Perf: faster final exponentiation (hard part) for BW6 curves

ecc/bw6-633/bw6-633.go

@@ -15,6 +15,8 @@
 //	𝔽p³[u] = 𝔽p/u³-2
 //	𝔽p⁶[v] = 𝔽p²/v²-u
 //
+// case t % r % u = 0
+//
 // optimal Ate loops:
 //
 //	x₀+1, x₀^5-x₀^4-x₀

ecc/bw6-633/internal/fptower/e6_pairing.go

@@ -2,14 +2,46 @@ package fptower
 
 import "github.com/consensys/gnark-crypto/ecc/bw6-633/fp"
 
+func (z *E6) nSquare(n int) {
+	for i := 0; i < n; i++ {
+		z.CyclotomicSquare(z)
+	}
+}
+
 func (z *E6) nSquareCompressed(n int) {
 	for i := 0; i < n; i++ {
 		z.CyclotomicSquareCompressed(z)
 	}
 }
 
+// Expc1 set z to z^c1 in E6 and return z
+// ht, hy = -7, -1
+// c1 = (ht-hy)/2 = -3
+func (z *E6) Expc1(x *E6) *E6 {
+	var result E6
+	result.CyclotomicSquare(x)
+	result.Mul(x, &result)
+	z.Conjugate(&result)
+
+	return z
+}
+
+// Expc2 set z to z^c2 in E6 and return z
+// ht, hy = -7, -1
+// c2 = (ht**2+3*hy**2)/4 = 13
+func (z *E6) Expc2(x *E6) *E6 {
+	var result E6
+	result.CyclotomicSquare(x)
+	result.Mul(x, &result)
+	result.nSquare(2)
+	result.Mul(x, &result)
+	z.Set(&result)
+
+	return z
+}
+
 // Expt set z to x^t in E6 and return z (t is the seed of the curve)
-// -2**32+2**30+2**22-2**20+1
+// t = -3218079743 = -2**32+2**30+2**22-2**20+1
 func (z *E6) Expt(x *E6) *E6 {
 
 	var result, x20, x22, x30, x32 E6
@@ -36,6 +68,108 @@ func (z *E6) Expt(x *E6) *E6 {
 	return z
 }
 
+// ExptMinus1 set z to x^(t-1) in E6 and return z
+// t-1 = -3218079744
+func (z *E6) ExptMinus1(x *E6) *E6 {
+	var result, t E6
+	result.Expt(x)
+	t.Conjugate(x)
+	result.Mul(&result, &t)
+	z.Set(&result)
+
+	return z
+}
+
+// ExptMinus1Squared set z to x^(t-1)^2 in E6 and return z
+// (t-1)^2 = 10356037238743105536
+func (z *E6) ExptMinus1Squared(x *E6) *E6 {
+	var result, t0, t1, t2 E6
+	result.CyclotomicSquare(x)
+	result.CyclotomicSquare(&result)
+	t1.Mul(x, &result)
+	result.Mul(&result, &t1)
+	t0.CyclotomicSquare(&result)
+	t0.Mul(&t1, &t0)
+	t2.CyclotomicSquare(&t0)
+	t2.Mul(&t0, &t2)
+	t2.CyclotomicSquare(&t2)
+	t1.Mul(&t1, &t2)
+	t1.nSquare(5)
+	t0.Mul(&t0, &t1)
+	t0.nSquare(11)
+	result.Mul(&result, &t0)
+	result.nSquareCompressed(40)
+	result.DecompressKarabina(&result)
+	z.Set(&result)
+	return z
+}
+
+// ExptPlus1 set z to x^(t+1) in E6 and return z
+// t + 1 = -3218079742
+func (z *E6) ExptPlus1(x *E6) *E6 {
+	var result E6
+	result.Expt(x)
+	result.Mul(&result, x)
+	z.Set(&result)
+
+	return z
+}
+
+// ExptSquarePlus1 set z to x^(t^2+1) in E6 and return z
+// t^2 + 1 = 10356037232306946050
+func (z *E6) ExptSquarePlus1(x *E6) *E6 {
+	var result, t0, t1, t2, t3 E6
+	t0.CyclotomicSquare(x)
+	result.Mul(x, &t0)
+	t0.Mul(&t0, &result)
+	t1.Mul(x, &t0)
+	t0.Mul(&t0, &t1)
+	t0.CyclotomicSquare(&t0)
+	t2.Mul(x, &t0)
+	t0.Mul(&t1, &t2)
+	t1.Mul(&t1, &t0)
+	t1.nSquare(2)
+	t1.Mul(&result, &t1)
+	t3.CyclotomicSquare(&t1)
+	t3.nSquare(4)
+	t2.Mul(&t2, &t3)
+	t2.nSquareCompressed(15)
+	t2.DecompressKarabina(&t2)
+	t1.Mul(&t1, &t2)
+	t1.nSquare(5)
+	t0.Mul(&t0, &t1)
+	t0.nSquare(10)
+	result.Mul(&result, &t0)
+	result.nSquareCompressed(20)
+	result.DecompressKarabina(&result)
+	result.Mul(x, &result)
+	result.CyclotomicSquare(&result)
+	z.Set(&result)
+
+	return z
+}
+
+// ExptMinus1Div3 set z to x^((t-1)/3) in E6 and return z
+// (t-1)/3 = -1072693248
+func (z *E6) ExptMinus1Div3(x *E6) *E6 {
+	var result, t0, t1 E6
+	result.CyclotomicSquare(x)
+	result.Mul(x, &result)
+	t0.CyclotomicSquare(&result)
+	t0.CyclotomicSquare(&t0)
+	t0.Mul(&result, &t0)
+	t1.CyclotomicSquare(&t0)
+	t1.nSquare(3)
+	t0.Mul(&t0, &t1)
+	t0.nSquare(2)
+	result.Mul(&result, &t0)
+	result.nSquareCompressed(20)
+	result.DecompressKarabina(&result)
+	z.Conjugate(&result)
+
+	return z
+}
+
 // MulBy034 multiplication by sparse element (c0,0,0,c3,c4,0)
 func (z *E6) MulBy034(c0, c3, c4 *fp.Element) *E6 {