From bd6879ac518431174a490ba42f7e6e822dcb3ee1 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?P=C3=A9ter=20Szil=C3=A1gyi?= Date: Mon, 5 Mar 2018 14:33:45 +0200 Subject: core/vm, crypto/bn256: switch over to cloudflare library (#16203) * core/vm, crypto/bn256: switch over to cloudflare library * crypto/bn256: unmarshal constraint + start pure go impl * crypto/bn256: combo cloudflare and google lib * travis: drop 386 test job --- crypto/bn256/bn256.go | 434 ---------------------------- crypto/bn256/bn256_amd64.go | 63 +++++ crypto/bn256/bn256_other.go | 63 +++++ crypto/bn256/bn256_test.go | 304 -------------------- crypto/bn256/cloudflare/bn256.go | 481 ++++++++++++++++++++++++++++++++ crypto/bn256/cloudflare/bn256_test.go | 118 ++++++++ crypto/bn256/cloudflare/constants.go | 59 ++++ crypto/bn256/cloudflare/curve.go | 229 +++++++++++++++ crypto/bn256/cloudflare/example_test.go | 45 +++ crypto/bn256/cloudflare/gfp.go | 81 ++++++ crypto/bn256/cloudflare/gfp.h | 32 +++ crypto/bn256/cloudflare/gfp12.go | 160 +++++++++++ crypto/bn256/cloudflare/gfp2.go | 156 +++++++++++ crypto/bn256/cloudflare/gfp6.go | 213 ++++++++++++++ crypto/bn256/cloudflare/gfp_amd64.go | 15 + crypto/bn256/cloudflare/gfp_amd64.s | 97 +++++++ crypto/bn256/cloudflare/gfp_pure.go | 19 ++ crypto/bn256/cloudflare/gfp_test.go | 62 ++++ crypto/bn256/cloudflare/main_test.go | 73 +++++ crypto/bn256/cloudflare/mul.h | 181 ++++++++++++ crypto/bn256/cloudflare/mul_bmi2.h | 112 ++++++++ crypto/bn256/cloudflare/optate.go | 271 ++++++++++++++++++ crypto/bn256/cloudflare/twist.go | 204 ++++++++++++++ crypto/bn256/constants.go | 44 --- crypto/bn256/curve.go | 278 ------------------ crypto/bn256/example_test.go | 43 --- crypto/bn256/gfp12.go | 200 ------------- crypto/bn256/gfp2.go | 227 --------------- crypto/bn256/gfp6.go | 296 -------------------- crypto/bn256/google/bn256.go | 447 +++++++++++++++++++++++++++++ crypto/bn256/google/bn256_test.go | 311 +++++++++++++++++++++ crypto/bn256/google/constants.go | 44 +++ crypto/bn256/google/curve.go | 278 ++++++++++++++++++ crypto/bn256/google/example_test.go | 43 +++ crypto/bn256/google/gfp12.go | 200 +++++++++++++ crypto/bn256/google/gfp2.go | 227 +++++++++++++++ crypto/bn256/google/gfp6.go | 296 ++++++++++++++++++++ crypto/bn256/google/main_test.go | 71 +++++ crypto/bn256/google/optate.go | 397 ++++++++++++++++++++++++++ crypto/bn256/google/twist.go | 255 +++++++++++++++++ crypto/bn256/main_test.go | 71 ----- crypto/bn256/optate.go | 397 -------------------------- crypto/bn256/twist.go | 249 ----------------- 43 files changed, 5303 insertions(+), 2543 deletions(-) delete mode 100644 crypto/bn256/bn256.go create mode 100644 crypto/bn256/bn256_amd64.go create mode 100644 crypto/bn256/bn256_other.go delete mode 100644 crypto/bn256/bn256_test.go create mode 100644 crypto/bn256/cloudflare/bn256.go create mode 100644 crypto/bn256/cloudflare/bn256_test.go create mode 100644 crypto/bn256/cloudflare/constants.go create mode 100644 crypto/bn256/cloudflare/curve.go create mode 100644 crypto/bn256/cloudflare/example_test.go create mode 100644 crypto/bn256/cloudflare/gfp.go create mode 100644 crypto/bn256/cloudflare/gfp.h create mode 100644 crypto/bn256/cloudflare/gfp12.go create mode 100644 crypto/bn256/cloudflare/gfp2.go create mode 100644 crypto/bn256/cloudflare/gfp6.go create mode 100644 crypto/bn256/cloudflare/gfp_amd64.go create mode 100644 crypto/bn256/cloudflare/gfp_amd64.s create mode 100644 crypto/bn256/cloudflare/gfp_pure.go create mode 100644 crypto/bn256/cloudflare/gfp_test.go create mode 100644 crypto/bn256/cloudflare/main_test.go create mode 100644 crypto/bn256/cloudflare/mul.h create mode 100644 crypto/bn256/cloudflare/mul_bmi2.h create mode 100644 crypto/bn256/cloudflare/optate.go create mode 100644 crypto/bn256/cloudflare/twist.go delete mode 100644 crypto/bn256/constants.go delete mode 100644 crypto/bn256/curve.go delete mode 100644 crypto/bn256/example_test.go delete mode 100644 crypto/bn256/gfp12.go delete mode 100644 crypto/bn256/gfp2.go delete mode 100644 crypto/bn256/gfp6.go create mode 100644 crypto/bn256/google/bn256.go create mode 100644 crypto/bn256/google/bn256_test.go create mode 100644 crypto/bn256/google/constants.go create mode 100644 crypto/bn256/google/curve.go create mode 100644 crypto/bn256/google/example_test.go create mode 100644 crypto/bn256/google/gfp12.go create mode 100644 crypto/bn256/google/gfp2.go create mode 100644 crypto/bn256/google/gfp6.go create mode 100644 crypto/bn256/google/main_test.go create mode 100644 crypto/bn256/google/optate.go create mode 100644 crypto/bn256/google/twist.go delete mode 100644 crypto/bn256/main_test.go delete mode 100644 crypto/bn256/optate.go delete mode 100644 crypto/bn256/twist.go (limited to 'crypto') diff --git a/crypto/bn256/bn256.go b/crypto/bn256/bn256.go deleted file mode 100644 index 7144c31a8..000000000 --- a/crypto/bn256/bn256.go +++ /dev/null @@ -1,434 +0,0 @@ -// 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 implements a particular bilinear group at the 128-bit security level. -// -// Bilinear groups are the basis of many of the new cryptographic protocols -// that have been proposed over the past decade. They consist of a triplet of -// groups (G₁, G₂ and GT) such that there exists a function e(g₁ˣ,g₂ʸ)=gTˣʸ -// (where gₓ is a generator of the respective group). That function is called -// a pairing function. -// -// This package specifically implements the Optimal Ate pairing over a 256-bit -// Barreto-Naehrig curve as described in -// http://cryptojedi.org/papers/dclxvi-20100714.pdf. Its output is compatible -// with the implementation described in that paper. -package bn256 - -import ( - "crypto/rand" - "io" - "math/big" -) - -// BUG(agl): this implementation is not constant time. -// TODO(agl): keep GF(p²) elements in Mongomery form. - -// G1 is an abstract cyclic group. The zero value is suitable for use as the -// output of an operation, but cannot be used as an input. -type G1 struct { - p *curvePoint -} - -// RandomG1 returns x and g₁ˣ where x is a random, non-zero number read from r. -func RandomG1(r io.Reader) (*big.Int, *G1, error) { - var k *big.Int - var err error - - for { - k, err = rand.Int(r, Order) - if err != nil { - return nil, nil, err - } - if k.Sign() > 0 { - break - } - } - - return k, new(G1).ScalarBaseMult(k), nil -} - -func (g *G1) String() string { - return "bn256.G1" + g.p.String() -} - -// CurvePoints returns p's curve points in big integer -func (e *G1) CurvePoints() (*big.Int, *big.Int, *big.Int, *big.Int) { - return e.p.x, e.p.y, e.p.z, e.p.t -} - -// ScalarBaseMult sets e to g*k where g is the generator of the group and -// then returns e. -func (e *G1) ScalarBaseMult(k *big.Int) *G1 { - if e.p == nil { - e.p = newCurvePoint(nil) - } - e.p.Mul(curveGen, k, new(bnPool)) - return e -} - -// ScalarMult sets e to a*k and then returns e. -func (e *G1) ScalarMult(a *G1, k *big.Int) *G1 { - if e.p == nil { - e.p = newCurvePoint(nil) - } - e.p.Mul(a.p, k, new(bnPool)) - return e -} - -// Add sets e to a+b and then returns e. -// BUG(agl): this function is not complete: a==b fails. -func (e *G1) Add(a, b *G1) *G1 { - if e.p == nil { - e.p = newCurvePoint(nil) - } - e.p.Add(a.p, b.p, new(bnPool)) - return e -} - -// Neg sets e to -a and then returns e. -func (e *G1) Neg(a *G1) *G1 { - if e.p == nil { - e.p = newCurvePoint(nil) - } - e.p.Negative(a.p) - return e -} - -// Marshal converts n to a byte slice. -func (n *G1) Marshal() []byte { - n.p.MakeAffine(nil) - - xBytes := new(big.Int).Mod(n.p.x, P).Bytes() - yBytes := new(big.Int).Mod(n.p.y, P).Bytes() - - // Each value is a 256-bit number. - const numBytes = 256 / 8 - - ret := make([]byte, numBytes*2) - copy(ret[1*numBytes-len(xBytes):], xBytes) - copy(ret[2*numBytes-len(yBytes):], yBytes) - - return ret -} - -// Unmarshal sets e to the result of converting the output of Marshal back into -// a group element and then returns e. -func (e *G1) Unmarshal(m []byte) (*G1, bool) { - // Each value is a 256-bit number. - const numBytes = 256 / 8 - - if len(m) != 2*numBytes { - return nil, false - } - - if e.p == nil { - e.p = newCurvePoint(nil) - } - - e.p.x.SetBytes(m[0*numBytes : 1*numBytes]) - e.p.y.SetBytes(m[1*numBytes : 2*numBytes]) - - if e.p.x.Sign() == 0 && e.p.y.Sign() == 0 { - // This is the point at infinity. - e.p.y.SetInt64(1) - e.p.z.SetInt64(0) - e.p.t.SetInt64(0) - } else { - e.p.z.SetInt64(1) - e.p.t.SetInt64(1) - - if !e.p.IsOnCurve() { - return nil, false - } - } - - return e, true -} - -// G2 is an abstract cyclic group. The zero value is suitable for use as the -// output of an operation, but cannot be used as an input. -type G2 struct { - p *twistPoint -} - -// RandomG1 returns x and g₂ˣ where x is a random, non-zero number read from r. -func RandomG2(r io.Reader) (*big.Int, *G2, error) { - var k *big.Int - var err error - - for { - k, err = rand.Int(r, Order) - if err != nil { - return nil, nil, err - } - if k.Sign() > 0 { - break - } - } - - return k, new(G2).ScalarBaseMult(k), nil -} - -func (g *G2) String() string { - return "bn256.G2" + g.p.String() -} - -// CurvePoints returns the curve points of p which includes the real -// and imaginary parts of the curve point. -func (e *G2) CurvePoints() (*gfP2, *gfP2, *gfP2, *gfP2) { - return e.p.x, e.p.y, e.p.z, e.p.t -} - -// ScalarBaseMult sets e to g*k where g is the generator of the group and -// then returns out. -func (e *G2) ScalarBaseMult(k *big.Int) *G2 { - if e.p == nil { - e.p = newTwistPoint(nil) - } - e.p.Mul(twistGen, k, new(bnPool)) - return e -} - -// ScalarMult sets e to a*k and then returns e. -func (e *G2) ScalarMult(a *G2, k *big.Int) *G2 { - if e.p == nil { - e.p = newTwistPoint(nil) - } - e.p.Mul(a.p, k, new(bnPool)) - return e -} - -// Add sets e to a+b and then returns e. -// BUG(agl): this function is not complete: a==b fails. -func (e *G2) Add(a, b *G2) *G2 { - if e.p == nil { - e.p = newTwistPoint(nil) - } - e.p.Add(a.p, b.p, new(bnPool)) - return e -} - -// Marshal converts n into a byte slice. -func (n *G2) Marshal() []byte { - n.p.MakeAffine(nil) - - xxBytes := new(big.Int).Mod(n.p.x.x, P).Bytes() - xyBytes := new(big.Int).Mod(n.p.x.y, P).Bytes() - yxBytes := new(big.Int).Mod(n.p.y.x, P).Bytes() - yyBytes := new(big.Int).Mod(n.p.y.y, P).Bytes() - - // Each value is a 256-bit number. - const numBytes = 256 / 8 - - ret := make([]byte, numBytes*4) - copy(ret[1*numBytes-len(xxBytes):], xxBytes) - copy(ret[2*numBytes-len(xyBytes):], xyBytes) - copy(ret[3*numBytes-len(yxBytes):], yxBytes) - copy(ret[4*numBytes-len(yyBytes):], yyBytes) - - return ret -} - -// Unmarshal sets e to the result of converting the output of Marshal back into -// a group element and then returns e. -func (e *G2) Unmarshal(m []byte) (*G2, bool) { - // Each value is a 256-bit number. - const numBytes = 256 / 8 - - if len(m) != 4*numBytes { - return nil, false - } - - if e.p == nil { - e.p = newTwistPoint(nil) - } - - e.p.x.x.SetBytes(m[0*numBytes : 1*numBytes]) - e.p.x.y.SetBytes(m[1*numBytes : 2*numBytes]) - e.p.y.x.SetBytes(m[2*numBytes : 3*numBytes]) - e.p.y.y.SetBytes(m[3*numBytes : 4*numBytes]) - - if e.p.x.x.Sign() == 0 && - e.p.x.y.Sign() == 0 && - e.p.y.x.Sign() == 0 && - e.p.y.y.Sign() == 0 { - // This is the point at infinity. - e.p.y.SetOne() - e.p.z.SetZero() - e.p.t.SetZero() - } else { - e.p.z.SetOne() - e.p.t.SetOne() - - if !e.p.IsOnCurve() { - return nil, false - } - } - - return e, true -} - -// GT is an abstract cyclic group. The zero value is suitable for use as the -// output of an operation, but cannot be used as an input. -type GT struct { - p *gfP12 -} - -func (g *GT) String() string { - return "bn256.GT" + g.p.String() -} - -// ScalarMult sets e to a*k and then returns e. -func (e *GT) ScalarMult(a *GT, k *big.Int) *GT { - if e.p == nil { - e.p = newGFp12(nil) - } - e.p.Exp(a.p, k, new(bnPool)) - return e -} - -// Add sets e to a+b and then returns e. -func (e *GT) Add(a, b *GT) *GT { - if e.p == nil { - e.p = newGFp12(nil) - } - e.p.Mul(a.p, b.p, new(bnPool)) - return e -} - -// Neg sets e to -a and then returns e. -func (e *GT) Neg(a *GT) *GT { - if e.p == nil { - e.p = newGFp12(nil) - } - e.p.Invert(a.p, new(bnPool)) - return e -} - -// Marshal converts n into a byte slice. -func (n *GT) Marshal() []byte { - n.p.Minimal() - - xxxBytes := n.p.x.x.x.Bytes() - xxyBytes := n.p.x.x.y.Bytes() - xyxBytes := n.p.x.y.x.Bytes() - xyyBytes := n.p.x.y.y.Bytes() - xzxBytes := n.p.x.z.x.Bytes() - xzyBytes := n.p.x.z.y.Bytes() - yxxBytes := n.p.y.x.x.Bytes() - yxyBytes := n.p.y.x.y.Bytes() - yyxBytes := n.p.y.y.x.Bytes() - yyyBytes := n.p.y.y.y.Bytes() - yzxBytes := n.p.y.z.x.Bytes() - yzyBytes := n.p.y.z.y.Bytes() - - // Each value is a 256-bit number. - const numBytes = 256 / 8 - - ret := make([]byte, numBytes*12) - copy(ret[1*numBytes-len(xxxBytes):], xxxBytes) - copy(ret[2*numBytes-len(xxyBytes):], xxyBytes) - copy(ret[3*numBytes-len(xyxBytes):], xyxBytes) - copy(ret[4*numBytes-len(xyyBytes):], xyyBytes) - copy(ret[5*numBytes-len(xzxBytes):], xzxBytes) - copy(ret[6*numBytes-len(xzyBytes):], xzyBytes) - copy(ret[7*numBytes-len(yxxBytes):], yxxBytes) - copy(ret[8*numBytes-len(yxyBytes):], yxyBytes) - copy(ret[9*numBytes-len(yyxBytes):], yyxBytes) - copy(ret[10*numBytes-len(yyyBytes):], yyyBytes) - copy(ret[11*numBytes-len(yzxBytes):], yzxBytes) - copy(ret[12*numBytes-len(yzyBytes):], yzyBytes) - - return ret -} - -// Unmarshal sets e to the result of