of value vi​ Gates RangeCheck0 RangeCheck0 RangeCheck1 Zero Rows 0 1 2 3 Cols 0 v0 v1 v2 crumb v2c9 MS:1 copy v0p0 copy v1p0 optional v12 crumb v2c10 2 copy v0p1 copy v1p1 crumb v2c0 crumb v2c11 3 plookup v0p2 plookup v1p2 plookup v2p0 plookup v0p0 4 plookup v0p3 plookup v1p3 plookup v2p1 plookup v0p1 5 plookup v0p4 plookup v1p4 plookup v2p2 plookup v1p0 6 plookup v0p5 plookup v1p5 plookup v2p3 plookup v1p1 7 crumb v0c0 crumb v1c0 crumb v2c1 crumb v2c12 8 crumb v0c1 crumb v1c1 crumb v2c2 crumb v2c13 9 crumb v0c2 crumb v1c2 crumb v2c3 crumb v2c14 10 crumb v0c3 crumb v1c3 crumb v2c4 crumb v2c15 11 crumb v0c4 crumb v1c4 crumb v2c5 crumb v2c16 12 crumb v0c5 crumb v1c5 crumb v2c6 crumb v2c17 13 crumb v0c6 crumb v1c6 crumb v2c7 crumb v2c18 LS:14 crumb v0c7 crumb v1c7 crumb v2c8 crumb v2c19 The 12-bit chunks are constrained with plookups and the 2-bit crumbs are constrained with degree-4 constraints of the form x(x−1)(x−2)(x−3). Note that copy denotes a plookup that is deferred to the 4th gate (i.e. Zero). This is because of the limitation that we have at most 4 lookups per row. The copies are constrained using the permutation argument. Gate types: Different rows are constrained using different CircuitGate types Row CircuitGate Purpose 0 RangeCheck0 Partially constrain v0​ 1 RangeCheck0 Partially constrain v1​ 2 RangeCheck1 Fully constrain v2​ (and trigger plookups constraints on row 3) 3 Zero Complete the constraining of v0​ and v1​ using lookups Note Each CircuitGate type corresponds to a unique polynomial and thus is assigned its own unique powers of alpha RangeCheck0 - Range check constraints This circuit gate is used to partially constrain values v0​ and v1​ Optionally, it can be used on its own as a single 64-bit range check by constraining columns 1 and 2 to zero The rest of v0​ and v1​ are constrained by the lookups in the Zero gate row This gate operates on the Curr row It uses three different types of constraints: copy - copy to another cell (12-bits) plookup - plookup (12-bits) crumb - degree-4 constraint (2-bits) Given value v the layout looks like this Column Curr 0 v 1 copy vp0 2 copy vp1 3 plookup vp2 4 plookup vp3 5 plookup vp4 6 plookup vp5 7 crumb vc0 8 crumb vc1 9 crumb vc2 10 crumb vc3 11 crumb vc4 12 crumb vc5 13 crumb vc6 14 crumb vc7 where the notation vpi and vci defined in the \"Layout\" section above. RangeCheck1 - Range check constraints This circuit gate is used to fully constrain v2​ It operates on the Curr and Next rows It uses two different types of constraints: plookup - plookup (12-bits) crumb - degree-4 constraint (2-bits) Given value v2 the layout looks like this Column Curr Next 0 v2 crumb v2c9 1 optional v12 crumb v2c10 2 crumb v2c0 crumb v2c11 3 plookup v2p0 (ignored) 4 plookup v2p1 (ignored) 5 plookup v2p2 (ignored) 6 plookup v2p3 (ignored) 7 crumb v2c1 crumb v2c12 8 crumb v2c2 crumb v2c13 9 crumb v2c3 crumb v2c14 10 crumb v2c4 crumb v2c15 11 crumb v2c5 crumb v2c16 12 crumb v2c6 crumb v2c17 13 crumb v2c7 crumb v2c18 14 crumb v2c8 crumb v2c19 where the notation v2ci and v2pi defined in the \"Layout\" section above. Foreign Field Addition These circuit gates are used to constrain that left_input +/- right_input = field_overflow * foreign_modulus + result Documentation For more details please see the Foreign Field Addition chapter. Mapping To make things clearer, the following mapping between the variable names used in the code and those of the RFC document can be helpful. left_input_lo -> a0 right_input_lo -> b0 result_lo -> r0 bound_lo -> u0 left_input_mi -> a1 right_input_mi -> b1 result_mi -> r1 bound_mi -> u1 left_input_hi -> a2 right_input_hi -> b2 result_hi -> r2 bound_hi -> u2 field_overflow -> q sign -> s carry_lo -> c0 carry_mi -> c1 bound_carry_lo -> k0 bound_carry_mi -> k1 Note: Our limbs are 88-bit long. We denote with: lo the least significant limb (in little-endian, this is from 0 to 87) mi the middle limb (in little-endian, this is from 88 to 175) hi the most significant limb (in little-endian, this is from 176 to 263) Let left_input_lo, left_input_mi, left_input_hi be 88-bit limbs of the left element Let right_input_lo, right_input_mi, right_input_hi be 88-bit limbs of the right element Let foreign_modulus_lo, foreign_modulus_mi, foreign_modulus_hi be 88-bit limbs of the foreign modulus Then the limbs of the result are result_lo = left_input_lo +/- right_input_lo - field_overflow * foreign_modulus_lo - 2^{88} * result_carry_lo result_mi = left_input_mi +/- right_input_mi - field_overflow * foreign_modulus_mi - 2^{88} * result_carry_mi + result_carry_lo result_hi = left_input_hi +/- right_input_hi - field_overflow * foreign_modulus_hi + result_carry_mi field_overflow =0 or 1 or −1 handles overflows in the field result_carry_i =−1,0,1 are auxiliary variables that handle carries between limbs Apart from the range checks of the chained inputs, we need to do an additional range check for a final bound to make sure that the result is less than the modulus, by adding 2^{3*88} - foreign_modulus to it. (This can be computed easily from the limbs of the modulus) Note that 2^{264} as limbs represents: (0, 0, 0, 1) then: The upper-bound check can be calculated as: bound_lo = result_lo - foreign_modulus_lo - bound_carry_lo * 2^{88} bound_mi = result_mi - foreign_modulus_mi - bound_carry_mi * 2^{88} + bound_carry_lo bound_hi = result_hi - foreign_modulus_hi + 2^{88} + bound_carry_mi Which is equivalent to another foreign field addition with right input 2^{264}, q = 1 and s = 1 bound_lo = result_lo + s * 0 - q * foreign_modulus_lo - bound_carry_lo * 2^{88} bound_mi = result_mi + s * 0 - q * foreign_modulus_mi - bound_carry_mi * 2^{88} + bound_carry_lo bound_hi = result_hi + s * 2^{88} - q * foreign_modulus_hi + bound_carry_mi bound_carry_i =0 or 1 or −1 are auxiliary variables that handle carries between limbs The range check of bound can be skipped until the end of the operations and result is an intermediate value that is unused elsewhere (since the final result must have had the right amount of moduli subtracted along the way, meaning a multiple of the modulus). In other words, intermediate results could potentially give a valid witness that satisfies the constraints but where the result is larger than the modulus (yet smaller than 2^{264}). The reason that we have a final bound check is to make sure that the final result (min_result) is indeed the minimum one (meaning less than the modulus). A more optimized version of these constraints is able to reduce by 2 the number of constraints and by 1 the number of witness cells needed. The idea is to condense the low and middle limbs in one longer limb of 176 bits (which fits inside our native field) and getting rid of the low carry flag. With this idea in mind, the sole carry flag we need is the one located between the middle and the high limbs. Layout The sign of the operation (whether it is an addition or a subtraction) is stored in the fourth coefficient as a value +1 (for addition) or -1 (for subtraction). The first 3 coefficients are the 3 limbs of the foreign modulus. One could lay this out as a double-width gate for chained foreign additions and a final row, e.g.: col ForeignFieldAdd chain ForeignFieldAdd final ForeignFieldAdd final Zero 0 left_input_lo (copy) result_lo (copy) min_result_lo (copy) bound_lo (copy) 1 left_input_mi (copy) result_mi (copy) min_result_mi (copy) bound_mi (copy) 2 left_input_hi (copy) result_hi (copy) min_result_hi (copy) bound_hi (copy) 3 right_input_lo (copy) 0 (check) 4 right_input_mi (copy) 0 (check) 5 right_input_hi (copy) 2^88 (check) 6 field_overflow (copy?) 1 (check) 7 carry bound_carry 8 9 10 11 12 13 14 We reuse the foreign field addition gate for the final bound check since this is an addition with a specific parameter structure. Checking that the correct right input, overflow, and overflow are used shall be done by copy constraining these values with a public input value. One could have a specific gate for just this check requiring less constrains, but the cost of adding one more selector gate outweights the savings of one row and a few constraints of difference. Integration Copy final overflow bit from public input containing value 1 * Range check the final bound Foreign Field Multiplication This gadget is used to constrain that left_input * right_input = quotient * foreign_field_modulus + remainder Documentation For more details please see the Foreign Field Multiplication chapter or the original Foreign Field Multiplication RFC Notations For clarity, we use more descriptive variable names in the code than in the RFC, which uses mathematical notations. In order to relate the two documents, the following mapping between the variable names used in the code and those of the RFC can be helpful. left_input0 => a0 right_input0 => b0 quotient0 => q0 remainder01 => r01\nleft_input1 => a1 right_input1 => b1 quotient1 => q1\nleft_input2 => a2 right_input2 => b2 quotient2 => q2 remainder2 => r2 product1_lo => p10 product1_hi_0 => p110 product1_hi_1 => p111 carry0 => v0 carry1_lo => v10 carry1_hi => v11 quotient_hi_bound => q'2 Suffixes The variable names in this code uses descriptive suffixes to convey information about the positions of the bits referred to. When a word is split into up to n parts we use: 0, 1 ... n (where n is the most significant). For example, if we split word x into three limbs, we'd name them x0, x1 and x2 or x[0], x[1] and x[2]. Continuing in this fashion, when one of those words is subsequently split in half, then we add the suffixes _lo and _hi, where hi corresponds to the most significant bits. For our running example, x1 would become x1_lo and x1_hi. If we are splitting into more than two things, then we pick meaningful names for each. So far we've explained our conventions for a splitting depth of up to 2. For splitting deeper than two, we simply cycle back to our depth 1 suffixes again. So for example, x1_lo would be split into x1_lo_0 and x1_lo_1. Parameters hi_foreign_field_modulus := high limb of foreign field modulus $f$ (stored in gate coefficient 0) neg_foreign_field_modulus := negated foreign field modulus $f'$ (stored in gate coefficients 1-3) n := the native field modulus is obtainable from F, the native field's trait bound Witness left_input := left foreign field element multiplicand $ ~\\in F_f$ right_input := right foreign field element multiplicand $ ~\\in F_f$ quotient := foreign field quotient $ ~\\in F_f$ remainder := foreign field remainder $ ~\\in F_f$ carry0 := 2 bit carry carry1_lo := low 88 bits of carry1 carry1_hi := high 3 bits of carry1 product1_lo := lowest 88 bits of middle intermediate product product1_hi_0 := lowest 88 bits of middle intermediate product's highest 88 + 2 bits product1_hi_1 := highest 2 bits of middle intermediate product quotient_hi_bound := quotient high bound for checking q2 ≤ f2 Layout The foreign field multiplication gate's rows are laid out like this col ForeignFieldMul Zero 0 left_input0 (copy) remainder01 (copy) 1 left_input1 (copy) remainder2 (copy) 2 left_input2 (copy) quotient0 (copy) 3 right_input0 (copy) quotient1 (copy) 4 right_input1 (copy) quotient2 (copy) 5 right_input2 (copy) quotient_hi_bound (copy) 6 product1_lo (copy) product1_hi_0 (copy) 7 carry1_0 (plookup) product1_hi_1 (dummy) 8 `carry1_12 (plookup) carry1_48 (plookup) 9 carry1_24 (plookup) carry1_60 (plookup) 10 carry1_36 (plookup) carry1_72 (plookup) 11 carry1_84 carry0 12 carry1_86 13 carry1_88 14 carry1_90 Rotation Rotation of a 64-bit word by a known offset Rot64 onstrains known-length rotation of 64-bit words: This circuit gate is used to constrain that a 64-bit word is rotated by $r < 64$ bits to the \"left\". The rotation is performed towards the most significant side (thus, the new LSB is fed with the old MSB). This gate operates on the Curr and Next rows. The idea is to split the rotation operation into two parts: Shift to the left Add the excess bits to the right We represent shifting with multiplication modulo $2^{64}$. That is, for each word to be rotated, we provide in the witness a quotient and a remainder, similarly to ForeignFieldMul such that the following operation holds: $$word \\cdot 2^{rot} = quotient \\cdot 2^{64} + remainder$$ Then, the remainder corresponds to the shifted word, and the quotient corresponds to the excess bits. $$word \\cdot 2^{rot} = excess \\cdot 2^{64} + shifted$$ Thus, in order to obtain the rotated word, we need to add the quotient and the remainder as follows: $$rotated = shifted + excess$$ The input word is known to be of length 64 bits. All we need for soundness is check that the shifted and excess parts of the word have the correct size as well. That means, we need to range check that: $$ \\begin{aligned} excess &< 2^{rot}\\ shifted &< 2^{64} \\end{aligned} $$ The latter can be obtained with a RangeCheck0 gate setting the two most significant limbs to zero. The former is equivalent to the following check: $$bound = excess - 2^{rot} + 2^{64} < 2^{64}$$ which is doable with the constraints in a RangeCheck0 gate. Since our current row within the Rot64 gate is almost empty, we can use it to perform the range check within the same gate. Then, using the following layout and assuming that the gate has a coefficient storing the value $2^{rot}$, which is publicly known Gate Rot64 RangeCheck0 gadgets (designer's duty) Column Curr Next ------ ------------------- ---------------- 0 copy word shifted 1 copy rotated 0 2 excess 0 3 bound_limb0 shifted_limb0 4 bound_limb1 shifted_limb1 5 bound_limb2 shifted_limb2 6 bound_limb3 shifted_limb3 7 bound_crumb0 shifted_crumb0 8 bound_crumb1 shifted_crumb1 9 bound_crumb2 shifted_crumb2 10 bound_crumb3 shifted_crumb3 11 bound_crumb4 shifted_crumb4 12 bound_crumb5 shifted_crumb5 13 bound_crumb6 shifted_crumb6 14 bound_crumb7 shifted_crumb7 In Keccak, rotations are performed over a 5x5 matrix state of w-bit words each cell. The values used to perform the rotation are fixed, public, and known in advance, according to the following table, depending on the coordinate of each cell within the 5x5 matrix state: y \\ x 0 1 2 3 4 0 0 36 3 105 210 1 1 300 10 45 66 2 190 6 171 15 253 3 28 55 153 21 120 4 91 276 231 136 78 But since we will always be using 64-bit words in our Keccak usecase ($w = 64$), we can have an equivalent table with these values modulo 64 to avoid needing multiple passes of the rotation gate (a single step would cause overflows otherwise): y \\ x 0 1 2 3 4 0 0 36 3 41 18 1 1 44 10 45 2 2 62 6 43 15 61 3 28 55 25 21 56 4 27 20 39 8 14 Since there is one value of the coordinates (x, y) where the rotation is 0 bits, we can skip that step in the gadget. This will save us one gate, and thus the whole 25-1=24 rotations will be performed in just 48 rows. Xor Xor16 - Chainable XOR constraints for words of multiples of 16 bits. This circuit gate is used to constrain that in1 xored with in2 equals out The length of in1, in2 and out must be the same and a multiple of 16bits. This gate operates on the Curr and Next rows. It uses three different types of constraints: copy - copy to another cell (32-bits) plookup - xor-table plookup (4-bits) decomposition - the constraints inside the gate The 4-bit nybbles are assumed to be laid out with 0 column being the least significant nybble. Given values in1, in2 and out, the layout looks like this: Column Curr Next 0 copy in1 copy in1' 1 copy in2 copy in2' 2 copy out copy out' 3 plookup0 in1_0 4 plookup1 in1_1 5 plookup2 in1_2 6 plookup3 in1_3 7 plookup0 in2_0 8 plookup1 in2_1 9 plookup2 in2_2 10 plookup3 in2_3 11 plookup0 out_0 12 plookup1 out_1 13 plookup2 out_2 14 plookup3 out_3 One single gate with next values of in1', in2' and out' being zero can be used to check that the original in1, in2 and out had 16-bits. We can chain this gate 4 times as follows to obtain a gadget for 64-bit words XOR: Row CircuitGate Purpose 0 Xor16 Xor 2 least significant bytes of the words 1 Xor16 Xor next 2 bytes of the words 2 Xor16 Xor next 2 bytes of the words 3 Xor16 Xor 2 most significant bytes of the words 4 Generic Zero values, can be reused as generic gate Info We could halve the number of rows of the 64-bit XOR gadget by having lookups for 8 bits at a time, but for now we will use the 4-bit XOR table that we have. Rough computations show that if we run 8 or more Keccaks in one circuit we should use the 8-bit XOR table.","breadcrumbs":"Kimchi » Gates","id":"254","title":"Gates"},"255":{"body":"Here we describe basic gadgets that we build using a combination of the gates described above. Not We implement NOT, i.e. bitwise negation, as a gadget in two different ways, needing no new gate type for it. Instead, it reuses the XOR gadget and the Generic gate. The first version of the NOT gadget reuses Xor16 by making the following observation: the bitwise NOT operation is equivalent to the bitwise XOR operation with the all one words of a certain length . In other words, $$\\neg x = x \\oplus 1^ $$ where $1^ $ denotes a bitstring of all ones of length $|x|$. Let $x_i$ be the $i$-th bit of $x$, the intuition is that if $x_i = 0$ then XOR with $1$ outputs $1$, thus negating $x_i$. Similarly, if $x_i = 1$ then XOR with 1 outputs 0, again negating $x_i$. Thus, bitwise XOR with $1^*$ is equivalent to bitwise negation (i.e. NOT). Then, if we take the XOR gadget with a second input to be the all one word of the same length, that gives us the NOT gadget. The correct length can be imposed by having a public input containing the 2^bits - 1 value and wiring it to the second input of the XOR gate. This approach needs as many rows as an XOR would need, for a single negation, but it comes with the advantage of making sure the input is of a certain length. The other approach can be more efficient if we already know the length of the inputs. For example, the input may be the input of a range check gate, or the output of a previous XOR gadget (which will be the case in our Keccak usecase). In this case, we simply perform the negation as a subtraction of the input word from the all one word (which again can be copied from a public input). This comes with the advantage of holding up to 2 word negations per row (an eight-times improvement over the XOR approach), but it requires the user to know the length of the input. NOT Layout using XOR Here we show the layout of the NOT gadget using the XOR approach. The gadget needs a row with a public input containing the all-one word of the given length. Then, a number of XORs follow, and a final Zero row is needed. In this case, the NOT gadget needs $\\ceil(\\frac{|x|}{16})$ Xor16 gates, that means one XOR row for every 16 bits of the input word. Row CircuitGate Purpose pub Generic Leading row with the public $1^*$ value i...i+n-1 Xor16 Negate every 4 nybbles of the word, from least to most significant i+n Generic Constrain that the final row is all zeros for correctness of Xor gate NOT Layout using Generic gates Here we show the layout of the NOT gadget using the Generic approach. The gadget needs a row with a public input containing the all-one word of the given length, exactly as above. Then, one Generic gate reusing the all-one word as left inputs can be used to negate up to two words per row. This approach requires that the input word is known (or constrained) to have a given length. Row CircuitGate Purpose pub Generic Leading row with the public $1^*$ value i Generic Negate one or two words of the length given by the length of the all-one word And We implement the AND gadget making use of the XOR gadget and the Generic gate. A new gate type is not needed, but we could potentially add an And16 gate type reusing the same ideas of Xor16 so as to save one final generic gate, at the cost of one additional AND lookup table that would have the same size as that of the Xor. For now, we are willing to pay this small overhead and produce AND gadget as follows: We observe that we can express bitwise addition as follows: $$A + B = (A \\oplus B) + 2 \\cdot (A \\wedge B)$$ where $\\oplus$ is the bitwise XOR operation, $\\wedge$ is the bitwise AND operation, and $+$ is the addition operation. In other words, the value of the addition is nothing but the XOR of its operands, plus the carry bit if both operands are 1. Thus, we can rewrite the above equation to obtain a definition of the AND operation as follows: $$A \\wedge B = \\frac{A + B - (A \\oplus B)}{2}$$ Let us define the following operations for better readability: a + b = sum\na x b = xor\na ^ b = and Then, we can rewrite the above equation as follows: $$ 2 \\cdot and = sum - xor $$ which can be expressed as a double generic gate. Then, our AND gadget for $n$ bytes looks as follows: $n/8$ Xor16 gates 1 (single) Generic gate to check that the final row of the XOR chain is all zeros. 