forked from mit-plv/fiat-crypto
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Basic.v
203 lines (183 loc) · 8.54 KB
/
Basic.v
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
Require Import Coq.Classes.Morphisms.
Require Import Crypto.Spec.CompleteEdwardsCurve Crypto.Curves.Edwards.AffineProofs.
Require Import Crypto.Util.Notations Crypto.Util.GlobalSettings.
Require Export Crypto.Util.FixCoqMistakes.
Require Import Crypto.Util.Decidable.
Require Import Crypto.Util.Tactics.DestructHead.
Require Import Crypto.Util.Tactics.UniquePose.
Section ExtendedCoordinates.
Context {F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}
{field:@Algebra.Hierarchy.field F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}
{char_ge_3 : @Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul (BinNat.N.succ_pos BinNat.N.two)}
{Feq_dec:DecidableRel Feq}.
Local Infix "=" := Feq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope.
Local Notation "0" := Fzero. Local Notation "1" := Fone.
Local Infix "+" := Fadd. Local Infix "*" := Fmul.
Local Infix "-" := Fsub. Local Infix "/" := Fdiv.
Local Notation "x ^ 2" := (x*x).
Context {a d: F}
{nonzero_a : a <> 0}
{square_a : exists sqrt_a, sqrt_a^2 = a}
{nonsquare_d : forall x, x^2 <> d}.
Local Notation Epoint := (@E.point F Feq Fone Fadd Fmul a d).
Local Notation Eadd := (E.add(nonzero_a:=nonzero_a)(square_a:=square_a)(nonsquare_d:=nonsquare_d)).
Local Notation onCurve x y := (a*x^2 + y^2 = 1 + d*x^2*y^2) (only parsing).
(** [Extended.point] represents a point on an elliptic curve using extended projective
* Edwards coordinates 1 (see <https://eprint.iacr.org/2008/522.pdf>). *)
Definition point := { P | let '(X,Y,Z,Ta,Tb) := P in
a * X^2*Z^2 + Y^2*Z^2 = (Z^2)^2 + d * X^2 * Y^2
/\ X * Y = Z * Ta * Tb
/\ Z <> 0 }.
Definition coordinates (P:point) : F*F*F*F*F := proj1_sig P.
Definition eq (P1 P2:point) :=
let '(X1, Y1, Z1, _, _) := coordinates P1 in
let '(X2, Y2, Z2, _, _) := coordinates P2 in
Z2*X1 = Z1*X2 /\ Z2*Y1 = Z1*Y2.
Ltac t_step :=
match goal with
| |- Proper _ _ => intro
| _ => progress intros
| _ => progress destruct_head' prod
| _ => progress destruct_head' @E.point
| _ => progress destruct_head' point
| _ => progress destruct_head' and
| _ => progress cbv [eq CompleteEdwardsCurve.E.eq E.eq E.zero E.add E.opp fst snd coordinates E.coordinates proj1_sig] in *
| |- _ /\ _ => split | |- _ <-> _ => split
end.
Ltac t := repeat t_step; try Field.fsatz.
Global Instance Equivalence_eq : Equivalence eq.
Proof using Feq_dec field nonzero_a. split; repeat intro; t. Qed.
Global Instance DecidableRel_eq : Decidable.DecidableRel eq.
Proof. intros P Q; destruct P as [ [ [ [ ] ? ] ? ] ?], Q as [ [ [ [ ] ? ] ? ] ? ]; exact _. Defined.
Program Definition from_twisted (P:Epoint) : point :=
let xy := E.coordinates P in (fst xy, snd xy, 1, fst xy, snd xy).
Next Obligation. t. Qed.
Global Instance Proper_from_twisted : Proper (E.eq==>eq) from_twisted.
Proof using Type. cbv [from_twisted]; t. Qed.
Program Definition to_twisted (P:point) : Epoint :=
let XYZTT := coordinates P in let Ta := snd XYZTT in
let XYZT := fst XYZTT in
let Tb := snd XYZT in
let XYZ := fst XYZT in let Z := snd XYZ in
let XY := fst XYZ in let Y := snd XY in
let X := fst XY in
let iZ := Finv Z in ((X*iZ), (Y*iZ)).
Next Obligation. t. Qed.
Global Instance Proper_to_twisted : Proper (eq==>E.eq) to_twisted.
Proof using Type. cbv [to_twisted]; t. Qed.
Lemma to_twisted_from_twisted P : E.eq (to_twisted (from_twisted P)) P.
Proof using Type. cbv [to_twisted from_twisted]; t. Qed.
Lemma from_twisted_to_twisted P : eq (from_twisted (to_twisted P)) P.
Proof using Type. cbv [to_twisted from_twisted]; t. Qed.
Program Definition zero : point := (0, 1, 1, 0, 1).
Next Obligation. t. Qed.
Program Definition opp P : point :=
match coordinates P return F*F*F*F*F with (X,Y,Z,Ta,Tb) => (Fopp X, Y, Z, Fopp Ta, Tb) end.
Next Obligation. t. Qed.
Section TwistMinusOne.
