532 lines
12 KiB
Go
532 lines
12 KiB
Go
// Copyright 2012 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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// Package bn256 implements a particular bilinear group at the 128-bit security level.
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//
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// Bilinear groups are the basis of many of the new cryptographic protocols
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// that have been proposed over the past decade. They consist of a triplet of
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// groups (G₁, G₂ and GT) such that there exists a function e(g₁ˣ,g₂ʸ)=gTˣʸ
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// (where gₓ is a generator of the respective group). That function is called
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// a pairing function.
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//
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// This package specifically implements the Optimal Ate pairing over a 256-bit
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// Barreto-Naehrig curve as described in
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// http://cryptojedi.org/papers/dclxvi-20100714.pdf. Its output is compatible
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// with the implementation described in that paper.
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package bn256
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import (
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"crypto/rand"
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"io"
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"math/big"
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)
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// BUG(agl): this implementation is not constant time.
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// TODO(agl): keep GF(p²) elements in Mongomery form.
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// G1 is an abstract cyclic group. The zero value is suitable for use as the
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// output of an operation, but cannot be used as an input.
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type G1 struct {
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p *curvePoint
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}
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// RandomG1 returns x and g₁ˣ where x is a random, non-zero number read from r.
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func RandomG1(r io.Reader) (*big.Int, *G1, error) {
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var k *big.Int
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var err error
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for {
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k, err = rand.Int(r, Order)
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if err != nil {
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return nil, nil, err
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}
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if k.Sign() > 0 {
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break
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}
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}
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return k, new(G1).ScalarBaseMult(k), nil
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}
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func (g *G1) String() string {
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return "bn256.G1" + g.p.String()
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}
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// CurvePoints returns p's curve points in big integer
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func (e *G1) CurvePoints() (*big.Int, *big.Int, *big.Int, *big.Int) {
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return e.p.x, e.p.y, e.p.z, e.p.t
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}
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// Set to identity element on the group.
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func (e *G1) SetInfinity() *G1 {
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e.p = newCurvePoint(new(bnPool))
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e.p.SetInfinity()
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return e
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}
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// ScalarBaseMult sets e to g*k where g is the generator of the group and
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// then returns e.
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// This method was updated to deal with negative numbers.
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func (e *G1) ScalarBaseMult(k *big.Int) *G1 {
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if e.p == nil {
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e.p = newCurvePoint(nil)
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}
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cmp := k.Cmp(big.NewInt(0))
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if cmp >=0 {
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if cmp == 0 {
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e.p.SetInfinity()
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} else {
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e.p.Mul(curveGen, k, new(bnPool))
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}
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} else {
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e.p.Negative(e.p.Mul(curveGen, new(big.Int).Abs(k), new(bnPool)))
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}
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return e
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}
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// ScalarMult sets e to a*k and then returns e.
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// This method was updated to deal with negative numbers.
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func (e *G1) ScalarMult(a *G1, k *big.Int) *G1 {
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if e.p == nil {
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e.p = newCurvePoint(nil)
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}
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cmp := k.Cmp(big.NewInt(0))
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if cmp >=0 {
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if cmp == 0 {
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e.p.SetInfinity()
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} else {
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e.p.Mul(a.p, k, new(bnPool))
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}
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} else {
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e.p.Negative(e.p.Mul(a.p, new(big.Int).Abs(k), new(bnPool)))
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}
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return e
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}
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// Add sets e to a+b and then returns e.
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// BUG(agl): this function is not complete: a==b fails.
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func (e *G1) Add(a, b *G1) *G1 {
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if e.p == nil {
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e.p = newCurvePoint(nil)
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}
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e.p.Add(a.p, b.p, new(bnPool))
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return e
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}
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// Neg sets e to -a and then returns e.
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func (e *G1) Neg(a *G1) *G1 {
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if e.p == nil {
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e.p = newCurvePoint(nil)
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}
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e.p.Negative(a.p)
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return e
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}
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// Marshal converts n to a byte slice.
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func (n *G1) Marshal() []byte {
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n.p.MakeAffine(nil)
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xBytes := new(big.Int).Mod(n.p.x, P).Bytes()
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yBytes := new(big.Int).Mod(n.p.y, P).Bytes()
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// Each value is a 256-bit number.
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const numBytes = 256 / 8
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ret := make([]byte, numBytes*2)
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copy(ret[1*numBytes-len(xBytes):], xBytes)
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copy(ret[2*numBytes-len(yBytes):], yBytes)
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return ret
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}
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// Unmarshal sets e to the result of converting the output of Marshal back into
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// a group element and then returns e.
