1 files changed, 213 insertions, 0 deletions
diff --git a/crypto/bn256/cloudflare/gfp6.go b/crypto/bn256/cloudflare/gfp6.go
new file mode 100644
index 000000000..83d61b781
--- /dev/null
+++ b/crypto/bn256/cloudflare/gfp6.go
@@ -0,0 +1,213 @@
+package bn256
+
+// For details of the algorithms used, see "Multiplication and Squaring on
+// Pairing-Friendly Fields, Devegili et al.
+// http://eprint.iacr.org/2006/471.pdf.
+
+// gfP6 implements the field of size p⁶ as a cubic extension of gfP2 where τ³=ξ
+// and ξ=i+3.
+type gfP6 struct {
+	x, y, z gfP2 // value is xτ² + yτ + z
+}
+
+func (e *gfP6) String() string {
+	return "(" + e.x.String() + ", " + e.y.String() + ", " + e.z.String() + ")"
+}
+
+func (e *gfP6) Set(a *gfP6) *gfP6 {
+	e.x.Set(&a.x)
+	e.y.Set(&a.y)
+	e.z.Set(&a.z)
+	return e
+}
+
+func (e *gfP6) SetZero() *gfP6 {
+	e.x.SetZero()
+	e.y.SetZero()
+	e.z.SetZero()
+	return e
+}
+
+func (e *gfP6) SetOne() *gfP6 {
+	e.x.SetZero()
+	e.y.SetZero()
+	e.z.SetOne()
+	return e
+}
+
+func (e *gfP6) IsZero() bool {
+	return e.x.IsZero() && e.y.IsZero() && e.z.IsZero()
+}
+
+func (e *gfP6) IsOne() bool {
+	return e.x.IsZero() && e.y.IsZero() && e.z.IsOne()
+}
+
+func (e *gfP6) Neg(a *gfP6) *gfP6 {
+	e.x.Neg(&a.x)
+	e.y.Neg(&a.y)
+	e.z.Neg(&a.z)
+	return e
+}
+
+func (e *gfP6) Frobenius(a *gfP6) *gfP6 {
+	e.x.Conjugate(&a.x)
+	e.y.Conjugate(&a.y)
+	e.z.Conjugate(&a.z)
+
+	e.x.Mul(&e.x, xiTo2PMinus2Over3)
+	e.y.Mul(&e.y, xiToPMinus1Over3)
+	return e
+}
+
+// FrobeniusP2 computes (xτ²+yτ+z)^(p²) = xτ^(2p²) + yτ^(p²) + z
+func (e *gfP6) FrobeniusP2(a *gfP6) *gfP6 {
+	// τ^(2p²) = τ²τ^(2p²-2) = τ²ξ^((2p²-2)/3)
+	e.x.MulScalar(&a.x, xiTo2PSquaredMinus2Over3)
+	// τ^(p²) = ττ^(p²-1) = τξ^((p²-1)/3)
+	e.y.MulScalar(&a.y, xiToPSquaredMinus1Over3)
+	e.z.Set(&a.z)
+	return e
+}
+
+func (e *gfP6) FrobeniusP4(a *gfP6) *gfP6 {
+	e.x.MulScalar(&a.x, xiToPSquaredMinus1Over3)
+	e.y.MulScalar(&a.y, xiTo2PSquaredMinus2Over3)
+	e.z.Set(&a.z)
+	return e
+}
+
+func (e *gfP6) Add(a, b *gfP6) *gfP6 {
+	e.x.Add(&a.x, &b.x)
+	e.y.Add(&a.y, &b.y)
+	e.z.Add(&a.z, &b.z)
+	return e
+}
+
+func (e *gfP6) Sub(a, b *gfP6) *gfP6 {
+	e.x.Sub(&a.x, &b.x)
+	e.y.Sub(&a.y, &b.y)
+	e.z.Sub(&a.z, &b.z)
+	return e
+}
+
+func (e *gfP6) Mul(a, b *gfP6) *gfP6 {
+	// "Multiplication and Squaring on Pairing-Friendly Fields"
+	// Section 4, Karatsuba method.