Context {a_eq_minus1:a = Fopp 1} {twice_d} {k_eq_2d:twice_d = d+d}.
Program Definition m1add
(P1 P2:point) : point :=
match coordinates P1, coordinates P2 return F*F*F*F*F with
(X1, Y1, Z1, Ta1, Tb1), (X2, Y2, Z2, Ta2, Tb2) =>
let A := (Y1-X1)*(Y2-X2) in
let B := (Y1+X1)*(Y2+X2) in
let C := (Ta1*Tb1)*twice_d*(Ta2*Tb2) in
let D := Z1*(Z2+Z2) in
let E := B-A in
let F := D-C in
let G := D+C in
let H := B+A in
let X3 := E*F in
let Y3 := G*H in
let Z3 := F*G in
(X3, Y3, Z3, E, H)
end.
Next Obligation.
match goal with
| [ |- match (let (_, _) := coordinates ?P1 in let (_, _) := _ in let (_, _) := _ in let (_, _) := _ in let (_, _) := coordinates ?P2 in _) with _ => _ end ]
=> pose proof (E.denominator_nonzero _ nonzero_a square_a _ nonsquare_d _ _ (proj2_sig (to_twisted P1)) _ _ (proj2_sig (to_twisted P2)))
end; t.
Qed.
Global Instance isomorphic_commutative_group_m1 :
@Group.isomorphic_commutative_groups
Epoint E.eq
Eadd
(E.zero(nonzero_a:=nonzero_a))
(E.opp(nonzero_a:=nonzero_a))
point eq m1add zero opp
from_twisted to_twisted.
Proof.
eapply Group.commutative_group_by_isomorphism; try exact _.
par: abstract
(cbv [to_twisted from_twisted zero opp m1add]; intros;
repeat match goal with
| |- context[E.add ?P ?Q] =>
unique pose proof (E.denominator_nonzero _ nonzero_a square_a _ nonsquare_d _ _ (proj2_sig P) _ _ (proj2_sig Q)) end;
t).
Qed.
Lemma to_twisted_m1add P Q : E.eq (to_twisted (m1add P Q)) (Eadd (to_twisted P) (to_twisted Q)).
Proof. pose proof isomorphic_commutative_group_m1 as H; destruct H as [ [] [] [] [] ]; trivial. Qed.
Program Definition m1double (P : point) : point :=
match coordinates P return F*F*F*F*F with
(X, Y, Z, _, _) =>
let trX := X^2 in
let trZ := Y^2 in
let trT := (let t0 := Z^2 in t0+t0) in
let rY := X+Y in
let t0 := rY^2 in
let cY := trZ+trX in
let cZ := trZ-trX in
let cX := t0-cY in
let cT := trT-cZ in
let X3 := cX*cT in
let Y3 := cY*cZ in
let Z3 := cZ*cT in
(X3, Y3, Z3, cX, cY)
end.
Next Obligation.
match goal with
| [ |- context [let (_, _) := coordinates ?P in _] ]
=> pose proof (E.denominator_nonzero _ nonzero_a square_a _ nonsquare_d _ _ (proj2_sig (to_twisted P)) _ _ (proj2_sig (to_twisted P)))
end; t.
Qed.
Lemma m1double_correct P : eq (m1double P) (m1add P P).
Proof. intros; progress destruct_head' @point; cbv [m1add m1double]; t. Qed.
Lemma to_twisted_m1double P : E.eq (to_twisted (m1double P)) (Eadd (to_twisted P) (to_twisted P)).
Proof. setoid_rewrite m1double_correct; trivial. eapply to_twisted_m1add. Qed.
End TwistMinusOne.
(* https://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#doubling-double-2008-hwcd *)
Program Definition double (P : point) : point :=
match coordinates P return F*F*F*F*F with
(X1, Y1, Z1, _, _) =>
let A := X1^2 in
let B := Y1^2 in
let t0 := Z1^2 in
let C := t0+t0 in
let D := a*A in
let t1 := X1+Y1 in
let t2 := t1^2 in
let t3 := t2-A in
let E := t3-B in
let G := D+B in
let F := G-C in
let H := D-B in
let X3 := E*F in
let Y3 := G*H in
let Z3 := F*G in
(X3, Y3, Z3, E, H)
end.
Next Obligation.
match goal with
| [ |- context [let (_, _) := coordinates ?P in _] ]
=> pose proof (E.denominator_nonzero _ nonzero_a square_a _ nonsquare_d _ _ (proj2_sig (to_twisted P)) _ _ (proj2_sig (to_twisted P)))
end; t.
Qed.
Lemma to_twisted_double P : E.eq (to_twisted (double P)) (Eadd (to_twisted P) (to_twisted P)).
Proof.
cbv beta delta [double].
match goal with
| [ |- context [let (_, _) := coordinates ?P in _] ]
=> pose proof (E.denominator_nonzero _ nonzero_a square_a _ nonsquare_d _ _ (proj2_sig (to_twisted P)) _ _ (proj2_sig (to_twisted P)))
end; progress destruct_head' @point; cbv [E.add double to_twisted]; t. Qed.
End ExtendedCoordinates.