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func (e *G1) Unmarshal(m []byte) (*G1, bool) {
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// Each value is a 256-bit number.
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const numBytes = 256 / 8
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if len(m) != 2*numBytes {
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return nil, false
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}
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if e.p == nil {
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e.p = newCurvePoint(nil)
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}
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e.p.x.SetBytes(m[0*numBytes : 1*numBytes])
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e.p.y.SetBytes(m[1*numBytes : 2*numBytes])
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if e.p.x.Sign() == 0 && e.p.y.Sign() == 0 {
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// This is the point at infinity.
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e.p.y.SetInt64(1)
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e.p.z.SetInt64(0)
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e.p.t.SetInt64(0)
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} else {
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e.p.z.SetInt64(1)
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e.p.t.SetInt64(1)
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if !e.p.IsOnCurve() {
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return nil, false
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}
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}
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return e, true
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}
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// G2 is an abstract cyclic group. The zero value is suitable for use as the
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// output of an operation, but cannot be used as an input.
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type G2 struct {
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p *twistPoint
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}
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// RandomG1 returns x and g₂ˣ where x is a random, non-zero number read from r.
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func RandomG2(r io.Reader) (*big.Int, *G2, error) {
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var k *big.Int
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var err error
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for {
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k, err = rand.Int(r, Order)
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if err != nil {
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return nil, nil, err
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}
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if k.Sign() > 0 {
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break
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}
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}
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return k, new(G2).ScalarBaseMult(k), nil
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}
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func (g *G2) String() string {
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return "bn256.G2" + g.p.String()
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}
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// CurvePoints returns the curve points of p which includes the real
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// and imaginary parts of the curve point.
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func (e *G2) CurvePoints() (*gfP2, *gfP2, *gfP2, *gfP2) {
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return e.p.x, e.p.y, e.p.z, e.p.t
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}
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// Set to identity element on the group.
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func (e *G2) SetInfinity() *G2 {
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e.p = newTwistPoint(new(bnPool))
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e.p.SetInfinity()
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return e
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}
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// ScalarBaseMult sets e to g*k where g is the generator of the group and
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// then returns out.
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// This method was updated to deal with negative numbers.
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func (e *G2) ScalarBaseMult(k *big.Int) *G2 {
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if e.p == nil {
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e.p = newTwistPoint(nil)
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}
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if k.Cmp(big.NewInt(0)) >=0 {
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e.p.Mul(twistGen, k, new(bnPool))
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} else {
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e.p.Negative(e.p.Mul(twistGen, new(big.Int).Abs(k), new(bnPool)), new(bnPool))
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}
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return e
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}
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// ScalarMult sets e to a*k and then returns e.
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// This method was updated to deal with negative numbers.
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func (e *G2) ScalarMult(a *G2, k *big.Int) *G2 {
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if e.p == nil {
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e.p = newTwistPoint(nil)
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}
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if k.Cmp(big.NewInt(0)) >=0 {
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e.p.Mul(a.p, k, new(bnPool))
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} else {
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e.p.Negative(e.p.Mul(a.p, new(big.Int).Abs(k), new(bnPool)), new(bnPool))
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}
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return e
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}
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// Add sets e to a+b and then returns e.
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// BUG(agl): this function is not complete: a==b fails.
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func (e *G2) Add(a, b *G2) *G2 {
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if e.p == nil {
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e.p = newTwistPoint(nil)
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}
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e.p.Add(a.p, b.p, new(bnPool))
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return e
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}
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// Neg sets e to -a and then returns e.
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func (e *G2) Neg(a *G2) *G2 {
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if e.p == nil {
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e.p = newTwistPoint(nil)
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}
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e.p.Negative(a.p, new(bnPool))
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return e
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}
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// Marshal converts n into a byte slice.
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func (n *G2) Marshal() []byte {
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n.p.MakeAffine(nil)
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xxBytes := new(big.Int).Mod(n.p.x.x, P).Bytes()
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xyBytes := new(big.Int).Mod(n.p.x.y, P).Bytes()
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yxBytes := new(big.Int).Mod(n.p.y.x, P).Bytes()
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yyBytes := new(big.Int).Mod(n.p.y.y, P).Bytes()
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// Each value is a 256-bit number.