1 (double) Generic gate to check sum $a + b = sum$ and the conjunction equation $2\\cdot and = sum - xor$. Finally, we connect the wires in the following positions (apart from the ones already connected for the XOR gates): Column 2 of the first Xor16 row (the output of the XOR operation) is connected to the right input of the second generic operation of the last row. Column 2 of the first generic operation of the last row is connected to the left input of the second generic operation of the last row. Meaning, the xor in a x b = xor is connected to the xor in 2 \\cdot and = sum - xor the sum in a + b = sum is connected to the sum in 2 \\cdot and = sum - xor","breadcrumbs":"Kimchi » Gadgets","id":"255","title":"Gadgets"},"256":{"body":"In this section we specify the setup that goes into creating two indexes from a circuit: A prover index , necessary for the prover to create proofs. A verifier index , necessary for the verifier to verify proofs. Note The circuit creation part is not specified in this document. It might be specified in a separate document, or we might want to specify how to create the circuit description tables. As such, the transformation of a circuit into these two indexes can be seen as a compilation step. Note that the prover still needs access to the original circuit to create proofs, as they need to execute it to create the witness (register table).","breadcrumbs":"Kimchi » Setup","id":"256","title":"Setup"},"257":{"body":"In this section we describe data that both the prover and the verifier index share. URS (Uniform Reference String) The URS is a set of parameters that is generated once, and shared between the prover and the verifier. It is used for polynomial commitments, so refer to the poly-commitment specification for more details. Note Kimchi currently generates the URS based on the circuit, and attach it to the index. So each circuit can potentially be accompanied with a different URS. On the other hand, Mina reuses the same URS for multiple circuits ( see zkapps for more details ). Domain . A domain large enough to contain the circuit and the zero-knowledge rows (used to provide zero-knowledge to the protocol). Specifically, the smallest subgroup in our field that has order greater or equal to n + ZK_ROWS, with n is the number of gates in the circuit. TODO: what if the domain is larger than the URS? Ordering of elements in the domain Note that in this specification we always assume that the first element of a domain is $1$. Shifts . As part of the permutation, we need to create PERMUTS shifts. To do that, the following logic is followed (in pseudo code): (TODO: move shift creation within the permutation section?) shifts[0] = 1 # first shift is identity for i in 0..7: # generate 7 shifts i = 7 shift, i = sample(domain, i) while shifts.contains(shift) do: shift, i = sample(domain, i) shift[i] = shift def sample(domain, i): i += 1 shift = Field(Blake2b512(to_be_bytes(i))) while is_not_quadratic_non_residue(shift) || domain.contains(shift): i += 1 shift = Field(Blake2b512(to_be_bytes(i))) return shift, i Public . This variable simply contains the number of public inputs. (TODO: actually, it's not contained in the verifier index) The compilation steps to create the common index are as follow: If the circuit is less than 2 gates, abort. Compute the number of zero-knowledge rows (zk_rows) that will be required to achieve zero-knowledge. The following constraints apply to zk_rows: The number of chunks c results in an evaluation at zeta and zeta * omega in each column for 2*c evaluations per column, so zk_rows >= 2*c + 1. The permutation argument interacts with the c chunks in parallel, so it is possible to cross-correlate between them to compromise zero knowledge. We know that there is some c >= 1 such that zk_rows = 2*c + k from the above. Thus, attempting to find the evaluation at a new point, we find that: the evaluation of every witness column in the permutation contains k unknowns; the evaluations of the permutation argument aggregation has k-1 unknowns; the permutation argument applies on all but zk_rows - 3 rows; and thus we form the equation zk_rows - 3 < 7 * k + (k - 1) to ensure that we can construct fewer equations than we have unknowns. This simplifies to k > (2 * c - 2) / 7, giving zk_rows > (16 * c - 2) / 7. We can derive c from the max_poly_size supported by the URS, and thus we find zk_rows and domain_size satisfying the fixpoint zk_rows = (16 * (domain_size / max_poly_size) + 5) / 7\ndomain_size = circuit_size + zk_rows Create a domain for the circuit. That is, compute the smallest subgroup of the field that has order greater or equal to n + zk_rows elements. Pad the circuit: add zero gates to reach the domain size. sample the PERMUTS shifts.","breadcrumbs":"Kimchi » Common Index","id":"257","title":"Common Index"},"258":{"body":"If lookup is used, the following values are added to the common index: LookupSelectors . The list of lookup selectors used. In practice, this tells you which lookup tables are used. TableIds . This is a list of table ids used by the Lookup gate. MaxJointSize . This is the maximum number of columns appearing in the lookup tables used by the lookup selectors. For example, the XOR lookup has 3 columns. To create the index, follow these steps: If no lookup is used in the circuit, do not create a lookup index Get the lookup selectors and lookup tables that are specified implicitly Concatenate explicit runtime lookup tables with the ones (implicitly) used by gates. Get the highest number of columns max_table_width that a lookup table can have. Create the concatenated table of all the fixed lookup tables. It will be of height the size of the domain, and of width the maximum width of any of the lookup tables. In addition, create an additional column to store all the tables' table IDs. For example, if you have a table with ID 0 1 2 3 5 6 7 0 0 0 and another table with ID 1 8 9 the concatenated table in a domain of size 5 looks like this: 1 2 3 5 6 7 0 0 0 8 9 0 0 0 0 with the table id vector: table id 0 0 0 1 0 To do this, for each table: Update the corresponding entries in a table id vector (of size the domain as well) with the table ID of the table. Copy the entries from the table to new rows in the corresponding columns of the concatenated table. Fill in any unused columns with 0 (to match the dummy value) Pad the end of the concatened table with the dummy value. Pad the end of the table id vector with 0s. pre-compute polynomial and evaluation form for the look up tables pre-compute polynomial and evaluation form for the table IDs, only if a table with an ID different from zero was used.","breadcrumbs":"Kimchi » Lookup Index","id":"258","title":"Lookup Index"},"259":{"body":"Both the prover and the verifier index, besides the common parts described above, are made out of pre-computations which can be used to speed up the protocol. These pre-computations are optimizations, in the context of normal proofs, but they are necessary for recursion. pub struct ProverIndex> { /// constraints system polynomials #[serde(bound = \"ConstraintSystem: Serialize + DeserializeOwned\")] pub cs: ConstraintSystem, /// The symbolic linearization of our circuit, which can compile to concrete types once certain values are learned in the protocol. #[serde(skip)] pub linearization: Linearization>, Column>, /// The mapping between powers of alpha and constraints #[serde(skip)] pub powers_of_alpha: Alphas, /// polynomial commitment keys #[serde(skip)] #[serde(bound(deserialize = \"OpeningProof::SRS: Default\"))] pub srs: Arc, /// maximal size of polynomial section pub max_poly_size: usize, #[serde(bound = \"ColumnEvaluations: Serialize + DeserializeOwned\")] pub column_evaluations: ColumnEvaluations, /// The verifier index corresponding to this prover index #[serde(skip)] pub verifier_index: Option>, /// The verifier index digest corresponding to this prover index #[serde_as(as = \"Option\")] pub verifier_index_digest: Option,\n}","breadcrumbs":"Kimchi » Prover Index","id":"259","title":"Prover Index"},"26":{"body":"The homomorphic commitment commit(x,r)=xG+rH described above is known as a Pedersen commitment. If you remove the rH term you get a non-hiding commitment, called a Pedersen hash . Both rely on the ECDLP hardness assumption. This means that, at least theoretically, you might be lucky (or have a quantum computer) and figure out that H=hG, which would allow you to find different values x′ and h′ to open the commitment. We say that pedersen commitments are computationally binding and not unconditionally binding. For example, you could express c=xG+rH alternatively as c=xG+rhG=(x+rh)G and compute a satisfying opening pair x′=rh and r′=hx​. On the other hand, Pedersen commitments are unconditionally hiding , as there is no way (even with a magic computer) to reveal what x is without knowing r. Lack of perfect binding is the reason why most of the \"proofs\" we will see later in this book are not referred to as proofs, but instead are referred to as arguments of knowledge (although we may care little about this distinction). Just remember that you need perfect binding to be called a proof. Interestingly, it is impossible to have a commitment scheme that has both perfect hiding and perfect binding. To recap, in cryptography the following distinctions are important Perfect. The property that an algorithm is statistically sound without hardness assumptions, also known as unconditional or statistical soundness. Computational. The algorithm relies on a hardness assumption or computational limitation for soundness. Thus, said another way, Pedersen commitments provide perfect hiding and computational binding.","breadcrumbs":"Commitments » Pedersen commitments","id":"26","title":"Pedersen commitments"},"260":{"body":"Same as the prover index, we have a number of pre-computations as part of the verifier index. #[serde_as]\n#[derive(Serialize, Deserialize, Debug, Clone)]\npub struct LookupVerifierIndex { pub joint_lookup_used: bool, #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub lookup_table: Vec>, #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub lookup_selectors: LookupSelectors>, /// Table IDs for the lookup values. /// This may be `None` if all lookups originate from table 0. #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub table_ids: Option>, /// Information about the specific lookups used pub lookup_info: LookupInfo, /// An optional selector polynomial for runtime tables #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub runtime_tables_selector: Option>,\n} #[serde_as]\n#[derive(Serialize, Deserialize, Debug, Clone)]\npub struct VerifierIndex> { /// evaluation domain #[serde_as(as = \"o1_utils::serialization::SerdeAs\")] pub domain: D, /// maximal size of polynomial section pub max_poly_size: usize, /// the number of randomized rows to achieve zero knowledge pub zk_rows: u64, /// polynomial commitment keys #[serde(skip)] #[serde(bound(deserialize = \"OpeningProof::SRS: Default\"))] pub srs: Arc, /// number of public inputs pub public: usize, /// number of previous evaluation challenges, for recursive proving pub prev_challenges: usize, // index polynomial commitments /// permutation commitment array #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub sigma_comm: [PolyComm; PERMUTS], /// coefficient commitment array #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub coefficients_comm: [PolyComm; COLUMNS], /// coefficient commitment array #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub generic_comm: PolyComm, // poseidon polynomial commitments /// poseidon constraint selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub psm_comm: PolyComm, // ECC arithmetic polynomial commitments /// EC addition selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub complete_add_comm: PolyComm, /// EC variable base scalar multiplication selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub mul_comm: PolyComm, /// endoscalar multiplication selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub emul_comm: PolyComm, /// endoscalar multiplication scalar computation selector polynomial commitment #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub endomul_scalar_comm: PolyComm, /// RangeCheck0 polynomial commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub range_check0_comm: Option>, /// RangeCheck1 polynomial commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub range_check1_comm: Option>, /// Foreign field addition gates polynomial commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub foreign_field_add_comm: Option>, /// Foreign field multiplication gates polynomial commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub foreign_field_mul_comm: Option>, /// Xor commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub xor_comm: Option>, /// Rot commitments #[serde(bound = \"Option>: Serialize + DeserializeOwned\")] pub rot_comm: Option>, /// wire coordinate shifts #[serde_as(as = \"[o1_utils::serialization::SerdeAs; PERMUTS]\")] pub shift: [G::ScalarField; PERMUTS], /// zero-knowledge polynomial #[serde(skip)] pub permutation_vanishing_polynomial_m: OnceCell>, // TODO(mimoo): isn't this redundant with domain.d1.group_gen ? /// domain offset for zero-knowledge #[serde(skip)] pub w: OnceCell, /// endoscalar coefficient #[serde(skip)] pub endo: G::ScalarField, #[serde(bound = \"PolyComm: Serialize + DeserializeOwned\")] pub lookup_index: Option>, #[serde(skip)] pub linearization: Linearization>, Column>, /// The mapping between powers of alpha and constraints #[serde(skip)] pub powers_of_alpha: Alphas,\n}","breadcrumbs":"Kimchi » Verifier Index","id":"260","title":"Verifier Index"},"261":{"body":"Originally, kimchi is based on an interactive protocol that was transformed into a non-interactive one using the Fiat-Shamir transform. For this reason, it can be useful to visualize the high-level interactive protocol before the transformation: sequenceDiagram participant Prover participant Verifier Note over Prover,Verifier: Prover produces commitments to secret polynomials Prover->>Verifier: public input & witness commitment Verifier->>Prover: beta & gamma Prover->>Verifier: permutation commitment opt lookup Prover->>Verifier: sorted Prover->>Verifier: aggreg end Note over Prover,Verifier: Prover produces commitment to quotient polynomial Verifier->>Prover: alpha Prover->>Verifier: quotient commitment Note over Prover,Verifier: Verifier produces an evaluation point Verifier->>Prover: zeta Note over Verifier: change of verifier (change of sponge) opt recursion Prover->>Verifier: recursion challenges end Note over Prover,Verifier: Prover provides needed evaluations for the linearization Prover->>Verifier: ft(zeta * omega) Prover->>Verifier: negated public input p(zeta) & p(zeta * omega) Prover->>Verifier: the 15 registers w_i(zeta) & w_i(zeta * omega) Prover->>Verifier: permutation poly z(zeta) & z(zeta * omega) Prover->>Verifier: the 6 sigmas s_i(zeta) & s_i(zeta * omega) Prover->>Verifier: the 15 coefficients c_i(zeta) & c_i(zeta * omega) opt lookup Prover->>Verifier: sorted(zeta) & sorted(zeta * omega) Prover->>Verifier: aggreg(zeta) & aggreg(zeta * omega) Prover->>Verifier: table(zeta) & table(zeta * omega) end Prover->>Verifier: the generic selector gen(zeta) & gen(zeta * omega) Prover->>Verifier: the poseidon selector pos(zeta) & pos(zeta * omega) Note over Prover,Verifier: Batch verification of evaluation proofs Verifier->>Prover: v, u Note over Verifier: change of verifier (change of sponge) Prover->>Verifier: aggregated evaluation proof (involves more interaction) The Fiat-Shamir transform simulates the verifier messages via a hash function that hashes the transcript of the protocol so far before outputing verifier messages. You can find these operations under the proof creation and proof verification algorithms as absorption and squeezing of values with the sponge.","breadcrumbs":"Kimchi » Proof Construction & Verification","id":"261","title":"Proof Construction & Verification"},"262":{"body":"A proof consists of the following data structures: /// Evaluations of a polynomial at 2 points\n#[serde_as]\n#[derive(Copy, Clone, Serialize, Deserialize, Default, Debug)]\n#[cfg_attr( feature = \"ocaml_types\", derive(ocaml::IntoValue, ocaml::FromValue, ocaml_gen::Struct)\n)]\n#[serde(bound( serialize = \"Vec: serde_with::SerializeAs\", deserialize = \"Vec: serde_with::DeserializeAs<'de, Evals>\"\n))]\npub struct PointEvaluations { /// Evaluation at the challenge point zeta. #[serde_as(as = \"Vec\")] pub zeta: Evals, /// Evaluation at `zeta . omega`, the product of the challenge point and the group generator. #[serde_as(as = \"Vec\")] pub zeta_omega: Evals,\n} // TODO: this should really be vectors here, perhaps create another type for chunked evaluations?\n/// Polynomial evaluations contained in a `ProverProof`.\n/// - **Chunked evaluations** `Field` is instantiated with vectors with a length\n/// that equals the length of the chunk\n/// - **Non chunked evaluations** `Field` is instantiated with a field, so they\n/// are single-sized#[serde_as]\n#[serde_as]\n#[derive(Debug, Clone, Serialize, Deserialize)]\npub struct ProofEvaluations { /// public input polynomials pub public: Option, /// witness polynomials pub w: [Evals; COLUMNS], /// permutation polynomial pub z: Evals, /// permutation polynomials /// (PERMUTS-1 evaluations because the last permutation is only used in /// commitment form) pub s: [Evals; PERMUTS - 1], /// coefficient polynomials pub coefficients: [Evals; COLUMNS], /// evaluation of the generic selector polynomial pub generic_selector: Evals, /// evaluation of the poseidon selector polynomial pub poseidon_selector: Evals, /// evaluation of the elliptic curve addition selector polynomial pub complete_add_selector: Evals, /// evaluation of the elliptic curve variable base scalar multiplication /// selector polynomial pub mul_selector: Evals, /// evaluation of the endoscalar multiplication selector polynomial pub emul_selector: Evals, /// evaluation of the endoscalar multiplication scalar computation selector /// polynomial pub endomul_scalar_selector: Evals, // Optional gates /// evaluation of the RangeCheck0 selector polynomial pub range_check0_selector: Option, /// evaluation of the RangeCheck1 selector polynomial pub range_check1_selector: Option, /// evaluation of the ForeignFieldAdd selector polynomial pub foreign_field_add_selector: Option, /// evaluation of the ForeignFieldMul selector polynomial pub foreign_field_mul_selector: Option, /// evaluation of the Xor selector polynomial pub xor_selector: Option, /// evaluation of the Rot selector polynomial pub rot_selector: Option, // lookup-related evaluations /// evaluation of lookup aggregation polynomial pub lookup_aggregation: Option, /// evaluation of lookup table polynomial pub lookup_table: Option, /// evaluation of lookup sorted polynomials pub lookup_sorted: [Option; 5], /// evaluation of runtime lookup table polynomial pub runtime_lookup_table: Option, // lookup selectors /// evaluation of the runtime lookup table selector polynomial pub runtime_lookup_table_selector: Option, /// evaluation of the Xor range check pattern selector polynomial pub xor_lookup_selector: Option, /// evaluation of the Lookup range check pattern selector polynomial pub lookup_gate_lookup_selector: Option, /// evaluation of the RangeCheck range check pattern selector polynomial pub range_check_lookup_selector: Option, /// evaluation of the ForeignFieldMul range check pattern selector /// polynomial pub foreign_field_mul_lookup_selector: Option,\n} /// Commitments linked to the lookup feature\n#[serde_as]\n#[derive(Debug, Clone, Serialize, Deserialize)]\n#[serde(bound = \"G: ark_serialize::CanonicalDeserialize + ark_serialize::CanonicalSerialize\")]\npub struct LookupCommitments { /// Commitments to the sorted lookup table polynomial (may have chunks) pub sorted: Vec>, /// Commitment to the lookup aggregation polynomial pub aggreg: PolyComm, /// Optional commitment to concatenated runtime tables pub runtime: Option>,\n} /// All the commitments that the prover creates as part of the proof.\n#[serde_as]\n#[derive(Debug, Clone, Serialize, Deserialize)]\n#[serde(bound = \"G: ark_serialize::CanonicalDeserialize + ark_serialize::CanonicalSerialize\")]\npub struct ProverCommitments { /// The commitments to the witness (execution trace) pub w_comm: [PolyComm; COLUMNS], /// The commitment to the permutation polynomial pub z_comm: PolyComm, /// The commitment to the quotient polynomial pub t_comm: PolyComm, /// Commitments related to the lookup argument pub lookup: Option>,\n} /// The proof that the prover creates from a\n/// [ProverIndex](super::prover_index::ProverIndex) and a `witness`.\n#[serde_as]\n#[derive(Debug, Clone, Serialize, Deserialize)]\n#[serde(bound = \"G: ark_serialize::CanonicalDeserialize + ark_serialize::CanonicalSerialize\")]\npub struct ProverProof { /// All the polynomial commitments required in the proof pub commitments: ProverCommitments, /// batched commitment opening proof #[serde(bound( serialize = \"OpeningProof: Serialize\", deserialize = \"OpeningProof: Deserialize<'de>\" ))] pub proof: OpeningProof, /// Two evaluations over a number of committed polynomials pub evals: ProofEvaluations>>, /// Required evaluation for [Maller's /// optimization](https://o1-labs.github.io/proof-systems/kimchi/maller_15.html#the-evaluation-of-l) #[serde_as(as = \"o1_utils::serialization::SerdeAs\")] pub ft_eval1: G::ScalarField, /// The challenges underlying the optional polynomials folded into the proof pub prev_challenges: Vec>,\n} /// A struct to store the challenges inside a `ProverProof`\n#[serde_as]\n#[derive(Debug, Clone, Deserialize, Serialize)]\n#[serde(bound = \"G: ark_serialize::CanonicalDeserialize + ark_serialize::CanonicalSerialize\")]\npub struct RecursionChallenge\nwhere G: AffineRepr,\n{ /// Vector of scalar field elements #[serde_as(as = \"Vec\")] pub chals: Vec, /// Polynomial commitment pub comm: PolyComm,\n} The following sections specify how a prover creates a proof, and how a verifier validates a number of proofs.","breadcrumbs":"Kimchi » Proof Structure","id":"262","title":"Proof Structure"},"263":{"body":"To create a proof, the prover expects: A prover index, containing a representation of the circuit (and optionaly pre-computed values to be used in the proof creation). The (filled) registers table, representing parts of the execution trace of the circuit. Note The public input is expected to be passed in the first Public rows of the registers table. The following constants are set: EVAL_POINTS = 2. This is the number of points that the prover has to evaluate their polynomials at. ($\\zeta$ and $\\zeta\\omega$ where $\\zeta$ will be deterministically generated.) ZK_ROWS = 3. This is the number of rows that will be randomized to provide zero-knowledgeness. Note that it only needs to be greater or equal to the number of evaluations (2) in the protocol. Yet, it contains one extra row to take into account the last constraint (final value of the permutation accumulator). (TODO: treat the final constraint separately so that ZK_ROWS = 2) The prover then follows the following steps to create the proof: Ensure we have room in the witness for the zero-knowledge rows. We currently expect the witness not to be of the same length as the domain, but