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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
}
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