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Diffstat (limited to 'crypto/bn256/gfp6.go')
-rw-r--r-- | crypto/bn256/gfp6.go | 296 |
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..218856617 --- /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 +} |