instead be of the length of the (smaller) circuit. If we cannot add zk_rows rows to the columns of the witness before reaching the size of the domain, abort. Pad the witness columns with Zero gates to make them the same length as the domain. Then, randomize the last zk_rows of each columns. Setup the Fq-Sponge. Absorb the digest of the VerifierIndex. Absorb the commitments of the previous challenges with the Fq-sponge. Compute the negated public input polynomial as the polynomial that evaluates to $-p_i$ for the first public_input_size values of the domain, and $0$ for the rest. Commit (non-hiding) to the negated public input polynomial. Absorb the commitment to the public polynomial with the Fq-Sponge. Note: unlike the original PLONK protocol, the prover also provides evaluations of the public polynomial to help the verifier circuit. This is why we need to absorb the commitment to the public polynomial at this point. Commit to the witness columns by creating COLUMNS hidding commitments. Note: since the witness is in evaluation form, we can use the commit_evaluation optimization. Absorb the witness commitments with the Fq-Sponge. Compute the witness polynomials by interpolating each COLUMNS of the witness. As mentioned above, we commit using the evaluations form rather than the coefficients form so we can take advantage of the sparsity of the evaluations (i.e., there are many 0 entries and entries that have less-than-full-size field elemnts.) If using lookup: if using runtime table: check that all the provided runtime tables have length and IDs that match the runtime table configuration of the index we expect the given runtime tables to be sorted as configured, this makes it easier afterwards calculate the contribution to the second column of the lookup table (the runtime vector) If queries involve a lookup table with multiple columns then squeeze the Fq-Sponge to obtain the joint combiner challenge $j'$, otherwise set the joint combiner challenge $j'$ to $0$. Derive the scalar joint combiner $j$ from $j'$ using the endomorphism (TODO: specify) If multiple lookup tables are involved, set the table_id_combiner as the $j^i$ with $i$ the maximum width of any used table. Essentially, this is to add a last column of table ids to the concatenated lookup tables. Compute the dummy lookup value as the combination of the last entry of the XOR table (so (0, 0, 0)). Warning: This assumes that we always use the XOR table when using lookups. Compute the lookup table values as the combination of the lookup table entries. Compute the sorted evaluations. Randomize the last EVALS rows in each of the sorted polynomials in order to add zero-knowledge to the protocol. Commit each of the sorted polynomials. Absorb each commitments to the sorted polynomials. Sample $\\beta$ with the Fq-Sponge. Sample $\\gamma$ with the Fq-Sponge. If using lookup: Compute the lookup aggregation polynomial. Commit to the aggregation polynomial. Absorb the commitment to the aggregation polynomial with the Fq-Sponge. Compute the permutation aggregation polynomial $z$. Commit (hidding) to the permutation aggregation polynomial $z$. Absorb the permutation aggregation polynomial $z$ with the Fq-Sponge. Sample $\\alpha'$ with the Fq-Sponge. Derive $\\alpha$ from $\\alpha'$ using the endomorphism (TODO: details) TODO: instantiate alpha? Compute the quotient polynomial (the $t$ in $f = Z_H \\cdot t$). The quotient polynomial is computed by adding all these polynomials together: the combined constraints for all the gates the combined constraints for the permutation TODO: lookup the negated public polynomial and by then dividing the resulting polynomial with the vanishing polynomial $Z_H$. TODO: specify the split of the permutation polynomial into perm and bnd? commit (hiding) to the quotient polynomial $t$ Absorb the commitment of the quotient polynomial with the Fq-Sponge. Sample $\\zeta'$ with the Fq-Sponge. Derive $\\zeta$ from $\\zeta'$ using the endomorphism (TODO: specify) If lookup is used, evaluate the following polynomials at $\\zeta$ and $\\zeta \\omega$: the aggregation polynomial the sorted polynomials the table polynonial Chunk evaluate the following polynomials at both $\\zeta$ and $\\zeta \\omega$: $s_i$ $w_i$ $z$ lookup (TODO, see this issue ) generic selector poseidon selector By \"chunk evaluate\" we mean that the evaluation of each polynomial can potentially be a vector of values. This is because the index's max_poly_size parameter dictates the maximum size of a polynomial in the protocol. If a polynomial $f$ exceeds this size, it must be split into several polynomials like so: $$f(x) = f_0(x) + x^n f_1(x) + x^{2n} f_2(x) + \\cdots$$ And the evaluation of such a polynomial is the following list for $x \\in {\\zeta, \\zeta\\omega}$: $$(f_0(x), f_1(x), f_2(x), \\ldots)$$ TODO: do we want to specify more on that? It seems unnecessary except for the t polynomial (or if for some reason someone sets that to a low value) Evaluate the same polynomials without chunking them (so that each polynomial should correspond to a single value this time). Compute the ft polynomial. This is to implement Maller's optimization . construct the blinding part of the ft polynomial commitment see this section Evaluate the ft polynomial at $\\zeta\\omega$ only. Setup the Fr-Sponge Squeeze the Fq-sponge and absorb the result with the Fr-Sponge. Absorb the previous recursion challenges. Compute evaluations for the previous recursion challenges. Absorb the unique evaluation of ft: $ft(\\zeta\\omega)$. Absorb all the polynomial evaluations in $\\zeta$ and $\\zeta\\omega$: the public polynomial z generic selector poseidon selector the 15 register/witness 6 sigmas evaluations (the last one is not evaluated) Sample $v'$ with the Fr-Sponge Derive $v$ from $v'$ using the endomorphism (TODO: specify) Sample $u'$ with the Fr-Sponge Derive $u$ from $u'$ using the endomorphism (TODO: specify) Create a list of all polynomials that will require evaluations (and evaluation proofs) in the protocol. First, include the previous challenges, in case we are in a recursive prover. Then, include: the negated public polynomial the ft polynomial the permutation aggregation polynomial z polynomial the generic selector the poseidon selector the 15 registers/witness columns the 6 sigmas the optional gates optionally, the runtime table if using lookup: add the lookup sorted polynomials add the lookup aggreg polynomial add the combined table polynomial if present, add the runtime table polynomial the lookup selectors Create an aggregated evaluation proof for all of these polynomials at $\\zeta$ and $\\zeta\\omega$ using $u$ and $v$.","breadcrumbs":"Kimchi » Proof Creation","id":"263","title":"Proof Creation"},"264":{"body":"TODO: we talk about batch verification, but is there an actual batch operation? It seems like we're just verifying an aggregated opening proof We define two helper algorithms below, used in the batch verification of proofs. Fiat-Shamir argument We run the following algorithm: Setup the Fq-Sponge. This sponge mostly absorbs group Absorb the digest of the VerifierIndex. Absorb the commitments of the previous challenges with the Fq-sponge. Absorb the commitment of the public input polynomial with the Fq-Sponge. Absorb the commitments to the registers / witness columns with the Fq-Sponge. If lookup is used: If it involves queries to a multiple-column lookup table, then squeeze the Fq-Sponge to obtain the joint combiner challenge $j'$, otherwise set the joint combiner challenge $j'$ to $0$. Derive the scalar joint combiner challenge $j$ from $j'$ using the endomorphism. (TODO: specify endomorphism) absorb the commitments to the sorted polynomials. Sample the first permutation challenge $\\beta$ with the Fq-Sponge. Sample the second permutation challenge $\\gamma$ with the Fq-Sponge. If using lookup, absorb the commitment to the aggregation lookup polynomial. Absorb the commitment to the permutation trace with the Fq-Sponge. Sample the quotient challenge $\\alpha'$ with the Fq-Sponge. Derive $\\alpha$ from $\\alpha'$ using the endomorphism (TODO: details). Enforce that the length of the $t$ commitment is of size 7. Absorb the commitment to the quotient polynomial $t$ into the argument. Sample $\\zeta'$ with the Fq-Sponge. Derive $\\zeta$ from $\\zeta'$ using the endomorphism (TODO: specify). Setup the Fr-Sponge. This sponge absorbs elements from Squeeze the Fq-sponge and absorb the result with the Fr-Sponge. Absorb the previous recursion challenges. Compute evaluations for the previous recursion challenges. Evaluate the negated public polynomial (if present) at $\\zeta$ and $\\zeta\\omega$. NOTE: this works only in the case when the poly segment size is not smaller than that of the domain. Absorb the unique evaluation of ft: $ft(\\zeta\\omega)$. Absorb all the polynomial evaluations in $\\zeta$ and $\\zeta\\omega$: the public polynomial z generic selector poseidon selector the 15 register/witness 6 sigmas evaluations (the last one is not evaluated) Sample the \"polyscale\" $v'$ with the Fr-Sponge. Derive $v$ from $v'$ using the endomorphism (TODO: specify). Sample the \"evalscale\" $u'$ with the Fr-Sponge. Derive $u$ from $u'$ using the endomorphism (TODO: specify). Create a list of all polynomials that have an evaluation proof. Compute the evaluation of $ft(\\zeta)$. Partial verification For every proof we want to verify, we defer the proof opening to the very end. This allows us to potentially batch verify a number of partially verified proofs. Essentially, this steps verifies that $f(\\zeta) = t(\\zeta) * Z_H(\\zeta)$. Check the length of evaluations inside the proof. Commit to the negated public input polynomial. Run the Fiat-Shamir argument . Combine the chunked polynomials' evaluations (TODO: most likely only the quotient polynomial is chunked) with the right powers of $\\zeta^n$ and $(\\zeta * \\omega)^n$. Compute the commitment to the linearized polynomial $f$. To do this, add the constraints of all of the gates, of the permutation, and optionally of the lookup. (See the separate sections in the constraints section.) Any polynomial should be replaced by its associated commitment, contained in the verifier index or in the proof, unless a polynomial has its evaluation provided by the proof in which case the evaluation should be used in place of the commitment. Compute the (chuncked) commitment of $ft$ (see Maller's optimization ). List the polynomial commitments, and their associated evaluations, that are associated to the aggregated evaluation proof in the proof: recursion public input commitment ft commitment (chunks of it) permutation commitment index commitments that use the coefficients witness commitments coefficient commitments sigma commitments optional gate commitments lookup commitments Batch verification of proofs Below, we define the steps to verify a number of proofs (each associated to a verifier index ). You can, of course, use it to verify a single proof. If there's no proof to verify, the proof validates