converting the output of Marshal back into -// a group element and then returns e. -func (e *GT) Unmarshal(m []byte) (*GT, bool) { - // Each value is a 256-bit number. - const numBytes = 256 / 8 - - if len(m) != 12*numBytes { - return nil, false - } - - if e.p == nil { - e.p = newGFp12(nil) - } - - e.p.x.x.x.SetBytes(m[0*numBytes : 1*numBytes]) - e.p.x.x.y.SetBytes(m[1*numBytes : 2*numBytes]) - e.p.x.y.x.SetBytes(m[2*numBytes : 3*numBytes]) - e.p.x.y.y.SetBytes(m[3*numBytes : 4*numBytes]) - e.p.x.z.x.SetBytes(m[4*numBytes : 5*numBytes]) - e.p.x.z.y.SetBytes(m[5*numBytes : 6*numBytes]) - e.p.y.x.x.SetBytes(m[6*numBytes : 7*numBytes]) - e.p.y.x.y.SetBytes(m[7*numBytes : 8*numBytes]) - e.p.y.y.x.SetBytes(m[8*numBytes : 9*numBytes]) - e.p.y.y.y.SetBytes(m[9*numBytes : 10*numBytes]) - e.p.y.z.x.SetBytes(m[10*numBytes : 11*numBytes]) - e.p.y.z.y.SetBytes(m[11*numBytes : 12*numBytes]) - - return e, true -} - -// Pair calculates an Optimal Ate pairing. -func Pair(g1 *G1, g2 *G2) *GT { - return >{optimalAte(g2.p, g1.p, new(bnPool))} -} - -// PairingCheck calculates the Optimal Ate pairing for a set of points. -func PairingCheck(a []*G1, b []*G2) bool { - pool := new(bnPool) - - acc := newGFp12(pool) - acc.SetOne() - - for i := 0; i < len(a); i++ { - if a[i].p.IsInfinity() || b[i].p.IsInfinity() { - continue - } - 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 -} diff --git a/crypto/bn256/bn256_amd64.go b/crypto/bn256/bn256_amd64.go new file mode 100644 index 000000000..35b4839c2 --- /dev/null +++ b/crypto/bn256/bn256_amd64.go @@ -0,0 +1,63 @@ +// Copyright 2018 The go-ethereum Authors +// This file is part of the go-ethereum library. +// +// The go-ethereum library is free software: you can redistribute it and/or modify +// it under the terms of the GNU Lesser General Public License as published by +// the Free Software Foundation, either version 3 of the License, or +// (at your option) any later version. +// +// The go-ethereum library is distributed in the hope that it will be useful, +// but WITHOUT ANY WARRANTY; without even the implied warranty of +// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +// GNU Lesser General Public License for more details. +// +// You should have received a copy of the GNU Lesser General Public License +// along with the go-ethereum library. If not, see . + +// +build amd64,!appengine,!gccgo + +// Package bn256 implements the Optimal Ate pairing over a 256-bit Barreto-Naehrig curve. +package bn256 + +import ( + "math/big" + + "github.com/ethereum/go-ethereum/crypto/bn256/cloudflare" +) + +// G1 is an abstract cyclic group. The zero value is suitable for use as the +// output of an operation, but cannot be used as an input. +type G1 struct { + bn256.G1 +} + +// Add sets e to a+b and then returns e. +func (e *G1) Add(a, b *G1) *G1 { + e.G1.Add(&a.G1, &b.G1) + return e +} + +// ScalarMult sets e to a*k and then returns e. +func (e *G1) ScalarMult(a *G1, k *big.Int) *G1 { + e.G1.ScalarMult(&a.G1, k) + return e +} + +// G2 is an abstract cyclic group. The zero value is suitable for use as the +// output of an operation, but cannot be used as an input. +type G2 struct { + bn256.G2 +} + +// PairingCheck calculates the Optimal Ate pairing for a set of points. +func PairingCheck(a []*G1, b []*G2) bool { + as := make([]*bn256.G1, len(a)) + for i, p := range a { + as[i] = &p.G1 + } + bs := make([]*bn256.G2, len(b)) + for i, p := range b { + bs[i] = &p.G2 + } + return bn256.PairingCheck(as, bs) +} diff --git a/crypto/bn256/bn256_other.go b/crypto/bn256/bn256_other.go new file mode 100644 index 000000000..81977a0a8 --- /dev/null +++ b/crypto/bn256/bn256_other.go @@ -0,0 +1,63 @@ +// Copyright 2018 The go-ethereum Authors +// This file is part of the go-ethereum library. +// +// The go-ethereum library is free software: you can redistribute it and/or modify +// it under the terms of the GNU Lesser General Public License as published by +// the Free Software Foundation, either version 3 of the License, or +// (at your option) any later version. +// +// The go-ethereum library is distributed in the hope that it will be useful, +// but WITHOUT ANY WARRANTY; without even the implied warranty of +// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +// GNU Lesser General Public License for more details. +// +// You should have received a copy of the GNU Lesser General Public License +// along with the go-ethereum library. If not, see . + +// +build !amd64 appengine gccgo + +// Package bn256 implements the Optimal Ate pairing over a 256-bit Barreto-Naehrig curve. +package bn256 + +import ( + "math/big" + + "github.com/ethereum/go-ethereum/crypto/bn256/google" +) + +// G1 is an abstract cyclic group. The zero value is suitable for use as the +// output of an operation, but cannot be used as an input. +type G1 struct { + bn256.G1 +} + +// Add sets e to a+b and then returns e. +func (e *G1) Add(a, b *G1) *G1 { + e.G1.Add(&a.G1, &b.G1) + return e +} + +// ScalarMult sets e to a*k and then returns e. +func (e *G1) ScalarMult(a *G1, k *big.Int) *G1 { + e.G1.ScalarMult(&a.G1, k) + return e +} + +// G2 is an abstract cyclic group. The zero value is suitable for use as the +// output of an operation, but cannot be used as an input. +type G2 struct { + bn256.G2 +} + +// PairingCheck calculates the Optimal Ate pairing for a set of points. +func PairingCheck(a []*G1, b []*G2) bool { + as := make([]*bn256.G1, len(a)) + for i, p := range a { + as[i] = &p.G1 + } + bs := make([]*bn256.G2, len(b)) + for i, p := range b { + bs[i] = &p.G2 + } + return bn256.PairingCheck(as, bs) +} diff --git a/crypto/bn256/bn256_test.go b/crypto/bn256/bn256_test.go deleted file mode 100644 index 866065d0c..000000000 --- a/crypto/bn256/bn256_test.go +++ /dev/null @@ -1,304 +0,0 @@ -// 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 - -import ( - "bytes" - "crypto/rand" - "math/big" - "testing" -) - -func TestGFp2Invert(t *testing.T) { - pool := new(bnPool) - - a := newGFp2(pool) - a.x.SetString("23423492374", 10) - a.y.SetString("12934872398472394827398470", 10) - - inv := newGFp2(pool) - inv.Invert(a, pool) - - b := newGFp2(pool).Mul(inv, a, pool) - if b.x.Int64() != 0 || b.y.Int64() != 1 { - t.Fatalf("bad result for a^-1*a: %s %s", b.x, b.y) - } - - a.Put(pool) - b.Put(pool) - inv.Put(pool) - - if c := pool.Count(); c > 0 { - t.Errorf("Pool count non-zero: %d\n", c) - } -} - -func isZero(n *big.Int) bool { - return new(big.Int).Mod(n, P).Int64() == 0 -} - -func isOne(n *big.Int) bool { - return new(big.Int).Mod(n, P).Int64() == 1 -} - -func TestGFp6Invert(t *testing.T) { - pool := new(bnPool) - - a := newGFp6(pool) - a.x.x.SetString("239487238491", 10) - a.x.y.SetString("2356249827341", 10) - a.y.x.SetString("082659782", 