1 files changed, 296 insertions, 0 deletions
diff --git a/crypto/bn256/gfp6.go b/crypto/bn256/gfp6.go
new file mode 100644
index 000000000..8fd777d52
--- /dev/null
+++ b/crypto/bn256/gfp6.go
@@ -0,0 +1,296 @@
+// Copyright 2012 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+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.
+
+import (
+	"math/big"
+)
+
+// gfP6 implements the field of size p⁶ as a cubic extension of gfP2 where τ³=ξ
+// and ξ=i+9.
+type gfP6 struct {
+	x, y, z *gfP2 // value is xτ² + yτ + z
+}
+
+func newGFp6(pool *bnPool) *gfP6 {
+	return &gfP6{newGFp2(pool), newGFp2(pool), newGFp2(pool)}
+}
+
+func (e *gfP6) String() string {
+	return "(" + e.x.String() + "," + e.y.String() + "," + e.z.String() + ")"
+}
+
+func (e *gfP6) Put(pool *bnPool) {
+	e.x.Put(pool)
+	e.y.Put(pool)
+	e.z.Put(pool)
+}
+
+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) Minimal() {
+	e.x.Minimal()
+	e.y.Minimal()
+	e.z.Minimal()
+}
+
+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) Negative(a *gfP6) *gfP6 {
+	e.x.Negative(a.x)
+	e.y.Negative(a.y)
+	e.z.Negative(a.z)
+	return e
+}
+
+func (e *gfP6) Frobenius(a *gfP6, pool *bnPool) *gfP6 {
+	e.x.Conjugate(a.x)
+	e.y.Conjugate(a.y)
+	e.z.Conjugate(a.z)
+
+	e.x.Mul(e.x, xiTo2PMinus2Over3, pool)
+	e.y.Mul(e.y, xiToPMinus1Over3, pool)
+	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) 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) Double(a *gfP6) *gfP6 {
+	e.x.Double(a.x)
+	e.y.Double(a.y)
+	e.z.Double(a.z)
+	return e
+}
+
+func (e *gfP6) Mul(a, b *gfP6, pool *bnPool) *gfP6 {
+	// "Multiplication and Squaring on Pairing-Friendly Fields"
+	// Section 4, Karatsuba method.
+	// http://eprint.iacr.org/2006/471.pdf
+
+	v0 := newGFp2(pool)
+	v0.Mul(a.z, b.z, pool)
+	v1 := newGFp2(pool)
+	v1.Mul(a.y, b.y, pool)
+	v2 := newGFp2(pool)
+	v2.Mul(a.x, b.x, pool)
+
+	t0 := newGFp2(pool)
+	t0.Add(a.x, a.y)
+	t1 := newGFp2(pool)
+	t1.Add(b.x, b.y)
+	tz := newGFp2(pool)
+	tz.Mul(t0, t1, pool)
+
+	tz.Sub(tz, v1)
+	tz.Sub(tz, v2)
+	tz.MulXi(tz, pool)
+	tz.Add(tz, v0)
+
+	t0.Add(a.y, a.z)
+	t1.Add(b.y, b.z)
+	ty := newGFp2(pool)
+	ty.Mul(t0, t1, pool)
+	ty.Sub(ty, v0)
+	ty.Sub(ty, v1)
+	t0.MulXi(v2, pool)
+	ty.Add(ty, t0)
+
+	t0.Add(a.x, a.z)
+	t1.Add(b.x, b.z)
+	tx := newGFp2(pool)
+	tx.Mul(t0, t1, pool)
+	tx.Sub(tx, v0)
+	tx.Add(tx, v1)
+	tx.Sub(tx, v2)
+
+	e.x.Set(tx)
+	e.y.Set(ty)
+	e.z.Set(tz)
+
+	t0.Put(pool)
+	t1.Put(pool)
+	tx.Put(pool)
+	ty.Put(pool)
+	tz.Put(pool)
+	v0.Put(pool)
+	v1.Put(pool)
+	v2.Put(pool)
+	return e
+}
+
+func (e *gfP6) MulScalar(a *gfP6, b *gfP2, pool *bnPool) *gfP6 {
+	e.x.Mul(a.x, b, pool)
+	e.y.Mul(a.y, b, pool)
+	e.z.Mul(a.z, b, pool)
+	return e
+}
+
+func (e *gfP6) MulGFP(a *gfP6, b *big.Int) *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, pool *bnPool) {
+	tz := newGFp2(pool)
+	tz.MulXi(a.x, pool)
+	ty := newGFp2(pool)
+	ty.Set(a.y)
+	e.y.Set(a.z)
+	e.x.Set(ty)
+	e.z.Set(tz)
+	tz.Put(pool)
+	ty.Put(pool)
+}
+
+func (e *gfP6) Square(a *gfP6, pool *bnPool) *gfP6 {
+	v0 := newGFp2(pool).Square(a.z, pool)
+	v1 := newGFp2(pool).Square(a.y, pool)
+	v2 := newGFp2(pool).Square(a.x, pool)
+
+	c0 := newGFp2(pool).Add(a.x, a.y)
+	c0.Square(c0, pool)
+	c0.Sub(c0, v1)
+	c0.Sub(c0, v2)
+	c0.MulXi(c0, pool)
+	c0.Add(c0, v0)
+
+	c1 := newGFp2(pool).Add(a.y, a.z)
+	c1.Square(c1, pool)
+	c1.Sub(c1, v0)
+	c1.Sub(c1, v1)
+	xiV2 := newGFp2(pool).MulXi(v2, pool)
+	c1.Add(c1, xiV2)
+
+	c2 := newGFp2(pool).Add(a.x, a.z)
+	c2.Square(c2, pool)
+	c2.Sub(c2, v0)
+	c2.Add(c2, v1)
+	c2.Sub(c2, v2)
+
+	e.x.Set(c2)
+	e.y.Set(c1)
+	e.z.Set(c0)
+
+	v0.Put(pool)
+	v1.Put(pool)
+	v2.Put(pool)
+	c0.Put(pool)
+	c1.Put(pool)
+	c2.Put(pool)
+	xiV2.Put(pool)
+
+	return e
+}
+
+func (e *gfP6) Invert(a *gfP6, pool *bnPool) *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 := newGFp2(pool)
+
+	A := newGFp2(pool)
+	A.Square(a.z, pool)
+	t1.Mul(a.x, a.y, pool)
+	t1.MulXi(t1, pool)
+	A.Sub(A, t1)
+
+	B := newGFp2(pool)
+	B.Square(a.x, pool)
+	B.MulXi(B, pool)
+	t1.Mul(a.y, a.z, pool)
+	B.Sub(B, t1)
+
+	C := newGFp2(pool)
+	C.Square(a.y, pool)
+	t1.Mul(a.x, a.z, pool)
+	C.Sub(C, t1)
+
+	F := newGFp2(pool)
+	F.Mul(C, a.y, pool)
+	F.MulXi(F, pool)
+	t1.Mul(A, a.z, pool)
+	F.Add(F, t1)
+	t1.Mul(B, a.x, pool)
+	t1.MulXi(t1, pool)
+	F.Add(F, t1)
+
+	F.Invert(F, pool)
+
+	e.x.Mul(C, F, pool)
+	e.y.Mul(B, F, pool)
+	e.z.Mul(A, F, pool)
+
+	t1.Put(pool)
+	A.Put(pool)
+	B.Put(pool)
+	C.Put(pool)
+	F.Put(pool)
+
+	return e
+}