trivially. Ensure that all the proof's verifier index have a URS of the same length. (TODO: do they have to be the same URS though? should we check for that?) Validate each proof separately following the partial verification steps. Use the PolyCom.verify to verify the partially evaluated proofs.","breadcrumbs":"Kimchi » Proof Verification","id":"264","title":"Proof Verification"},"265":{"body":"commit_evaluation: TODO","breadcrumbs":"Kimchi » Optimizations","id":"265","title":"Optimizations"},"266":{"body":"TODO","breadcrumbs":"Kimchi » Security Considerations","id":"266","title":"Security Considerations"},"267":{"body":"Note Note that the URS is called SRS in our codebase at the moment. SRS is incorrect, as it is used in the presence of a trusted setup (which kimchi does not have). This needs to be fixed. The URS comprises of: Gs: an arbitrary ordered list of curve points that can be used to commit to a polynomial in a non-hiding way. H: a blinding curve point that can be used to add hidding to a polynomial commitment scheme. The URS is generated deterministically, and thus can be rederived if not stored. As different circuits of different size do not share the same domain, and the size of the domain is directly equal to the number of Gs curve points we use to commit to polynomials, we often want to share the same URS of maximum size across many circuits (constraint systems). Circuit associated with smaller domains will simply use a truncation of the shared URS. Note A circuit's domain is generated as the first power of 2 that is equal, or larger, to the number of gates used in the circuit (including zero-knowledge gates).","breadcrumbs":"Universal Reference String (URS) » Universal Reference String (URS)","id":"267","title":"Universal Reference String (URS)"},"268":{"body":"TODO: specify this part","breadcrumbs":"Universal Reference String (URS) » Group map","id":"268","title":"Group map"},"269":{"body":"The following pseudo-code specifies how the Gs curve points, as well as the H curve point, are generated. def create(depth): # generators Gs = [] for i in range(0, depth): digest = Blake2b512.hash(be_encode(i)) Gs.push(point_of_random_bytes(digest)) # blinding point digest = (b\"srs_misc\" || be_encode(0)) H = point_of_random_bytes(digest) # return (Gs, H) def point_of_random_bytes(random_bytes): # packing in bit-representation const N: usize = 31 let mut bits = [false; 8 * N] for i in range(0, N) { for j in range(0, 8) { bits[8 * i + j] = (random_bytes[i] >> j) & 1 == 1; } } let n = BigInt::from_bits_be(&bits); let t = G::BaseField::from_repr(n) return map.to_group(t) TODO: specify map.to_group in the previous section.","breadcrumbs":"Universal Reference String (URS) » Curve points to commit to a polynomial","id":"269","title":"Curve points to commit to a polynomial"},"27":{"body":"We can commit to several values x1​,⋯,xn​ by sending separate Pedersen commitments to all of these values as such: x1​G+r1​H,⋮xn​G+rn​H But we can instead batch/aggregate all of these commitments together as a single commitment: x1​G1​+⋯+xn​Gn​+rH with G1​,⋯,Gn​,H independent bases with unknown discrete logarithms. If you represent xs and the Gs as two vectors x=(x1​,⋯,xn​) and G=(G1​,⋯,Gn​), we can quickly write the previous statement as an inner product xG+rH Vector commitments (sometimes referred to as multi-commitments) are a powerful construction because an arbitrarily large vector can be committed with a single curve point. The naive approach to constructing an opening proof for a length n vector commitment has size O(n). It is simply the tuple (x1​,…,xn​,r). As we will see later, opening proofs for vector commitments is an interesting topic and there is a much more efficient algorithm.","breadcrumbs":"Commitments » Vector commitments","id":"27","title":"Vector commitments"},"270":{"body":"As there is no limit to the number of commitment curve points you can generate, we only provide the first three ones to serve as test vectors. TODO: specify the encoding","breadcrumbs":"Universal Reference String (URS) » URS values in practice","id":"270","title":"URS values in practice"},"271":{"body":"Gs . G0 = (x=121C4426885FD5A9701385AAF8D43E52E7660F1FC5AFC5F6468CC55312FC60F8, y=21B439C01247EA3518C5DDEB324E4CB108AF617780DDF766D96D3FD8AB028B70)\nG1 = (x=26C9349FF7FB4AB230A6F6AEF045F451FBBE9B37C43C3274E2AA4B82D131FD26, y=1996274D67EC0464C51F79CCFA1F511C2AABB666ABE67733EE8185B71B27A504)\nG2 = (x=26985F27306586711466C5B2C28754AA62FE33516D75CEF1F7751F1A169713FD, y=2E8930092FE6A18B331CE0E6E27B413AA18E76394F18A2835DA9FAE10AA3229D) H : H = (x=092060386301C999AAB4F263757836369CA27975E28BC7A8E5B2CE5B26262201, y=314FC4D83AE66A509F9D41BE6165F2606A209A9B5805EE85CE20249C5EBCBE26)","breadcrumbs":"Universal Reference String (URS) » Vesta","id":"271","title":"Vesta"},"272":{"body":"G0 = (x363D83141FD1E0540718FADBA7278ABAEEDB46D7A3F050F2CFF1DF4F300C9C30, y=034C68F4079B4F338A19BE2D7BFA44B395C65B9790DD273F361327446C778764)\nG1 = (x2CC40B77D87665244AE5EB5304E8744004C80061AD08476A0F0656C13134EA45, y=28146EC860159DB55CB5EA5B14F0AA2F8751DEDFE0DDAFD1C313B15575C4B4AC)\nG2 = (x2808BC21BEB90314377BF6130285FABE6CE4B8A4457FB25BC95EBA0083DF27E3, y=1E04E53DD6395FAB8018D7FE98F9C7FAB39C40BFBE48589626A7B8532728B002) H : H = (x221B959DACD2052AAE26193FCA36B53279866A4FBBAB0D5A2F828B5FD7778201, y=058C8F1105CAE57F4891EADC9B85C8954E5067190E155E61D66855ACE69C16C0)","breadcrumbs":"Universal Reference String (URS) » Pallas","id":"272","title":"Pallas"},"273":{"body":"Pickles is Mina’s inductive zk-SNARK composition system. It lets you construct proofs with zk-SNARKs and combine them in flexible ways to deliver incremental verifiable computation. Pickles uses a pair of amicable curves called Pasta in order to deliver incremental verifiable computation efficiently. These curves are referred to as \"tick\" and \"tock\" within the Mina source code. Tick - Vesta (a.k.a. Step), constraint domain size 218 (block and transaction proofs) Tock - Pallas (a.k.a. Wrap), constraint domain size 217 (signatures) The Tock prover does less (only performs recursive verifications and no other logic), so it requires fewer constraints and has a smaller domain size. Internally Pickles refers to Tick and Tock as Step and Wrap, respectively. Tock is used to prove the verification of a Tick proof and outputs a Tick proof. Tick is used to prove the verification of a Tock proof and outputs a Tock proof. In other words, Provetock​(Verify(Tick))=Tickproof​ Provetick​(Verify(Tock))=Tockproof​ Both Tick and Tock can verify at most 2 proofs of the opposite kind, though, theoretically more is possible. Currently, in Mina we have the following situation. Every Tock always wraps 1 Tick proof, such as Tick proofs can verify 2 Tock proofs Blockchain SNARK takes previous blockchain SNARK proof and a transaction proof Verifying two Tock transaction proofs","breadcrumbs":"Pickles » Pickles","id":"273","title":"Pickles"},"274":{"body":"vrf is appendix C of https://eprint.iacr.org/2017/573.pdf","breadcrumbs":"Consensus » Consensus","id":"274","title":"Consensus"},"28":{"body":"To construct SNARKs we use polynomial commitments. A polynomial commitment scheme for a field F (or it could even be a ring) is a way of committing to a polynomial f∈F[x] to get a commitment c, in such a way that for any α∈F, you can provide y=f(α), along with an \"opening proof\" π that proves that the polynomial committed to in c equals y when evaluated at α. In other words, it is a type of commitment C, a type of randomness R, a type of opening proof P along with algorithms commit:F[x]×R→Copen:C×F×(F[x]×R)→F×Pverify:C×(F×P)→Bool such that for any f∈F[x],r∈R,α∈F , we have c:=commit(f,r)verify(c,open(c,α,(f,r)))=true and if b=f(α) then it is not possible to compute π∈P such that verify(c,(b,π))=true In other words, if b=f(α) then every π which is feasible to compute results in verify(c,(b,π))=false. One thing that's pretty cool is that because polynomial commitment schemes let you construct zk-SNARKs, polynomial commitment schemes imply commitment schemes with arbitrary opening functionality. TODO","breadcrumbs":"Commitments » Polynomial commitments","id":"28","title":"Polynomial commitments"},"29":{"body":"All known constructions of polynomial commitment schemes are a bit complicated. The easiest to describe is called the Kate (pronounced \"kah-TAY\") scheme, also known as \"KZG\". It requires a \"prime-order group with a pairing\", which is three groups G1​,G2​,GT​ of prime order p (hence, all isomorphic cyclic groups) together with a function e:G1​×G2​→GT​ such that for any a1​,a2​∈Z, g1​∈G1​, g2​∈G2​, we have e(a1​⋅g1​,a2​⋅g2​)=a1​a2​⋅e(g1​,g2​) e is called a \"pairing\" or a \"bilinear pairing\". What this lets us do is \"multiply in the scalar\" but only once. Fix a degree bound d on the polynomials we would like to be able to commit to. The KZG scheme, will let us commit to polynomials in Fp​[x]| |<------ b | OT Prot. | | |-------> x_b +----------+ Due to a result of Impagliazzo and Rudich in this paper , it is very unlikely that OT is possible without the use of public-key cryptography. However, OT can be efficiently extended. That is, starting with a small number of base OTs, one could create many more OTs with only symmetric primitives. The seminal work of IKNP03 presented a very efficient protocol for extending OTs, requiring only black-box use of symmetric primitives and λ base OTs, where λ is the security parameter. This doc focuses on the family of protocols inspired by IKNP.","breadcrumbs":"Two Party Computation » Oblivious Transfer » Oblivious Transfer","id":"66","title":"Oblivious Transfer"},"67":{"body":"We introduce the CO15 OT as the base OT. Notation . The protocol is described over an additive group (G,G,p,+) of prime order p generated by G. Denote H:(G×G)×G→{0,1}λ as a key-derivation function to extract a λ-bit key from group elements. Note that H here is different from the tweakable hash function in garbled circuit. In the CO15 protocol, the sender holds n 128-bit strings {(Xi0​,Xi1​)}i∈[n]​ and the receiver holds n-bit string b∈{0,1}n. The protocol is as follows. The sender samples y←Fp​, and computes S=y⋅G and T=y⋅S. The sender sends S to the receiver, who aborts if S∈/G. For i∈[n], the receiver samples xi​←Fp​, and computes: Ri​=bi​S+xi​G The receiver sends {Ri​}i∈[n]​ to the sender, who aborts if Ri​∈/G. For i∈[n], the sender computes ki0​=H(S,Ri​,yRi​), ki1​=H(S,Ri​,yRi​−T) and sends {(ci0​=ki0​⊕Xi0​,ci1​=ki1​⊕Xi1​)}i∈[n]​ to the receiver. For i∈[n], the receiver computes ki​=H(S,Ri​,xi​S), and outputs cibi​​⊕ki​ Correctness . The receiver always computes the hash of xi​S=(xi​y)G. The sender sends the hashes of yRi​ and yRi​−T. If bi​=0, ki0​ is the hash of yRi​=(yxi​)G, then the receive will get Xi0​. if bi​=1, ki1​ is the hash of yRi​−T=y(S+xi​G)−T=(yxi​)G, then the receiver will get Xi1​. Security . We refer the security analysis to CO15 .","breadcrumbs":"Two Party Computation » Oblivious Transfer » Base OT » Base OT","id":"67","title":"Base OT"},"68":{"body":"To meet the large-scale demand of OTs, OT extensions can be used. An OT extension protocol works by running a small number of base OTs that are used as a base for obtaining many OTs via the use of cheap symmetric cryptographic operations only. This is conceptually similar to hybrid encryption. In this subsection, we focus on the maliciously secure OT extension presented in KOS15 . To be compatible with the garbled circuit optimizations, Correlated OT (COT) is considered here. COT is a variant of OT, where the sender chooses a secret string Δ and the receiver has a choice bit b. After the protocol, then sender obtains two strings X0​=X and X1​=X⊕Δ and the receiver obtains Xb​, where X is a random string. Sender Receiver +----------+ Delta ------->| |<------ b | COT Prot.