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const numBytes = 256 / 8
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ret := make([]byte, numBytes*4)
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copy(ret[1*numBytes-len(xxBytes):], xxBytes)
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copy(ret[2*numBytes-len(xyBytes):], xyBytes)
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copy(ret[3*numBytes-len(yxBytes):], yxBytes)
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copy(ret[4*numBytes-len(yyBytes):], yyBytes)
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return ret
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}
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// Unmarshal sets e to the result of converting the output of Marshal back into
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// a group element and then returns e.
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func (e *G2) Unmarshal(m []byte) (*G2, bool) {
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// Each value is a 256-bit number.
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const numBytes = 256 / 8
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if len(m) != 4*numBytes {
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return nil, false
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}
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if e.p == nil {
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e.p = newTwistPoint(nil)
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}
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e.p.x.x.SetBytes(m[0*numBytes : 1*numBytes])
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e.p.x.y.SetBytes(m[1*numBytes : 2*numBytes])
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e.p.y.x.SetBytes(m[2*numBytes : 3*numBytes])
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e.p.y.y.SetBytes(m[3*numBytes : 4*numBytes])
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if e.p.x.x.Sign() == 0 &&
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e.p.x.y.Sign() == 0 &&
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e.p.y.x.Sign() == 0 &&
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e.p.y.y.Sign() == 0 {
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// This is the point at infinity.
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e.p.y.SetOne()
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e.p.z.SetZero()
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e.p.t.SetZero()
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} else {
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e.p.z.SetOne()
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e.p.t.SetOne()
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if !e.p.IsOnCurve() {
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return nil, false
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}
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}
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return e, true
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}
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// GT is an abstract cyclic group. The zero value is suitable for use as the
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// output of an operation, but cannot be used as an input.
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type GT struct {
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p *gfP12
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}
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func (g *GT) String() string {
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return "bn256.GT" + g.p.String()
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}
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// ScalarMult sets e to a*k and then returns e.
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func (e *GT) ScalarMult(a *GT, k *big.Int) *GT {
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if e.p == nil {
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e.p = newGFp12(nil)
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}
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e.p.Exp(a.p, k, new(bnPool))
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return e
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}
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func (e *GT) Exp(a *GT, k *big.Int) *GT {
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var returnValue *GT
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if k.Cmp(big.NewInt(0)) >=0 {
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returnValue = a.ScalarMult(a, k)
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} else {
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returnValue = a.Invert(a.ScalarMult(a, new(big.Int).Abs(k)))
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}
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return returnValue
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}
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func (e *GT) Invert(a *GT) *GT {
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if e.p == nil {
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e.p = newGFp12(nil)
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}
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e.p.Invert(a.p, new(bnPool))
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return e
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}
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// SetZero returns true iff a = 0.
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func (e *G1) SetZero() {
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e.p.SetInfinity()
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}
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// IsZero returns true iff a = 0.
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func (e *G1) IsZero() bool {
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return e.p.IsInfinity()
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}
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// IsZero returns true iff a = 0.
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func (e *G2) IsZero() bool {
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return e.p.IsInfinity()
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}
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// IsZero returns true iff a = 0.
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func (e *GT) IsZero() bool {
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return e.p.IsZero()
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}
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// IsOne returns true iff a = 0.
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func (e *GT) IsOne() bool {
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return e.p.IsOne()
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}
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// Add sets e to a+b and then returns e.
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func (e *GT) Add(a, b *GT) *GT {
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if e.p == nil {
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e.p = newGFp12(nil)
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}
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e.p.Mul(a.p, b.p, new(bnPool))
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return e
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}
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// Neg sets e to -a and then returns e.
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func (e *GT) Neg(a *GT) *GT {
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if e.p == nil {
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e.p = newGFp12(nil)
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}
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e.p.Invert(a.p, new(bnPool))
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return e
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}
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// Marshal converts n into a byte slice.
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func (n *GT) Marshal() []byte {
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n.p.Minimal()
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xxxBytes := n.p.x.x.x.Bytes()
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xxyBytes := n.p.x.x.y.Bytes()
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xyxBytes := n.p.x.y.x.Bytes()
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xyyBytes := n.p.x.y.y.Bytes()
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xzxBytes := n.p.x.z.x.Bytes()
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xzyBytes := n.p.x.z.y.Bytes()
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yxxBytes := n.p.y.x.x.Bytes()
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yxyBytes := n.p.y.x.y.Bytes()
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yyxBytes := n.p.y.y.x.Bytes()
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yyyBytes := n.p.y.y.y.Bytes()
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yzxBytes := n.p.y.z.x.Bytes()
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yzyBytes := n.p.y.z.y.Bytes()
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// Each value is a 256-bit number.
