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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" ) // gfP2 implements a field of size p² as a quadratic extension of the base // field where i²=-1. type gfP2 struct { x, y *big.Int // value is xi+y. } func newGFp2(pool *bnPool) *gfP2 { return &gfP2{pool.Get(), pool.Get()} } func (e *gfP2) String() string { x := new(big.Int).Mod(e.x, p) y := new(big.Int).Mod(e.y, p) return "(" + x.String() + "," + y.String() + ")" } func (e *gfP2) Put(pool *bnPool) { pool.Put(e.x) pool.Put(e.y) } func (e *gfP2) Set(a *gfP2) *gfP2 { e.x.Set(a.x) e.y.Set(a.y) return e } func (e *gfP2) SetZero() *gfP2 { e.x.SetInt64(0) e.y.SetInt64(0) return e } func (e *gfP2) SetOne() *gfP2 { e.x.SetInt64(0) e.y.SetInt64(1) return e } func (e *gfP2) Minimal() { if e.x.Sign() < 0 || e.x.Cmp(p) >= 0 { e.x.Mod(e.x, p) } if e.y.Sign() < 0 || e.y.Cmp(p) >= 0 { e.y.Mod(e.y, p) } } func (e *gfP2) IsZero() bool { return e.x.Sign() == 0 && e.y.Sign() == 0 } func (e *gfP2) IsOne() bool { if e.x.Sign() != 0 { return false } words := e.y.Bits() return len(words) == 1 && words[0] == 1 } func (e *gfP2) Conjugate(a *gfP2) *gfP2 { e.y.Set(a.y) e.x.Neg(a.x) return e } func (e *gfP2) Negative(a *gfP2) *gfP2 { e.x.Neg(a.x) e.y.Neg(a.y) return e } func (e *gfP2) Add(a, b *gfP2) *gfP2 { e.x.Add(a.x, b.x) e.y.Add(a.y, b.y) return e } func (e *gfP2) Sub(a, b *gfP2) *gfP2 { e.x.Sub(a.x, b.x) e.y.Sub(a.y, b.y) return e } func (e *gfP2) Double(a *gfP2) *gfP2 { e.x.Lsh(a.x, 1) e.y.Lsh(a.y, 1) return e } func (c *gfP2) Exp(a *gfP2, power *big.Int, pool *bnPool) *gfP2 { sum := newGFp2(pool) sum.SetOne() t := newGFp2(pool) for i := power.BitLen() - 1; i >= 0; i-- { t.Square(sum, pool) if power.Bit(i) != 0 { sum.Mul(t, a, pool) } else { sum.Set(t) } } c.Set(sum) sum.Put(pool) t.Put(pool) return c } // See "Multiplication and Squaring in Pairing-Friendly Fields", // http://eprint.iacr.org/2006/471.pdf func (e *gfP2) Mul(a, b *gfP2, pool *bnPool) *gfP2 { tx := pool.Get().Mul(a.x, b.y) t := pool.Get().Mul(b.x, a.y) tx.Add(tx, t) tx.Mod(tx, p) ty := pool.Get().Mul(a.y, b.y) t.Mul(a.x, b.x) ty.Sub(ty, t) e.y.Mod(ty, p) e.x.Set(tx) pool.Put(tx) pool.Put(ty) pool.Put(t) return e } func (e *gfP2) MulScalar(a *gfP2, b *big.Int) *gfP2 { e.x.Mul(a.x, b) e.y.Mul(a.y, b) return e } // MulXi sets e=ξa where ξ=i+3 and then returns e. func (e *gfP2) MulXi(a *gfP2, pool *bnPool) *gfP2 { // (xi+y)(i+3) = (3x+y)i+(3y-x) tx := pool.Get().Lsh(a.x, 1) tx.Add(tx, a.x) tx.Add(tx, a.y) ty := pool.Get().Lsh(a.y, 1) ty.Add(ty, a.y) ty.Sub(ty, a.x) e.x.Set(tx) e.y.Set(ty) pool.Put(tx) pool.Put(ty) return e } func (e *gfP2) Square(a *gfP2, pool *bnPool) *gfP2 { // Complex squaring algorithm: // (xi+b)² = (x+y)(y-x) + 2*i*x*y t1 := pool.Get().Sub(a.y, a.x) t2 := pool.Get().Add(a.x, a.y) ty := pool.Get().Mul(t1, t2) ty.Mod(ty, p) t1.Mul(a.x, a.y) t1.Lsh(t1, 1) e.x.Mod(t1, p) e.y.Set(ty) pool.Put(t1) pool.Put(t2) pool.Put(ty) return e } func (e *gfP2) Invert(a *gfP2, pool *bnPool) *gfP2 { // See "Implementing cryptographic pairings", M. Scott, section 3.2. // ftp://136.206.11.249/pub/crypto/pairings.pdf t := pool.Get() t.Mul(a.y, a.y) t2 := pool.Get() t2.Mul(a.x, a.x) t.Add(t, t2) inv := pool.Get() inv.ModInverse(t, p) e.x.Neg(a.x) e.x.Mul(e.x, inv) e.x.Mod(e.x, p) e.y.Mul(a.y, inv) e.y.Mod(e.y, p) pool.Put(t) pool.Put(t2) pool.Put(inv) return e }