| x_0 = x, x_1 = x + Delta <-------| |-------> x_b +----------+ Notation . Let λ be the security parameter, we use λ=128 in zkOracles. Identify F2λ​ with the finite field F2λ​. Use \"⋅\" for multiplication in F2λ​ and \"⋆\" for the component-wise product in F2λ​. Given a matrix A, denote its rows by subindices ai​ and its columns by superindices ak. Use v[i] to denote the i-th entry of v. The KOS15 OT protocol to generate ℓ(≫λ) OTs is described as follows. Initialize The receiver samples λ pairs of λ-bit seeds {(k0i​,k1i​)}i=1λ​. The sender choose a uniformly random λ-bit string Δ. The two parties calls the base OT with inputs Δ and {(k0i​,k1i​)}i=1λ​. Note that the sender acts as the receiver in the base OT, and the receiver acts as the sender in the base OT. The sender obtains kΔi​i​ for i∈[λ]. Extend The receiver takes as input the choice bits x1​,...,xℓ​, defines ℓ′=ℓ+λ+s, where s is the statistic parameter, and we set s=40 in zkOracles. Let x=x1​∥x2​∥...∥xℓ​∥x′∈F2ℓ′​, with x′∈F2ℓ′−ℓ​ uniformly chosen. The receiver defines vectors x1​,...,xℓ′​ as xi​=x[i]⋅(1,...,1)∈F2λ​ for i∈[ℓ′]. The receiver expands k0i​ and k1i​ with a pseudo random generator (PRG), letting t0i​=PRG(k0i​)∈F2ℓ′​ and t1i​=PRG(k1i​)∈F2ℓ′​ for i∈[λ] The sender expands kΔi​i​ with the same PRG and gets tΔi​i​=PRG(kΔi​i​) for i∈[λ] The receiver computes ui=t0i​⊕t1i​⊕x∈F2ℓ′​ for i∈[λ] and sends them to the sender. The sender computes qi=Δi​⋅ui⊕tΔi​i​=t0i​⊕Δi​⋅x∈F2ℓ′​ for i∈[λ] Let qj​ be the j-th row of the ℓ′×λ matrix Q=[q1∣q2∣⋯∣qλ], and similarly let tj​ be the j-th row of T=[t01​∣t02​∣⋯∣t0λ​]. Note that qj​=tj​⊕(xj​⋆Δ)=tj​⊕(x[j]⋅Δ) for j∈[ℓ′] Correlation Check The sender and receiver run a πRAND​(F2λℓ′​) protocol to obtain random elements χ1​,...,χℓ′​. πRAND​(F2λℓ′​) will be described later. The receiver computes x=j=1∑ℓ′​x[j]⋅χj​ and t=j=1∑ℓ′​tj​⋅χj​ and sends them to the sender. The sender computes q=j=1∑ℓ′​qj​⋅χj​ and checks t=q⊕x⋅Δ. If the check fails, abort. Output The sender computes vj​=H(j,qj​)⊕H(j,qj​⊕Δ)⊕Δ for j∈[ℓ] and sends them to the receiver. H is a tweakable hash function as used in garbled circuit. The sender outputs (H(j,qj​), H(j,qj​)⊕Δ) For j∈[ℓ], the receiver outputs the following: If xj​=0, outputs H(j,tj​); If xj​=1, outputs H(j,tj​)⊕vj​ This ends the description of the KOS15 protocol. We note that a very recent work pointed out that there is a flaw in the security proof of the main lemma of KOS15 and provided a fix, which is less efficient. They also pointed out that this the original KOS15 protocol is still secure in practice. Therefore, we choose the original KOS15 protocol in zkOracles in this version. We begin to describe the πRAND​(F2λℓ​) protocol as follows. The sender and receiver locally choose uniformly 128-bit random seeds s0​ and s1​, respectively. The sender and receiver compute commitments of the seeds by choosing 128-bit random strings r0​,r1​ and generate c0​=SHA256(s0​∥r0​) and c1​=SHA256(s1​∥r1​), respectively. The sender sends c0​ to the receiver, and the receiver sends c1​ to the sender. On receiving the commitments, the sender and receiver open the commitments by sending (s0​,r0​) and (s1​,r1​) to each other, respectively. The sender checks c1​ by re-computing it with s1​,r1​, and the receiver checks c0​ by re-computing it with s0​,r0​. If checks fail, abort. The sender and receiver locally compute s=s0​⊕s1​ and generate random elements with a PRG as χ1​,...,χℓ​←PRG(s)","breadcrumbs":"Two Party Computation » Oblivious Transfer » OT Extension » OT Extension","id":"68","title":"OT Extension"},"69":{"body":"With garbled circuit and oblivious transfer, we are ready to describe the well-known Yao's method to construct secure two-party computation protocols for any polynomial-size function. Given any function F(x,y), where x is the private input of Alice, and y is the private input of Bob. Let x=x1​∥x2​∥⋯∥xℓ1​​ and y=y1​∥y2​∥⋯∥yℓ2​​, where xi​ and yi​ are bits. Let Alice be the garbler and Bob be the evaluator, Yao's protocol is described as follows. Alice and Bob run a COT protocol to generate ℓ2​ OTs, where Alice is the sender and Bob is the receiver. Alice obtains (Y1​,Y1​⊕Δ),...,(Yℓ2​​,Yℓ2​​⊕Δ), and Bob obtains (Y1​⊕y1​⋅Δ,...,Yℓ2​​⊕yℓ2​​⋅Δ) Alice chooses uniformly random labels X1​,X2​,...,Xℓ1​​. Alice uses (X1​,...,Xℓ1​​,Y1​,...,Yℓ2​​) and global Δ to generate the garbled circuit of F and sends GC(F) to Bob. Alice also sends the decoding information to Bob. Alice encodes her inputs and sends X1​⊕x1​⋅Δ,...,Xℓ1​​⊕xℓ1​​⋅Δ to Bob. With the encoded X labels and Y labels, Bob evaluates GC(F) to get the output labels, and then decodes these output labels with decoding information and gets the output bits.","breadcrumbs":"Two Party Computation » Full Protocol » Full Protocol","id":"69","title":"Full Protocol"},"7":{"body":"When cryptographers talk about \"groups\", they usually mean a \"computational group\", which is a group equipped with efficient algorithms for computing the group operation and inversion. This is different because a group in the mathematical sense need not have its operations be computable at all.","breadcrumbs":"Groups » More things to know","id":"7","title":"More things to know"},"70":{"body":"Many modern proof systems (and I think all that are in use) are constructed according to the following recipe. You start out with a class of computations. You devise a way to arithmetize those computations. That is, to express your computation as a statement about polynomials. More specifically, you describe what is often called an \"algebraic interactive oracle proof\" (AIOP) that encodes your computation. An AIOP is a protocol describing an interaction between a prover and a verifier, in which the prover sends the verifier some \"polynomial oracles\" (basically a black box function that given a point evaluates a polynomial at that point), the verifier sends the prover random challenges, and at the end, the verifier queries the prover's polynomials at points of its choosing and makes a decision as to whether it has been satisfied by the proof. An AIOP is an imagined interaction between parties. It is an abstract description of the protocol that will be \"compiled\" into a SNARK. There are several \"non-realistic\" aspects about it. One is that the prover sends the verifier black-box polynomials that the verifier can evaluate. These polynomials have degree comparable to the size of the computation being verified. If we implemented these \"polynomial oracles\" by having the prover really send the O(n) size polynomials (say by sending all their coefficients), then we would not have a zk-SNARK at all, since the verifier would have to read this linearly sized polynomial so we would lose succinctness, and the polynomials would not be black-box functions, so we may lose zero-knowledge. Instead, when we concretely instantiate the AIOP, we have the prover send constant-sized, hiding polynomial commitments . Then, in the phase of the AIOP where the verifier queries the polynomials, the prover sends an opening proof for the polynomial commitments which the verifier can check, thus simulating the activity of evaluating the prover's polynomials on your own. So this is the next step of making a SNARK: instantiating the AIOP with a polynomial commitment scheme of one's choosing. There are several choices here and these affect the properties of the SNARK you are constructing, as the SNARK will inherit efficiency and setup properties of the polynomial commitment scheme used. An AIOP describes an interactive protocol between the verifier and the prover. In reality, typically, we also want our proofs to be non-interactive. This is accomplished by what is called the Fiat--Shamir transformation . The basic idea is this: all that the verifier is doing is sampling random values to send to the prover. Instead, to generate a \"random\" value, the prover simulates the verifier by hashing its messages. The resulting hash is used as the \"random\" challenge. At this point we have a fully non-interactive proof. Let's review our steps. Start with a computation. Translate the computation into a statement about polynomials and design a corresponding AIOP. Compile the AIOP into an interactive protocol by having the prover send hiding polynomial commitments instead of polynomial oracles. Get rid of the verifier-interaction by replacing it with a hash function. I.e., apply the Fiat--Shamir transform.","breadcrumbs":"Proof Systems » Proof Systems Design Overview","id":"70","title":"Proof Systems Design Overview"},"71":{"body":"Finally we can discuss zk-SNARKs. In these notes right now I will not start with intuitive motivations for SNARKs because I can't think of them right now, but maybe later. Let f:A×W→Bool be a program (in some yet to be specified programming language). For a:A, zk-SNARKs let you produce proofs for statements of the form I know w:W such that f(a,w)=true in such a way that no information about w is revealed by the proof (this is the zero-knowledge or ZK part of them). In other words, zk-SNARKs allow you to prove execution of programs, where you are also allowed to hide a selected portion (i.e., the W portion) of the data involved in the computation. We could also phrase zk-SNARKs in terms of computing functions f:A×W→B, which seems more general, but actually this would be taken care of by the previous sense where we can only use boolean-valued functions, by proving against the function g:(A×B)×W→Bool defined by g((a,b),w):=b=?