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const numBytes = 256 / 8
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ret := make([]byte, numBytes*12)
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copy(ret[1*numBytes-len(xxxBytes):], xxxBytes)
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copy(ret[2*numBytes-len(xxyBytes):], xxyBytes)
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copy(ret[3*numBytes-len(xyxBytes):], xyxBytes)
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copy(ret[4*numBytes-len(xyyBytes):], xyyBytes)
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copy(ret[5*numBytes-len(xzxBytes):], xzxBytes)
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copy(ret[6*numBytes-len(xzyBytes):], xzyBytes)
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copy(ret[7*numBytes-len(yxxBytes):], yxxBytes)
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copy(ret[8*numBytes-len(yxyBytes):], yxyBytes)
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copy(ret[9*numBytes-len(yyxBytes):], yyxBytes)
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copy(ret[10*numBytes-len(yyyBytes):], yyyBytes)
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copy(ret[11*numBytes-len(yzxBytes):], yzxBytes)
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copy(ret[12*numBytes-len(yzyBytes):], yzyBytes)
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return ret
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}
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// Unmarshal sets e to the result of converting the output of Marshal back into
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// a group element and then returns e.
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func (e *GT) Unmarshal(m []byte) (*GT, bool) {
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// Each value is a 256-bit number.
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const numBytes = 256 / 8
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if len(m) != 12*numBytes {
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return nil, false
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}
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if e.p == nil {
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e.p = newGFp12(nil)
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}
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e.p.x.x.x.SetBytes(m[0*numBytes : 1*numBytes])
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e.p.x.x.y.SetBytes(m[1*numBytes : 2*numBytes])
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e.p.x.y.x.SetBytes(m[2*numBytes : 3*numBytes])
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e.p.x.y.y.SetBytes(m[3*numBytes : 4*numBytes])
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e.p.x.z.x.SetBytes(m[4*numBytes : 5*numBytes])
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e.p.x.z.y.SetBytes(m[5*numBytes : 6*numBytes])
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e.p.y.x.x.SetBytes(m[6*numBytes : 7*numBytes])
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e.p.y.x.y.SetBytes(m[7*numBytes : 8*numBytes])
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e.p.y.y.x.SetBytes(m[8*numBytes : 9*numBytes])
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e.p.y.y.y.SetBytes(m[9*numBytes : 10*numBytes])
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e.p.y.z.x.SetBytes(m[10*numBytes : 11*numBytes])
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e.p.y.z.y.SetBytes(m[11*numBytes : 12*numBytes])
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return e, true
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}
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// Pair calculates an Optimal Ate pairing.
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func Pair(g1 *G1, g2 *G2) *GT {
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return >{optimalAte(g2.p, g1.p, new(bnPool))}
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}
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// PairingCheck calculates the Optimal Ate pairing for a set of points.
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func PairingCheck(a []*G1, b []*G2) bool {
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pool := new(bnPool)
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acc := newGFp12(pool)
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acc.SetOne()
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for i := 0; i < len(a); i++ {
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if a[i].p.IsInfinity() || b[i].p.IsInfinity() {
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continue
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}
|
|
acc.Mul(acc, miller(b[i].p, a[i].p, pool), pool)
|
|
}
|
|
ret := finalExponentiation(acc, pool)
|
|
acc.Put(pool)
|
|
|
|
return ret.IsOne()
|
|
}
|
|
|
|
// bnPool implements a tiny cache of *big.Int objects that's used to reduce the
|
|
// number of allocations made during processing.
|
|
type bnPool struct {
|
|
bns []*big.Int
|
|
count int
|
|
}
|
|
|
|
func (pool *bnPool) Get() *big.Int {
|
|
if pool == nil {
|
|
return new(big.Int)
|
|
}
|
|
|
|
pool.count++
|
|
l := len(pool.bns)
|
|
if l == 0 {
|
|
return new(big.Int)
|
|
}
|
|
|
|
bn := pool.bns[l-1]
|
|
pool.bns = pool.bns[:l-1]
|
|
return bn
|
|
}
|
|
|
|
func (pool *bnPool) Put(bn *big.Int) {
|
|
if pool == nil {
|
|
return
|
|
}
|
|
pool.bns = append(pool.bns, bn)
|
|
pool.count--
|
|
}
|
|
|
|
func (pool *bnPool) Count() int {
|
|
return pool.count
|
|
}
|