10) - a.y.y.SetString("182703523765", 10) - a.z.x.SetString("978236549263", 10) - a.z.y.SetString("64893242", 10) - - inv := newGFp6(pool) - inv.Invert(a, pool) - - b := newGFp6(pool).Mul(inv, a, pool) - if !isZero(b.x.x) || - !isZero(b.x.y) || - !isZero(b.y.x) || - !isZero(b.y.y) || - !isZero(b.z.x) || - !isOne(b.z.y) { - t.Fatalf("bad result for a^-1*a: %s", b) - } - - a.Put(pool) - b.Put(pool) - inv.Put(pool) - - if c := pool.Count(); c > 0 { - t.Errorf("Pool count non-zero: %d\n", c) - } -} - -func TestGFp12Invert(t *testing.T) { - pool := new(bnPool) - - a := newGFp12(pool) - a.x.x.x.SetString("239846234862342323958623", 10) - a.x.x.y.SetString("2359862352529835623", 10) - a.x.y.x.SetString("928836523", 10) - a.x.y.y.SetString("9856234", 10) - a.x.z.x.SetString("235635286", 10) - a.x.z.y.SetString("5628392833", 10) - a.y.x.x.SetString("252936598265329856238956532167968", 10) - a.y.x.y.SetString("23596239865236954178968", 10) - a.y.y.x.SetString("95421692834", 10) - a.y.y.y.SetString("236548", 10) - a.y.z.x.SetString("924523", 10) - a.y.z.y.SetString("12954623", 10) - - inv := newGFp12(pool) - inv.Invert(a, pool) - - b := newGFp12(pool).Mul(inv, a, pool) - if !isZero(b.x.x.x) || - !isZero(b.x.x.y) || - !isZero(b.x.y.x) || - !isZero(b.x.y.y) || - !isZero(b.x.z.x) || - !isZero(b.x.z.y) || - !isZero(b.y.x.x) || - !isZero(b.y.x.y) || - !isZero(b.y.y.x) || - !isZero(b.y.y.y) || - !isZero(b.y.z.x) || - !isOne(b.y.z.y) { - t.Fatalf("bad result for a^-1*a: %s", b) - } - - a.Put(pool) - b.Put(pool) - inv.Put(pool) - - if c := pool.Count(); c > 0 { - t.Errorf("Pool count non-zero: %d\n", c) - } -} - -func TestCurveImpl(t *testing.T) { - pool := new(bnPool) - - g := &curvePoint{ - pool.Get().SetInt64(1), - pool.Get().SetInt64(-2), - pool.Get().SetInt64(1), - pool.Get().SetInt64(0), - } - - x := pool.Get().SetInt64(32498273234) - X := newCurvePoint(pool).Mul(g, x, pool) - - y := pool.Get().SetInt64(98732423523) - Y := newCurvePoint(pool).Mul(g, y, pool) - - s1 := newCurvePoint(pool).Mul(X, y, pool).MakeAffine(pool) - s2 := newCurvePoint(pool).Mul(Y, x, pool).MakeAffine(pool) - - if s1.x.Cmp(s2.x) != 0 || - s2.x.Cmp(s1.x) != 0 { - t.Errorf("DH points don't match: (%s, %s) (%s, %s)", s1.x, s1.y, s2.x, s2.y) - } - - pool.Put(x) - X.Put(pool) - pool.Put(y) - Y.Put(pool) - s1.Put(pool) - s2.Put(pool) - g.Put(pool) - - if c := pool.Count(); c > 0 { - t.Errorf("Pool count non-zero: %d\n", c) - } -} - -func TestOrderG1(t *testing.T) { - g := new(G1).ScalarBaseMult(Order) - if !g.p.IsInfinity() { - t.Error("G1 has incorrect order") - } - - one := new(G1).ScalarBaseMult(new(big.Int).SetInt64(1)) - g.Add(g, one) - g.p.MakeAffine(nil) - if g.p.x.Cmp(one.p.x) != 0 || g.p.y.Cmp(one.p.y) != 0 { - t.Errorf("1+0 != 1 in G1") - } -} - -func TestOrderG2(t *testing.T) { - g := new(G2).ScalarBaseMult(Order) - if !g.p.IsInfinity() { - t.Error("G2 has incorrect order") - } - - one := new(G2).ScalarBaseMult(new(big.Int).SetInt64(1)) - g.Add(g, one) - g.p.MakeAffine(nil) - if g.p.x.x.Cmp(one.p.x.x) != 0 || - g.p.x.y.Cmp(one.p.x.y) != 0 || - g.p.y.x.Cmp(one.p.y.x) != 0 || - g.p.y.y.Cmp(one.p.y.y) != 0 { - t.Errorf("1+0 != 1 in G2") - } -} - -func TestOrderGT(t *testing.T) { - gt := Pair(&G1{curveGen}, &G2{twistGen}) - g := new(GT).ScalarMult(gt, Order) - if !g.p.IsOne() { - t.Error("GT has incorrect order") - } -} - -func TestBilinearity(t *testing.T) { - for i := 0; i < 2; i++ { - a, p1, _ := RandomG1(rand.Reader) - b, p2, _ := RandomG2(rand.Reader) - e1 := Pair(p1, p2) - - e2 := Pair(&G1{curveGen}, &G2{twistGen}) - e2.ScalarMult(e2, a) - e2.ScalarMult(e2, b) - - minusE2 := new(GT).Neg(e2) - e1.Add(e1, minusE2) - - if !e1.p.IsOne() { - t.Fatalf("bad pairing result: %s", e1) - } - } -} - -func TestG1Marshal(t *testing.T) { - g := new(G1).ScalarBaseMult(new(big.Int).SetInt64(1)) - form := g.Marshal() - _, ok := new(G1).Unmarshal(form) - if !ok { - t.Fatalf("failed to unmarshal") - } - - g.ScalarBaseMult(Order) - form = g.Marshal() - g2, ok := new(G1).Unmarshal(form) - if !ok { - t.Fatalf("failed to unmarshal ∞") - } - if !g2.p.IsInfinity() { - t.Fatalf("∞ unmarshaled incorrectly") - } -} - -func TestG2Marshal(t *testing.T) { - g := new(G2).ScalarBaseMult(new(big.Int).SetInt64(1)) - form := g.Marshal() - _, ok := new(G2).Unmarshal(form) - if !ok { - t.Fatalf("failed to unmarshal") - } - - g.ScalarBaseMult(Order) - form = g.Marshal() - g2, ok := new(G2).Unmarshal(form) - if !ok { - t.Fatalf("failed to unmarshal ∞") - } - if !g2.p.IsInfinity() { - t.Fatalf("∞ unmarshaled incorrectly") - } -} - -func TestG1Identity(t *testing.T) { - g := new(G1).ScalarBaseMult(new(big.Int).SetInt64(0)) - if !g.p.IsInfinity() { - t.Error("failure") - } -} - -func TestG2Identity(t *testing.T) { - g := new(G2).ScalarBaseMult(new(big.Int).SetInt64(0)) - if !g.p.IsInfinity() { - t.Error("failure") - } -} - -func TestTripartiteDiffieHellman(t *testing.T) { - a, _ := rand.Int(rand.Reader, Order) - b, _ := rand.Int(rand.Reader, Order) - c, _ := rand.Int(rand.Reader, Order) - - pa, _ := new(G1).Unmarshal(new(G1).ScalarBaseMult(a).Marshal()) - qa, _ := new(G2).Unmarshal(new(G2).ScalarBaseMult(a).Marshal()) - pb, _ := new(G1).Unmarshal(new(G1).ScalarBaseMult(b).Marshal()) - qb, _ := new(G2).Unmarshal(new(G2).ScalarBaseMult(b).Marshal()) - pc, _ := new(G1).Unmarshal(new(G1).ScalarBaseMult(c).Marshal()) - qc, _ := new(G2).Unmarshal(new(G2).ScalarBaseMult(c).Marshal()) - - k1 := Pair(pb, qc) - k1.ScalarMult(k1, a) - k1Bytes := k1.Marshal() - - k2 := Pair(pc, qa) - k2.ScalarMult(k2, b) - k2Bytes := k2.Marshal() - - k3 := Pair(pa, qb) - k3.ScalarMult(k3, c) - k3Bytes := k3.Marshal() - - if !bytes.Equal(k1Bytes, k2Bytes) || !bytes.Equal(k2Bytes, k3Bytes) { - t.Errorf("keys didn't agree") - } -} - -func BenchmarkPairing(b *testing.B) { - for i := 0; i < b.N; i++ { - Pair(&G1{curveGen}, &G2{twistGen}) - } -} diff --git a/crypto/bn256/cloudflare/bn256.go b/crypto/bn256/cloudflare/bn256.go new file mode 100644 index 000000000..c6ea2d07e --- /dev/null +++ b/crypto/bn256/cloudflare/bn256.go @@ -0,0 +1,481 @@ +// Package bn256 implements a particular bilinear group at the 128-bit security +// level. +// +// Bilinear groups are the basis of many of the new cryptographic protocols that +// have been proposed over the past decade. They consist of a triplet of groups +// (G₁, G₂ and GT) such that there exists a function e(g₁ˣ,g₂ʸ)=gTˣʸ (where gₓ +// is a generator of the respective group). That function is called a pairing +// function. +// +// This package specifically implements the Optimal Ate pairing over a 256-bit +// Barreto-Naehrig curve as described in +// http://cryptojedi.org/papers/dclxvi-20100714.pdf. Its output is compatible +// with the implementation