​f(a,w). SNARK stands for Succinct Non-interactive ARguments of Knowledge . If it also satisfies zero-knowledge, then it is called a zk-SNARK. Preprocessing SNARKs allow the verifier to precompute some encodings of the relation to run independently of the instance length. Definitions of these schemes owe the succinctness property to the fact that both the verifier and the proof length are (at most) polylogarithmic in the size of the computation (the witness length). Recent schemes such as Plonk, go beyond that definition and provide zkSNARKs with constant proof length. Unlike the verifier in preprocessing SNARKs, the prover can't be faster than linear, as it has to read the circuit at least.","breadcrumbs":"Proof Systems » zk-SNARKs » zk-SNARKs","id":"71","title":"zk-SNARKs"},"72":{"body":"We will define a notion of computation (a programming language basically) for which it is easy to construct zk-SNARKs.","breadcrumbs":"Proof Systems » zk-SNARKs » Expressing computations with \"iterating over trace tables\"","id":"72","title":"Expressing computations with \"iterating over trace tables\""},"73":{"body":"The proof system in Mina is a variant of PlonK . To understand PlonK, you can refer to our series of videos on the scheme. In this section we explain our variant, called kimchi . kimchi is not formally a zk-SNARK, as it is not succinct in the proof size. zk-SNARKs must have a log(n) proof size where n is the number of gates in the circuit. In kimchi, due to the Bootleproof polynomial commitment approach, our proofs are log(d) where d is the maximum degree of the committed polynomials (in the order of the number of rows). In practice, our proofs are in the order of dozens of kilobytes. Note that kimchi is a zk-SNARK in terms of verification complexity (log(n)) and the proving complexity is quasilinear (O(nlogn)) – recall that the prover cannot be faster than linear. Essentially what PlonK allows you to do is to, given a program with inputs and outputs, take a snapshot of its execution. Then, you can remove parts of the inputs, or outputs, or execution trace, and still prove to someone that the execution was performed correctly for the remaining inputs and outputs of the snapshot. There are a lot of steps involved in the transformation of \"I know an input to this program such that when executed with these other inputs it gives this output\" to \"here's a sequence of operations you can execute to convince yourself of that\". At the end of this chapter, you will understand that the protocol boils down to filling a number of tables (illustrated in the diagram below): tables that specify the circuit, and tables that describes an execution trace of the circuit (given some secret and public inputs). kimchi","breadcrumbs":"Overview » Overview","id":"73","title":"Overview"},"74":{"body":"Size: number of rows Columns: number of variables per rows Cell: a pair (row, column) Witness: the values assigned in all the cells Gate: polynomials that act on the variables in a row Selector vector: a vector of values 1 or 0 (not true for constant vector I think) that toggles gates and variables in a row Gadget: a series of contiguous rows with some specific gates set (via selector vectors)","breadcrumbs":"Overview » Glossary » Glossary","id":"74","title":"Glossary"},"75":{"body":"PlonK needs a domain to encode the circuit on (for each point we can encode a row/constraint/gate). We choose a domain H such that it is a multiplicative subgroup of the scalar field of our curve.","breadcrumbs":"Domain » Domain","id":"75","title":"Domain"},"76":{"body":"Furthermore, as FFT (used for interpolation, see the section on FFTs ) works best in domains of size 2k for some k, we choose H to be a subgroup of order 2k. Since the pasta curves both have scalar fields of prime orders p and q, their multiplicative subgroup is of order p−1 and q−1 respectively (without the zero element). The pasta curves were generated specifically so that both p−1 and q−1 are multiples of 232. We say that they have 2-adicity of 32. Looking at the definition of one of the pasta fields in Rust you can see that it is defined specifically for a trait related to FFTs: impl FftParameters for FqParameters { type BigInt = BigInteger; const TWO_ADICITY: u32 = 32; #[rustfmt::skip] const TWO_ADIC_ROOT_OF_UNITY: BigInteger = BigInteger([ 0x218077428c9942de, 0xcc49578921b60494, 0xac2e5d27b2efbee2, 0xb79fa897f2db056 ]); The 2-adicity of 32 means that there's a multiplicative subgroup of size 232 that exists in the field. The code above also defines a generator g for it, such that g232=1 and gi=1 for i∈[[1,232−1]] (so it's a primitive 232-th root of unity). Lagrange's theorem tells us that if we have a group of order n, then we'll have subgroups with orders dividing n. So in our case, we have subgroups with all the powers of 2, up to the 32-th power of 2. To find any of these groups, it is pretty straight forward as well. Notice that: let h=g2, then h231=g232=1 and so h generates a subgroup of order 31 let t=g22, then t230=g232=1 and so t generates a subgroup of order 30 and so on... In arkworks you can see how this is implemented: let size = n.next_power_of_two() as u64;\nlet log_size_of_group = ark_std::log2(usize::try_from(size).expect(\"too large\"));\nomega = Self::TWO_ADIC_ROOT_OF_UNITY;\nfor _ in log_size_of_group..Self::TWO_ADICITY { omega.square_in_place();\n} this allows you to easily find subgroups of different sizes of powers of 2, which is useful in zero-knowledge proof systems as FFT optimizations apply well on domains that are powers of 2. You can read more about that in the mina book .","breadcrumbs":"Domain » 2-adicity","id":"76","title":"2-adicity"},"77":{"body":"For each curve, we generate a domain H as the set H=1,ω,ω2,⋯,ωn−1. As we work in a multiplicative subgroup, the polynomial that vanishes in all of these points can be written and efficiently calculated as ZH​(x)=x∣H∣−1. This is because, by definition, ω∣H∣=1. If x∈H, then x=ωi for some i, and we have: x∣H∣=(ωi)∣H∣=(ω∣H∣)i=1i=1 This optimization is important, as without it calculating the vanishing polynomial would take a number of operations linear in ∣H∣ (which is not doable in a verifier circuit, for recursion). With the optimization, it takes us a logarithmic number of operation (using exponentiation by squaring ) to calculate the vanishing polynomial.","breadcrumbs":"Domain » Efficient computation of the vanishing polynomial","id":"77","title":"Efficient computation of the vanishing polynomial"},"78":{"body":"What's a lagrange base? Li​(x)=1 if x=gi, 0 otherwise.","breadcrumbs":"Lagrange Basis in Multiplicative Subgroups » Lagrange basis in multiplicative subgroups","id":"78","title":"Lagrange basis in multiplicative subgroups"},"79":{"body":"Arkworks has the formula to construct a lagrange base: Evaluate all Lagrange polynomials at τ to get the lagrange coefficients. Define the following as H: The coset we are in, with generator g and offset h m: The size of the coset H ZH​: The vanishing polynomial for H. ZH​(x)=∏i∈[m]​(x−h⋅gi)=xm−hm vi​: A sequence of values, where v0​=m∗hm−11​, and vi+1​=g⋅vi​ We then compute Li,H​(τ) as Li,H​(τ)=ZH​(τ)⋅τ−hgivi​​ However, if τ in H, both the numerator and denominator equal 0 when i corresponds to the value tau equals, and the coefficient is 0 everywhere else. We handle this case separately, and we can easily detect by checking if the vanishing poly is 0. following this, for h=1 we have: L0​(x)=m(x−1)ZH​(x)​ L1​(x)=m(x−g)ZH​(x)g​ L2​(x)=m(x−g2)ZH​(x)g2​ and so on","breadcrumbs":"Lagrange Basis in Multiplicative Subgroups » What's the formula?","id":"79","title":"What's the formula?"},"8":{"body":"Implement a type JacobianPoint and functions for computing the group operation Familiarize yourself with the types and traits in ark_ec. Specifically, todo Implement fn decompress(c: (F, bool)) -> (F, F)","breadcrumbs":"Groups » Exercises","id":"8","title":"Exercises"},"80":{"body":"https://en.wikipedia.org/wiki/Lagrange_polynomial#Barycentric_form","breadcrumbs":"Lagrange Basis in Multiplicative Subgroups » What's the logic here?","id":"80","title":"What's the logic here?"},"81":{"body":"So far we've talked about an interactive protocol between a prover and a verifier. The zero-knowledge proof was also in the honest verifier zero-knowedlge (HVZK) model, which is problematic. In practice, we want to remove the interaction and have the prover produce a proof by themselves, that anyone can verify.","breadcrumbs":"Non-Interactivity via Fiat-Shamir » Non-Interactivity via Fiat-Shamir","id":"81","title":"Non-Interactivity via Fiat-Shamir"},"82":{"body":"public-coin protocols are protocols were the messages of the verifier are simply random messages. This is important as our technique to transform an interactive protocol to a non-interactive protocol works on public-coin protocols.","breadcrumbs":"Non-Interactivity via Fiat-Shamir » Public-coin protocols","id":"82","title":"Public-coin protocols"},"83":{"body":"The whole idea is to replace the verifier by a random oracle, which in practice is a hash function. Note that by doing this, we remove potential leaks that can happen when the verifier acts dishonestly. Initially the Fiat-Shamir transformation was only applied to sigma protocols, named after the greek letter Σ due to its shape resembling the direction of messages (prover sends a commit to a verifier, verifier sends a challenge to a prover, prover sends the final proof to a verifier). A Z would have made more sense but here we are.","breadcrumbs":"Non-Interactivity via Fiat-Shamir » Fiat-Shamir trick","id":"83","title":"Fiat-Shamir trick"},"84":{"body":"As our protocol has more than three moves, where every even move is a challenge from the verifier, we need to generalize Fiat-Shamir. This is simple: every verifier move can be replaced by a hash of the transcript (every message sent and received so far) to obtain a challenge.","breadcrumbs":"Non-Interactivity via Fiat-Shamir » Generalization of Fiat-Shamir","id":"84","title":"Generalization of Fiat-Shamir"},"85":{"body":"While we use a hash function for that, a different construction called the duplex construction is particularly useful in such situations as they allow to continuously absorb the transcript and produce challenges, while automatically authenticating the fact that they produced a challenge. Merlin is a standardization of such a construction using the Strobe protocol framework (a framework to make use of a duplex construction). Note that the more recent Xoodyak (part of NIST's lightweight competition) could have been used for this as well. Note also that Mina uses none of these standards, instead it simply uses Poseidon (see section on poseidon).","breadcrumbs":"Non-Interactivity via Fiat-Shamir » In practice: a duplex construction as Merlin","id":"85","title":"In practice: a duplex construction as Merlin"},"86":{"body":"Plookup allows us to check if witness values belong to a look up table. This is usually useful for reducing the number of constraints needed for bit-wise operations. So in the rest of this document we'll use the XOR table as an example. 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\ No newline at end of file diff --git a/specs/poseidon.html b/specs/poseidon.html index ed18da4ff0..5f1136bb41 100644 --- a/specs/poseidon.html +++ b/specs/poseidon.html @@ -320,7 +320,7 @@

Poseidon-Fp

Poseidon-Fq

You can find the MDS matrix and round constants we use in /poseidon/src/pasta/fp_kimchi.rs.

Test vectors

-

We have test vectors contained in /poseidon/src/tests/test_vectors/kimchi.json.

+

We have test vectors contained in /poseidon/tests/test_vectors/kimchi.json.

Pointers to the OCaml code

  • our ocaml implementation: https://github.com/minaprotocol/mina/blob/develop/src/lib/random_oracle/random_oracle.mli