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 // 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+3. 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 }