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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" ) // gfP12 implements the field of size p¹² as a quadratic extension of gfP6 // where ω²=τ. type gfP12 struct { x, y *gfP6 // value is xω + y } func newGFp12(pool *bnPool) *gfP12 { return &gfP12{newGFp6(pool), newGFp6(pool)} } func (e *gfP12) String() string { return "(" + e.x.String() + "," + e.y.String() + ")" } func (e *gfP12) Put(pool *bnPool) { e.x.Put(pool) e.y.Put(pool) } func (e *gfP12) Set(a *gfP12) *gfP12 { e.x.Set(a.x) e.y.Set(a.y) return e } func (e *gfP12) SetZero() *gfP12 { e.x.SetZero() e.y.SetZero() return e } func (e *gfP12) SetOne() *gfP12 { e.x.SetZero() e.y.SetOne() return e } func (e *gfP12) Minimal() { e.x.Minimal() e.y.Minimal() } func (e *gfP12) IsZero() bool { e.Minimal() return e.x.IsZero() && e.y.IsZero() } func (e *gfP12) IsOne() bool { e.Minimal() return e.x.IsZero() && e.y.IsOne() } func (e *gfP12) Conjugate(a *gfP12) *gfP12 { e.x.Negative(a.x) e.y.Set(a.y) return a } func (e *gfP12) Negative(a *gfP12) *gfP12 { e.x.Negative(a.x) e.y.Negative(a.y) return e } // Frobenius computes (xω+y)^p = x^p ω·ξ^((p-1)/6) + y^p func (e *gfP12) Frobenius(a *gfP12, pool *bnPool) *gfP12 { e.x.Frobenius(a.x, pool) e.y.Frobenius(a.y, pool) e.x.MulScalar(e.x, xiToPMinus1Over6, pool) return e } // FrobeniusP2 computes (xω+y)^p² = x^p² ω·ξ^((p²-1)/6) + y^p² func (e *gfP12) FrobeniusP2(a *gfP12, pool *bnPool) *gfP12 { e.x.FrobeniusP2(a.x) e.x.MulGFP(e.x, xiToPSquaredMinus1Over6) e.y.FrobeniusP2(a.y) return e } func (e *gfP12) Add(a, b *gfP12) *gfP12 { e.x.Add(a.x, b.x) e.y.Add(a.y, b.y) return e } func (e *gfP12) Sub(a, b *gfP12) *gfP12 { e.x.Sub(a.x, b.x) e.y.Sub(a.y, b.y) return e } func (e *gfP12) Mul(a, b *gfP12, pool *bnPool) *gfP12 { tx := newGFp6(pool) tx.Mul(a.x, b.y, pool) t := newGFp6(pool) t.Mul(b.x, a.y, pool) tx.Add(tx, t) ty := newGFp6(pool) ty.Mul(a.y, b.y, pool) t.Mul(a.x, b.x, pool) t.MulTau(t, pool) e.y.Add(ty, t) e.x.Set(tx) tx.Put(pool) ty.Put(pool) t.Put(pool) return e } func (e *gfP12) MulScalar(a *gfP12, b *gfP6, pool *bnPool) *gfP12 { e.x.Mul(a.x, b, pool) e.y.Mul(a.y, b, pool) return e } func (c *gfP12) Exp(a *gfP12, power *big.Int, pool *bnPool) *gfP12 { sum := newGFp12(pool) sum.SetOne() t := newGFp12(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 } func (e *gfP12) Square(a *gfP12, pool *bnPool) *gfP12 { // Complex squaring algorithm v0 := newGFp6(pool) v0.Mul(a.x, a.y, pool) t := newGFp6(pool) t.MulTau(a.x, pool) t.Add(a.y, t) ty := newGFp6(pool) ty.Add(a.x, a.y) ty.Mul(ty, t, pool) ty.Sub(ty, v0) t.MulTau(v0, pool) ty.Sub(ty, t) e.y.Set(ty) e.x.Double(v0) v0.Put(pool) t.Put(pool) ty.Put(pool) return e } func (e *gfP12) Invert(a *gfP12, pool *bnPool) *gfP12 { // See "Implementing cryptographic pairings", M. Scott, section 3.2. // ftp://136.206.11.249/pub/crypto/pairings.pdf t1 := newGFp6(pool) t2 := newGFp6(pool) t1.Square(a.x, pool) t2.Square(a.y, pool) t1.MulTau(t1, pool) t1.Sub(t2, t1) t2.Invert(t1, pool) e.x.Negative(a.x) e.y.Set(a.y) e.MulScalar(e, t2, pool) t1.Put(pool) t2.Put(pool) return e }