+	// http://eprint.iacr.org/2006/471.pdf
+	v0 := (&gfP2{}).Mul(&a.z, &b.z)
+	v1 := (&gfP2{}).Mul(&a.y, &b.y)
+	v2 := (&gfP2{}).Mul(&a.x, &b.x)
+
+	t0 := (&gfP2{}).Add(&a.x, &a.y)
+	t1 := (&gfP2{}).Add(&b.x, &b.y)
+	tz := (&gfP2{}).Mul(t0, t1)
+	tz.Sub(tz, v1).Sub(tz, v2).MulXi(tz).Add(tz, v0)
+
+	t0.Add(&a.y, &a.z)
+	t1.Add(&b.y, &b.z)
+	ty := (&gfP2{}).Mul(t0, t1)
+	t0.MulXi(v2)
+	ty.Sub(ty, v0).Sub(ty, v1).Add(ty, t0)
+
+	t0.Add(&a.x, &a.z)
+	t1.Add(&b.x, &b.z)
+	tx := (&gfP2{}).Mul(t0, t1)
+	tx.Sub(tx, v0).Add(tx, v1).Sub(tx, v2)
+
+	e.x.Set(tx)
+	e.y.Set(ty)
+	e.z.Set(tz)
+	return e
+}
+
+func (e *gfP6) MulScalar(a *gfP6, b *gfP2) *gfP6 {
+	e.x.Mul(&a.x, b)
+	e.y.Mul(&a.y, b)
+	e.z.Mul(&a.z, b)
+	return e
+}
+
+func (e *gfP6) MulGFP(a *gfP6, b *gfP) *gfP6 {
+	e.x.MulScalar(&a.x, b)
+	e.y.MulScalar(&a.y, b)
+	e.z.MulScalar(&a.z, b)
+	return e
+}
+
+// MulTau computes τ·(aτ²+bτ+c) = bτ²+cτ+aξ
+func (e *gfP6) MulTau(a *gfP6) *gfP6 {
+	tz := (&gfP2{}).MulXi(&a.x)
+	ty := (&gfP2{}).Set(&a.y)
+
+	e.y.Set(&a.z)
+	e.x.Set(ty)
+	e.z.Set(tz)
+	return e
+}
+
+func (e *gfP6) Square(a *gfP6) *gfP6 {
+	v0 := (&gfP2{}).Square(&a.z)
+	v1 := (&gfP2{}).Square(&a.y)
+	v2 := (&gfP2{}).Square(&a.x)
+
+	c0 := (&gfP2{}).Add(&a.x, &a.y)
+	c0.Square(c0).Sub(c0, v1).Sub(c0, v2).MulXi(c0).Add(c0, v0)
+
+	c1 := (&gfP2{}).Add(&a.y, &a.z)
+	c1.Square(c1).Sub(c1, v0).Sub(c1, v1)
+	xiV2 := (&gfP2{}).MulXi(v2)
+	c1.Add(c1, xiV2)
+
+	c2 := (&gfP2{}).Add(&a.x, &a.z)
+	c2.Square(c2).Sub(c2, v0).Add(c2, v1).Sub(c2, v2)
+
+	e.x.Set(c2)
+	e.y.Set(c1)
+	e.z.Set(c0)
+	return e
+}
+
+func (e *gfP6) Invert(a *gfP6) *gfP6 {
+	// See "Implementing cryptographic pairings", M. Scott, section 3.2.
+	// ftp://136.206.11.249/pub/crypto/pairings.pdf
+
+	// Here we can give a short explanation of how it works: let j be a cubic root of
+	// unity in GF(p²) so that 1+j+j²=0.
+	// Then (xτ² + yτ + z)(xj²τ² + yjτ + z)(xjτ² + yj²τ + z)
+	// = (xτ² + yτ + z)(Cτ²+Bτ+A)
+	// = (x³ξ²+y³ξ+z³-3ξxyz) = F is an element of the base field (the norm).
+	//
+	// On the other hand (xj²τ² + yjτ + z)(xjτ² + yj²τ + z)
+	// = τ²(y²-ξxz) + τ(ξx²-yz) + (z²-ξxy)
+	//
+	// So that's why A = (z²-ξxy), B = (ξx²-yz), C = (y²-ξxz)
+	t1 := (&gfP2{}).Mul(&a.x, &a.y)
+	t1.MulXi(t1)
+
+	A := (&gfP2{}).Square(&a.z)
+	A.Sub(A, t1)
+
+	B := (&gfP2{}).Square(&a.x)
+	B.MulXi(B)
+	t1.Mul(&a.y, &a.z)
+	B.Sub(B, t1)
+
+	C := (&gfP2{}).Square(&a.y)
+	t1.Mul(&a.x, &a.z)
+	C.Sub(C, t1)
+
+	F := (&gfP2{}).Mul(C, &a.y)
+	F.MulXi(F)
+	t1.Mul(A, &a.z)
+	F.Add(F, t1)
+	t1.Mul(B, &a.x).MulXi(t1)
+	F.Add(F, t1)
+
+	F.Invert(F)
+
+	e.x.Mul(C, F)
+	e.y.Mul(B, F)
+	e.z.Mul(A, F)
+	return e
+}