described in that paper. +package bn256 + +import ( + "crypto/rand" + "errors" + "io" + "math/big" +) + +func randomK(r io.Reader) (k *big.Int, err error) { + for { + k, err = rand.Int(r, Order) + if k.Sign() > 0 || err != nil { + return + } + } +} + +// G1 is an abstract cyclic group. The zero value is suitable for use as the +// output of an operation, but cannot be used as an input. +type G1 struct { + p *curvePoint +} + +// RandomG1 returns x and g₁ˣ where x is a random, non-zero number read from r. +func RandomG1(r io.Reader) (*big.Int, *G1, error) { + k, err := randomK(r) + if err != nil { + return nil, nil, err + } + + return k, new(G1).ScalarBaseMult(k), nil +} + +func (g *G1) String() string { + return "bn256.G1" + g.p.String() +} + +// ScalarBaseMult sets e to g*k where g is the generator of the group and then +// returns e. +func (e *G1) ScalarBaseMult(k *big.Int) *G1 { + if e.p == nil { + e.p = &curvePoint{} + } + e.p.Mul(curveGen, k) + return e +} + +// ScalarMult sets e to a*k and then returns e. +func (e *G1) ScalarMult(a *G1, k *big.Int) *G1 { + if e.p == nil { + e.p = &curvePoint{} + } + e.p.Mul(a.p, k) + return e +} + +// Add sets e to a+b and then returns e. +func (e *G1) Add(a, b *G1) *G1 { + if e.p == nil { + e.p = &curvePoint{} + } + e.p.Add(a.p, b.p) + return e +} + +// Neg sets e to -a and then returns e. +func (e *G1) Neg(a *G1) *G1 { + if e.p == nil { + e.p = &curvePoint{} + } + e.p.Neg(a.p) + return e +} + +// Set sets e to a and then returns e. +func (e *G1) Set(a *G1) *G1 { + if e.p == nil { + e.p = &curvePoint{} + } + e.p.Set(a.p) + return e +} + +// Marshal converts e to a byte slice. +func (e *G1) Marshal() []byte { + // Each value is a 256-bit number. + const numBytes = 256 / 8 + + e.p.MakeAffine() + ret := make([]byte, numBytes*2) + if e.p.IsInfinity() { + return ret + } + temp := &gfP{} + + montDecode(temp, &e.p.x) + temp.Marshal(ret) + montDecode(temp, &e.p.y) + temp.Marshal(ret[numBytes:]) + + return ret +} + +// Unmarshal sets e to the result of converting the output of Marshal back into +// a group element and then returns e. +func (e *G1) Unmarshal(m []byte) ([]byte, error) { + // Each value is a 256-bit number. + const numBytes = 256 / 8 + if len(m) < 2*numBytes { + return nil, errors.New("bn256: not enough data") + } + // Unmarshal the points and check their caps + if e.p == nil { + e.p = &curvePoint{} + } else { + e.p.x, e.p.y = gfP{0}, gfP{0} + } + var err error + if err = e.p.x.Unmarshal(m); err != nil { + return nil, err + } + if err = e.p.y.Unmarshal(m[numBytes:]); err != nil { + return nil, err + } + // Encode into Montgomery form and ensure it's on the curve + montEncode(&e.p.x, &e.p.x) + montEncode(&e.p.y, &e.p.y) + + zero := gfP{0} + if e.p.x == zero && e.p.y == zero { + // This is the point at infinity. + e.p.y = *newGFp(1) + e.p.z = gfP{0} + e.p.t = gfP{0} + } else { + e.p.z = *newGFp(1) + e.p.t = *newGFp(1) + + if !e.p.IsOnCurve() { + return nil, errors.New("bn256: malformed point") + } + } + return m[2*numBytes:], nil +} + +// G2 is an abstract cyclic group. The zero value is suitable for use as the +// output of an operation, but cannot be used as an input. +type G2 struct { + p *twistPoint +} + +// RandomG2 returns x and g₂ˣ where x is a random, non-zero number read from r. +func RandomG2(r io.Reader) (*big.Int, *G2, error) { + k, err := randomK(r) + if err != nil { + return nil, nil, err + } + + return k, new(G2).ScalarBaseMult(k), nil +} + +func (e *G2) String() string { + return "bn256.G2" + e.p.String() +} + +// ScalarBaseMult sets e to g*k where g is the generator of the group and then +// returns out. +func (e *G2) ScalarBaseMult(k *big.Int) *G2 { + if e.p == nil { + e.p = &twistPoint{} + } + e.p.Mul(twistGen, k) + return e +} + +// ScalarMult sets e to a*k and then returns e. +func (e *G2) ScalarMult(a *G2, k *big.Int) *G2 { + if e.p == nil { + e.p = &twistPoint{} + } + e.p.Mul(a.p, k) + return e +} + +// Add sets e to a+b and then returns e. +func (e *G2) Add(a, b *G2) *G2 { + if e.p == nil { + e.p = &twistPoint{} + } + e.p.Add(a.p, b.p) + return e +} + +// Neg sets e to -a and then returns e. +func (e *G2) Neg(a *G2) *G2 { + if e.p == nil { + e.p = &twistPoint{} + } + e.p.Neg(a.p) + return e +} + +// Set sets e to a and then returns e. +func (e *G2) Set(a *G2) *G2 { + if e.p == nil { + e.p = &twistPoint{} + } + e.p.Set(a.p) + return e +} + +// Marshal converts e into a byte slice. +func (e *G2) Marshal() []byte { + // Each value is a 256-bit number. + const numBytes = 256 / 8 + + if e.p == nil { + e.p = &twistPoint{} + } + + e.p.MakeAffine() + ret := make([]byte, numBytes*4) + if e.p.IsInfinity() { + return ret + } + temp := &gfP{} + + montDecode(temp, &e.p.x.x) + temp.Marshal(ret) + montDecode(temp, &e.p.x.y) + temp.Marshal(ret[numBytes:]) + montDecode(temp, &e.p.y.x) + temp.Marshal(ret[2*numBytes:]) + montDecode(temp, &e.p.y.y) + temp.Marshal(ret[3*numBytes:]) + + return ret +} + +// Unmarshal sets e to the result of converting the output of Marshal back into +// a group element and then returns e. +func (e *G2) Unmarshal(m []byte) ([]byte, error) { + // Each value is a 256-bit number. + const numBytes = 256 / 8 + if len(m) < 4*numBytes { + return nil, errors.New("bn256: not enough data") + } + // Unmarshal the points and check their caps + if e.p == nil { + e.p = &twistPoint{} + } + var err error + if err = e.p.x.x.Unmarshal(m); err != nil { + return nil, err + } + if err = e.p.x.y.Unmarshal(m[numBytes:]); err != nil { + return nil, err + } + if err = e.p.y.x.Unmarshal(m[2*numBytes:]); err != nil { + return nil, err + } + if err = e.p.y.y.Unmarshal(m[3*numBytes:]); err != nil { + return nil, err + } + // Encode into Montgomery form and ensure it's on the curve + montEncode(&e.p.x.x, &e.p.x.x) + montEncode(&e.p.x.y, &e.p.x.y) + montEncode(&e.p.y.x, &e.p.y.x) + montEncode(&e.p.y.y, &e.p.y.y) + + if e.p.x.IsZero() && e.p.y.IsZero() { + // This is the point at infinity. + e.p.y.SetOne() + e.p.z.SetZero() + e.p.t.SetZero() + } else { + e.p.z.SetOne() + e.p.t.SetOne() + + if !e.p.IsOnCurve() { + return nil, errors.New("bn256: malformed point") + } + } + return m[4*numBytes:], nil +} + +// GT is an abstract cyclic group. The zero value is suitable for use as the +// output of an operation, but cannot be used as an input. +type GT struct { + p *gfP12 +} + +// Pair calculates an Optimal Ate pairing. +func Pair(g1 *G1, g2 *G2) *GT { + return >{optimalAte(g2.p, g1.p)} +} + +// PairingCheck calculates the Optimal Ate pairing for a set of points. +func PairingCheck(a []*G1, b []*G2) bool { + acc := new(gfP12) + acc.SetOne() + + for i := 0; i < len(a); i++ { + if a[i].p.IsInfinity() || b[i].p.IsInfinity() { + continue + } + acc.Mul(acc, miller(b[i].p, a[i].p)) + } + return finalExponentiation(acc).IsOne() +} + +// Miller applies Miller's algorithm, which is a bilinear function from the +// source groups to F_p^12. Miller(g1, g2).Finalize() is equivalent to Pair(g1, +// g2). +func Miller(g1 *G1, g2 *G2) *GT { + return >{miller(g2.p, g1.p)} +} + +func (g *GT) String() string { + return "bn256.GT" + g.p.String() +} + +// ScalarMult sets e to a*k and then returns e. +func (e *GT) ScalarMult(a *GT, k *big.Int) *GT { + if e.p == nil { + e.p = &gfP12{} + } + e.p.Exp(a.p, k) + return e +} + +// Add sets e to a+b and then returns e. +func (e *GT) Add(a, b *GT) *GT { + if e.p == nil { + e.p = &gfP12{} + } + e.p.Mul(a.p, b.p) + return e +} + +// Neg sets e to -a and then returns e. +func (e *GT) Neg(a *GT) *GT { + if e.p == nil { + e.p = &gfP12{} + } + e.p.Conjugate(a.p) + return e +} + +// Set sets e to a and then returns e. +func (e *GT) Set(a *GT) *GT { + if e.p == nil { + e.p = &gfP12{} + } + e.p.Set(a.p) + return e +} + +// Finalize is a linear function from F_p^12 to GT. +func (e *GT) Finalize() *GT { + ret := finalExponentiation(e.p) + e.p.Set(ret) + return e +} + +// Marshal converts e into a byte slice. +func (e *GT) Marshal() []byte { + // Each value is a 256-bit number. + const numBytes = 256 / 8 + + ret := make([]byte, numBytes*12) + temp := &gfP{} + + montDecode(temp, &e.p.x.x.x) + temp.Marshal(ret) + montDecode(temp, &e.p.x.x.y) + temp.Marshal(ret[numBytes:]) + montDecode(temp, &e.p.x.y.x) + temp.Marshal(ret[2*numBytes:]) + montDecode(temp, &e.p.x.y.y) + temp.Marshal(ret[3*numBytes:]) + montDecode(temp, &e.p.x.z.x) + temp.Marshal(ret[4*numBytes:]) + montDecode(temp, &e.p.x.z.y) + temp.Marshal(ret[5*numBytes:]) + montDecode(temp, &e.p.y.x.x) + temp.Marshal(ret[6*numBytes:]) + montDecode(temp, &e.p.y.x.y) + temp.Marshal(ret[7*numBytes:]) + montDecode(temp, &e.p.y.y.x) + temp.Marshal(ret[8*numBytes:]) + montDecode(temp, &e.p.y.y.y) + temp.Marshal(ret[9*numBytes:]) + montDecode(temp, &e.p.y.z.x) + temp.Marshal(ret[10*numBytes:]) + montDecode(temp, &e.p.y.z.y) + temp.Marshal(ret[11*numBytes:]) + + return ret +} + +// Unmarshal sets e to the result of converting the output of Marshal back into +// a group element and then returns e. +func (e *GT) Unmarshal(m []byte) ([]byte, error) { + // Each value is a 256-bit number. + const numBytes = 256 / 8 + + if len(m) < 12*numBytes { + return nil, errors.New("bn256: not enough data") + } + + if e.p == nil { + e.p = &gfP12{} + } + + var err error + if err = e.p.x.x.x.Unmarshal(m); err != nil { + return nil, err + } + if err = e.p.x.x.y.Unmarshal(m[numBytes:]); err != nil { + return nil, err + } + if err = e.p.x.y.x.Unmarshal(m[2*numBytes:]); err != nil { + return nil, err + } + if err = e.p.x.y.y.Unmarshal(m[3*numBytes:]); err != nil { + return nil, err + } + if err = e.p.x.z.x.Unmarshal(m[4*numBytes:]); err != nil { + return nil, err + } + if err = e.p.x.z.y.Unmarshal(m[5*numBytes:]); err != nil { + return nil, err + } + if err = e.p.y.x.x.Unmarshal(m[6*numBytes:]); err != nil { + return nil, err + } + if err = e.p.y.x.y.Unmarshal(m[7*numBytes:]); err != nil { + return nil, err + } + if err = e.p.y.y.x.Unmarshal(m[8*numBytes:]); err != nil { + return nil, err + } + if err = e.p.y.y.y.Unmarshal(m[9*numBytes:]); err != nil { + return nil, err + } + if err = e.p.y.z.x.Unmarshal(m[10*numBytes:]); err != nil { + return nil, err + } + if err = e.p.y.z.y.Unmarshal(m[11*numBytes:]); err != nil { + return nil, err + } + montEncode(&e.p.x.x.x, &e.p.x.x.x) + montEncode(&e.p.x.x.y, &e.p.x.x.y) + montEncode(&e.p.x.y.x, &e.p.x.y.x) + montEncode(&e.p.x.y.y, &e.p.x.y.y) + montEncode(&e.p.x.z.x, &e.p.x.z.x) + montEncode(&e.p.x.z.y, &e.p.x.z.y) + montEncode(&e.p.y.x.x, &e.p.y.x.x) + montEncode(&e.p.y.x.y, &e.p.y.x.y) + montEncode(&e.p.y.y.x, &e.p.y.y.x) + montEncode(&e.p.y.y.y, &e.p.y.y.y) + montEncode(&e.p.y.z.x, &e.p.y.z.x) + montEncode(&e.p.y.z.y, &e.p.y.z.y) + + return m[12*numBytes:], nil +} diff --git a/crypto/bn256/cloudflare/bn256_test.go b/crypto/bn256/cloudflare/bn256_test.go new file mode 100644 index 000000000..369a3edaa --- /dev/null +++ b/crypto/bn256/cloudflare/bn256_test.go @@ -0,0 +1,118 @@ +// +build amd64,!appengine,!gccgo + +package bn256 + +import ( + "bytes" + "crypto/rand" + "testing" +) + +func TestG1Marshal(t *testing.T) { + _, Ga, err := RandomG1(rand.Reader) + if err != nil { + t.Fatal(err) + } + ma := Ga.Marshal() + + Gb := new(G1) + _, err = Gb.Unmarshal(ma) + if err != nil { + t.Fatal(err) + } + mb := Gb.Marshal() + + if !bytes.Equal(ma, mb) { + t.Fatal("bytes are different") + } +} + +func TestG2Marshal(t *testing.T) { + _, Ga, err := RandomG2(rand.Reader) + if err != nil { + t.Fatal(err) + } + ma := Ga.Marshal() + + Gb := new(G2) + _, err = Gb.Unmarshal(ma) + if err != nil { + t.Fatal(err) + } + mb := Gb.Marshal() + + if !bytes.Equal(ma, mb) { + t.Fatal("bytes are different") + } +} + +func TestBilinearity(t *testing.T) { + for i := 0; i < 2; i++ { + a, p1, _ := RandomG1(rand.Reader) + b, p2, _ := RandomG2(rand.Reader) + e1 := Pair(p1, p2) + + e2 := Pair(&G1{curveGen}, &G2{twistGen}) + e2.ScalarMult(e2, a) + e2.ScalarMult(e2, b) + + if *e1.p != *e2.p { + t.Fatalf("bad pairing result: %s", e1) + } + } +} + +func TestTripartiteDiffieHellman(t *testing.T) { + a, _ := rand.Int(rand.Reader, Order) + b, _ := rand.Int(rand.Reader, Order) + c, _ := rand.Int(rand.Reader, Order) + + pa, pb, pc := new(G1), new(G1), new(G1) + qa, qb, qc := new(G2), new(G2), new(G2) + + pa.Unmarshal(new(G1).ScalarBaseMult(a).Marshal()) + qa.Unmarshal(new(G2).ScalarBaseMult(a).Marshal()) + pb.Unmarshal(new(G1).ScalarBaseMult(b).Marshal()) + qb.Unmarshal(new(G2).ScalarBaseMult(b).Marshal()) + pc.Unmarshal(new(G1).ScalarBaseMult(c).Marshal()) + qc.Unmarshal(new(G2).ScalarBaseMult(c).Marshal()) + + k1 := Pair(pb, qc) + k1.ScalarMult(k1, a) + k1Bytes := k1.Marshal() + + k2 := Pair(pc, qa) + k2.ScalarMult(k2, b) + k2Bytes := k2.Marshal() + + k3 := Pair(pa, qb) + k3.ScalarMult(k3, c) + k3Bytes := k3.Marshal() + + if !bytes.Equal(k1Bytes, k2Bytes) || !bytes.Equal(k2Bytes, k3Bytes) { + t.Errorf("keys didn't agree") + } +} + +func BenchmarkG1(b *testing.B) { + x, _ := rand.Int(rand.Reader, Order) + b.ResetTimer() + + for i := 0; i < b.N; i++ { + new(G1).ScalarBaseMult(x) + } +} + +func BenchmarkG2(b *testing.B) { + x, _ := rand.Int(rand.Reader, Order) + b.ResetTimer() + + for i := 0; i < b.N; i++ { + new(G2).ScalarBaseMult(x) + } +} +func BenchmarkPairing(b *testing.B) { + for i := 0; i < b.N; i++ { + Pair(&G1{curveGen}, &G2{twistGen}) + } +} diff --git a/crypto/bn256/cloudflare/constants.go b/crypto/bn256/cloudflare/constants.go new file mode 100644 index 000000000..5122aae64 --- /dev/null +++ b/crypto/bn256/cloudflare/constants.go @@ -0,0 +1,59 @@ +// 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 + +import ( + "math/big" +) + +func bigFromBase10(s string) *big.Int { + n, _ := new(big.Int).SetString(s, 10) + return n +} + +// u is the BN parameter that determines the prime: 1868033³. +var u = bigFromBase10("4965661367192848881") + +// Order is the number of elements in both G₁ and G₂: 36u⁴+36u³+18u²+6u+1. +var Order = bigFromBase10("21888242871839275222246405745257275088548364400416034343698204186575808495617") + +// P is a prime over which we form a basic field: 36u⁴+36u³+24u²+6u+1. +var P = bigFromBase10("21888242871839275222246405745257275088696311157297823662689037894645226208583") + +// p2 is p, represented as little-endian 64-bit words. +var p2 = [4]uint64{0x3c208c16d87cfd47, 0x97816a916871ca8d, 0xb85045b68181585d, 0x30644e72e131a029} + +// np is the negative inverse of p, mod 2^256. +var np = [4]uint64{0x87d20782e4866389, 0x9ede7d651eca6ac9, 0xd8afcbd01833da80, 0xf57a22b791888c6b} + +// rN1 is R^-1 where R = 2^256 mod p. +var rN1 = &gfP{0xed84884a014afa37, 0xeb2022850278edf8, 0xcf63e9cfb74492d9, 0x2e67157159e5c639} + +// r2 is R^2 where R = 2^256 mod p. +var r2 = &gfP{0xf32cfc5b538afa89, 0xb5e71911d44501fb, 0x47ab1eff0a417ff6, 0x06d89f71cab8351f} + +// r3 is R^3 where R = 2^256 mod p. +var r3 = &gfP{0xb1cd6dafda1530df, 0x62f210e6a7283db6, 0xef7f0b0c0ada0afb, 0x20fd6e902d592544} + +// xiToPMinus1Over6 is ξ^((p-1)/6) where ξ = i+9. +var xiToPMinus1Over6 = &gfP2{gfP{0xa222ae234c492d72, 0xd00f02a4565de15b, 0xdc2ff3a253dfc926, 0x10a75716b3899551}, gfP{0xaf9ba69633144907, 0xca6b1d7387afb78a, 0x11bded5ef08a2087, 0x02f34d751a1f3a7c}} + +// xiToPMinus1Over3 is ξ^((p-1)/3) where ξ = i+9. +var xiToPMinus1Over3 = &gfP2{gfP{0x6e849f1ea0aa4757, 0xaa1c7b6d89f89141, 0xb6e713cdfae0ca3a, 0x26694fbb4e82ebc3}, gfP{0xb5773b104563ab30, 0x347f91c8a9aa6454, 0x7a007127242e0991, 0x1956bcd8118214ec}} + +// xiToPMinus1Over2 is ξ^((p-1)/2) where ξ = i+9. +var xiToPMinus1Over2 = &gfP2{gfP{0xa1d77ce45ffe77c7, 0x07affd117826d1db, 0x6d16bd27bb7edc6b, 0x2c87200285defecc}, gfP{0xe4bbdd0c2936b629, 0xbb30f162e133bacb, 0x31a9d1b6f9645366, 0x253570bea500f8dd}} + +// xiToPSquaredMinus1Over3 is ξ^((p²-1)/3) where ξ = i+9. +var xiToPSquaredMinus1Over3 = &gfP{0x3350c88e13e80b9c, 0x7dce557cdb5e56b9, 0x6001b4b8b615564a, 0x2682e617020217e0} + +// xiTo2PSquaredMinus2Over3 is ξ^((2p²-2)/3) where ξ = i+9 (a cubic root of unity, mod p). +var xiTo2PSquaredMinus2Over3 = &gfP{0x71930c11d782e155, 0xa6bb947cffbe3323, 0xaa303344d4741444, 0x2c3b3f0d26594943} + +// xiToPSquaredMinus1Over6 is ξ^((1p²-1)/6) where ξ = i+9 (a cubic root of -1, mod p). +var xiToPSquaredMinus1Over6 = &gfP{0xca8d800500fa1bf2, 0xf0c5d61468b39769, 0x0e201271ad0d4418, 0x04290f65bad856e6} + +// xiTo2PMinus2Over3 is ξ^((2p-2)/3) where ξ = i+9. +var xiTo2PMinus2Over3 = &gfP2{gfP{0x5dddfd154bd8c949, 0x62cb29a5a4445b60, 0x37bc870a0c7dd2b9, 0x24830a9d3171f0fd}, gfP{0x7361d77f843abe92, 0xa5bb2bd3273411fb, 0x9c941f314b3e2399, 0x15df9cddbb9fd3ec}} diff --git a/crypto/bn256/cloudflare/curve.go b/crypto/bn256/cloudflare/curve.go new file mode 100644 index 000000000..b6aecc0a6 --- /dev/null +++ b/crypto/bn256/cloudflare/curve.go @@ -0,0 +1,229 @@ +package bn256 + +import ( + "math/big" +) + +// curvePoint implements the elliptic curve y²=x³+3. Points are kept in Jacobian +// form and t=z² when valid. G₁ is the set of points of this curve on GF(p). +type curvePoint struct { + x, y, z, t gfP +} + +var curveB = newGFp(3) + +// curveGen is the generator of G₁. +var curveGen = &curvePoint{ + x: *newGFp(1), + y: *newGFp(2), + z: *newGFp(1), + t: *newGFp(1), +} + +func (c *curvePoint) String() string { + c.MakeAffine() + x, y := &gfP{}, &gfP{} + montDecode(x, &c.x) + montDecode(y, &c.y) + return "(" + x.String() + ", " + y.String() + ")" +} + +func (c *curvePoint) Set(a *curvePoint) { + c.x.Set(&a.x) + c.y.Set(&a.y) + c.z.Set(&a.z) + c.t.Set(&a.t) +} + +// IsOnCurve returns true iff c is on the curve. +func (c *curvePoint) IsOnCurve() bool { + c.MakeAffine() + if c.IsInfinity() { + return true + } + + y2, x3 := &gfP{}, &gfP{} + gfpMul(y2, &c.y, &c.y) + gfpMul(x3, &c.x, &c.x) + gfpMul(x3, x3, &c.x) + gfpAdd(x3, x3, curveB) + + return *y2 == *x3 +} + +func (c *curvePoint) SetInfinity() { + c.x = gfP{0} + c.y = *newGFp(1) + c.z = gfP{0} + c.t = gfP{0} +} + +func (c *curvePoint) IsInfinity() bool { + return c.z == gfP{0} +} + +func (c *curvePoint) Add(a, b *curvePoint) { + if a.IsInfinity() { + c.Set(b) + return + } + if b.IsInfinity() { + c.Set(a) + return + } + + // See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/addition/add-2007-bl.op3 + + // Normalize the points by replacing a = [x1:y1:z1] and b = [x2:y2:z2] + // by [u1:s1:z1·z2] and [u2:s2:z1·z2] + // where u1 = x1·z2², s1 = y1·z2³ and u1 = x2·z1², s2 = y2·z1³ + z12, z22 := &gfP{}, &gfP{} + gfpMul(z12, &a.z, &a.z) + gfpMul(z22, &b.z, &b.z) + + u1, u2 := &gfP{}, &gfP{} + gfpMul(u1, &a.x, z22) + gfpMul(u2, &b.x, z12) + + t, s1 := &gfP{}, &gfP{} + gfpMul(t, &b.z, z22) + gfpMul(s1, &a.y, t) + + s2 := &gfP{} + gfpMul(t, &a.z, z12) + gfpMul(s2, &b.y, t) + + // Compute x = (2h)²(s²-u1-u2) + // where s = (s2-s1)/(u2-u1) is the slope of the line through + // (u1,s1) and (u2,s2). The extra factor 2h = 2(u2-u1) comes from the value of z below. + // This is also: + // 4(s2-s1)² - 4h²(u1+u2) = 4(s2-s1)² - 4h³ - 4h²(2u1) + // = r² - j - 2v + // with the notations below. + h := &gfP{} + gfpSub(h, u2, u1) + xEqual := *h == gfP{0} + + gfpAdd(t, h, h) + // i = 4h² + i := &gfP{} + gfpMul(i, t, t) + // j = 4h³ + j := &gfP{} + gfpMul(j, h, i) + + gfpSub(t, s2, s1) + yEqual := *t == gfP{0} + if xEqual && yEqual { + c.Double(a) + return + } + r := &gfP{} + gfpAdd(r, t, t) + + v := &gfP{} + gfpMul(v, u1, i) + + // t4 = 4(s2-s1)² + t4, t6 := &gfP{}, &gfP{} + gfpMul(t4, r, r) + gfpAdd(t, v, v) + gfpSub(t6, t4, j) + + gfpSub(&c.x, t6, t) + + // Set y = -(2h)³(s1 + s*(x/4h²-u1)) + // This is also + // y = - 2·s1·j - (s2-s1)(2x - 2i·u1) = r(v-x) - 2·s1·j + gfpSub(t, v, &c.x) // t7 + gfpMul(t4, s1, j) // t8 + gfpAdd(t6, t4, t4) // t9 + gfpMul(t4, r, t) // t10 + gfpSub(&c.y, t4, t6) + + // Set z = 2(u2-u1)·z1·z2 = 2h·z1·z2 + gfpAdd(t, &a.z, &b.z) // t11 + gfpMul(t4, t, t) // t12 + gfpSub(t, t4, z12) // t13 + gfpSub(t4, t, z22) // t14 + gfpMul(&c.z, t4, h) +} + +func (c *curvePoint) Double(a *curvePoint) { + // See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/doubling/dbl-2009-l.op3 + A, B, C := &gfP{}, &gfP{}, &gfP{} + gfpMul(A, &a.x, &a.x) + gfpMul(B, &a.y, &a.y) + gfpMul(C, B, B) + + t, t2 := &gfP{}, &gfP{} + gfpAdd(t, &a.x, B) + gfpMul(t2, t, t) + gfpSub(t, t2, A) + gfpSub(t2, t, C) + + d, e, f := &gfP{}, &gfP{}, &gfP{} + gfpAdd(d, t2, t2) + gfpAdd(t, A, A) + gfpAdd(e, t, A) + gfpMul(f, e, e) + + gfpAdd(t, d, d) + gfpSub(&c.x, f, t) + + gfpAdd(t, C, C) + gfpAdd(t2, t, t) + gfpAdd(t, t2, t2) + gfpSub(&c.y, d, &c.x) + gfpMul(t2, e, &c.y) + gfpSub(&c.y, t2, t) + + gfpMul(t, &a.y, &a.z) + gfpAdd(&c.z, t, t) +} + +func (c *curvePoint) Mul(a *curvePoint, scalar *big.Int) { + sum, t := &curvePoint{}, &curvePoint{} + sum.SetInfinity() + + for i := scalar.BitLen(); i >= 0; i-- { + t.Double(sum) + if scalar.Bit(i) != 0 { + sum.Add(t, a) + } else { + sum.Set(t) + } + } + c.Set(sum) +} + +func (c *curvePoint) MakeAffine() { + if c.z == *newGFp(1) { + return + } else if c.z == *newGFp(0) { + c.x = gfP{0} + c.y = *newGFp(1) + c.t = gfP{0} + return + } + + zInv := &gfP{} + zInv.Invert(&c.z) + + t, zInv2 := &gfP{}, &gfP{} + gfpMul(t, &c.y, zInv) + gfpMul(zInv2, zInv, zInv) + + gfpMul(&c.x, &c.x, zInv2) + gfpMul(&c.y, t, zInv2) + + c.z = *newGFp(1) + c.t = *newGFp(1) +} + +func (c *curvePoint) Neg(a *curvePoint) { + c.x.Set(&a.x) + gfpNeg(&c.y, &a.y) + c.z.Set(&a.z) + c.t = gfP{0} +} diff --git a/crypto/bn256/cloudflare/example_test.go b/crypto/bn256/cloudflare/example_test.go new file mode 100644 index 000000000..2ee545c67 --- /dev/null +++ b/crypto/bn256/cloudflare/example_test.go @@ -0,0 +1,45 @@ +// 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. + +// +build amd64,!appengine,!gccgo + +package bn256 + +import ( + "crypto/rand" +) + +func ExamplePair() { + // This implements the tripartite Diffie-Hellman algorithm from "A One + // Round Protocol for Tripartite Diffie-Hellman", A. Joux. + // http://www.springerlink.com/content/cddc57yyva0hburb/fulltext.pdf + + // Each of three parties, a, b and c, generate a private value. + a, _ := rand.Int(rand.Reader, Order) + b, _ := rand.Int(rand.Reader, Order) + c, _ := rand.Int(rand.Reader, Order) + + // Then each party calculates g₁ and g₂ times their private value. + pa := new(G1).ScalarBaseMult(a) + qa := new(G2).ScalarBaseMult(a) + + pb := new(G1).ScalarBaseMult(b) + qb := new(G2).ScalarBaseMult(b) + + pc := new(G1).ScalarBaseMult(c) + qc := new(G2).ScalarBaseMult(c) + + // Now each party exchanges its public values with the other two and + // all parties can calculate the shared key. + k1 := Pair(pb, qc) + k1.ScalarMult(k1, a) + + k2 := Pair(pc, qa) + k2.ScalarMult(k2, b) + + k3 := Pair(pa, qb) + k3.ScalarMult(k3, c) + + // k1, k2 and k3 will all be equal. +} diff --git a/crypto/bn256/cloudflare/gfp.go b/crypto/bn256/cloudflare/gfp.go new file mode 100644 index 000000000..e8e84e7b3 --- /dev/null +++ b/crypto/bn256/cloudflare/gfp.go @@ -0,0 +1,81 @@ +package bn256 + +import ( + "errors" + "fmt" +) + +type gfP [4]uint64 + +func newGFp(x int64) (out *gfP) { + if x >= 0 { + out = &gfP{uint64(x)} + } else { + out = &gfP{uint64(-x)} + gfpNeg(out, out) + } + + montEncode(out, out) + return out +} + +func (e *gfP) String() string { + return fmt.Sprintf("%16.16x%16.16x%16.16x%16.16x", e[3], e[2], e[1], e[0]) +} + +func (e *gfP) Set(f *gfP) { + e[0] = f[0] + e[1] = f[1] + e[2] = f[2] + e[3] = f[3] +} + +func (e *gfP) Invert(f *gfP) { + bits := [4]uint64{0x3c208c16d87cfd45, 0x97816a916871ca8d, 0xb85045b68181585d, 0x30644e72e131a029} + + sum, power := &gfP{}, &gfP{} + sum.Set(rN1) + power.Set(f) + + for word := 0; word < 4; word++ { + for bit := uint(0); bit < 64; bit++ { + if (bits[word]>>bit)&1 == 1 { + gfpMul(sum, sum, power) + } + gfpMul(power, power, power) + } + } + + gfpMul(sum, sum, r3) + e.Set(sum) +} + +func (e *gfP) Marshal(out []byte) { + for w := uint(0); w < 4; w++ { + for b := uint(0); b < 8; b++ { + out[8*w+b] = byte(e[3-w] >> (56 - 8*b)) + } + } +} + +func (e *gfP) Unmarshal(in []byte) error { + // Unmarshal the bytes into little endian form + for w := uint(0); w < 4; w++ { + for b := uint(0); b < 8; b++ { + e[3-w] += uint64(in[8*w+b]) << (56 - 8*b) + } + } + // Ensure the point respects the curve modulus + for i := 3; i >= 0; i-- { + if e[i] < p2[i] { + return nil + } + if e[i] > p2[i] { + return errors.New("bn256: coordinate exceeds modulus") + } + } + return errors.New("bn256: coordinate equals modulus") +} + +func montEncode(c, a *gfP) { gfpMul(c, a, r2) } +func montDecode(c, a *gfP) { gfpMul(c, a, &gfP{1}) } diff --git a/crypto/bn256/cloudflare/gfp.h b/crypto/bn256/cloudflare/gfp.h new file mode 100644 index 000000000..66f5a4d07 --- /dev/null +++ b/crypto/bn256/cloudflare/gfp.h @@ -0,0 +1,32 @@ +#define storeBlock(a0,a1,a2,a3, r) \ + MOVQ a0, 0+r \ + MOVQ a1, 8+r \ + MOVQ a2, 16+r \ + MOVQ a3, 24+r + +#define loadBlock(r, a0,a1,a2,a3) \ + MOVQ 0+r, a0 \ + MOVQ 8+r, a1 \ + MOVQ 16+r, a2 \ + MOVQ 24+r, a3 + +#define gfpCarry(a0,a1,a2,a3,a4, b0,b1,b2,b3,b4) \ + \ // b = a-p + MOVQ a0, b0 \ + MOVQ a1, b1 \ + MOVQ a2, b2 \ + MOVQ a3, b3 \ + MOVQ a4, b4 \ + \ + SUBQ ·p2+0(SB), b0 \ + SBBQ ·p2+8(SB), b1 \ + SBBQ ·p2+16(SB), b2 \ + SBBQ ·p2+24(SB), b3 \ + SBBQ $0, b4 \ + \ + \ // if b is negative then return a + \ // else return b + CMOVQCC b0, a0 \ + CMOVQCC b1, a1 \ + CMOVQCC b2, a2 \ + CMOVQCC b3, a3 diff --git a/crypto/bn256/cloudflare/gfp12.go b/crypto/bn256/cloudflare/gfp12.go new file mode 100644 index 000000000..93fb368a7 --- /dev/null +++ b/crypto/bn256/cloudflare/gfp12.go @@ -0,0 +1,160 @@ +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 (e *gfP12) String() string { + return "(" + e.x.String() + "," + e.y.String() + ")" +} + +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) IsZero() bool { + return e.x.IsZero() && e.y.IsZero() +} + +func (e *gfP12) IsOne() bool { + return e.x.IsZero() && e.y.IsOne() +} + +func (e *gfP12) Conjugate(a *gfP12) *gfP12 { + e.x.Neg(&a.x) + e.y.Set(&a.y) + return e +} + +func (e *gfP12) Neg(a *gfP12) *gfP12 { + e.x.Neg(&a.x) + e.y.Neg(&a.y) + return e +} + +// Frobenius computes (xω+y)^p = x^p ω·ξ^((p-1)/6) + y^p +func (e *gfP12) Frobenius(a *gfP12) *gfP12 { + e.x.Frobenius(&a.x) + e.y.Frobenius(&a.y) + e.x.MulScalar(&e.x, xiToPMinus1Over6) + return e +} + +// FrobeniusP2 computes (xω+y)^p² = x^p² ω·ξ^((p²-1)/6) + y^p² +func (e *gfP12) FrobeniusP2(a *gfP12) *gfP12 { + e.x.FrobeniusP2(&a.x) + e.x.MulGFP(&e.x, xiToPSquaredMinus1Over6) + e.y.FrobeniusP2(&a.y) + return e +} + +func (e *gfP12) FrobeniusP4(a *gfP12) *gfP12 { + e.x.FrobeniusP4(&a.x) + e.x.MulGFP(&e.x, xiToPSquaredMinus1Over3) + e.y.FrobeniusP4(&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) *gfP12 { + tx := (&gfP6{}).Mul(&a.x, &b.y) + t := (&gfP6{}).Mul(&b.x, &a.y) + tx.Add(tx, t) + + ty := (&gfP6{}).Mul(&a.y, &b.y) + t.Mul(&a.x, &b.x).MulTau(t) + + e.x.Set(tx) + e.y.Add(ty, t) + return e +} + +func (e *gfP12) MulScalar(a *gfP12, b *gfP6) *gfP12 { + e.x.Mul(&e.x, b) + e.y.Mul(&e.y, b) + return e +} + +func (c *gfP12) Exp(a *gfP12, power *big.Int) *gfP12 { + sum := (&gfP12{}).SetOne() + t := &gfP12{} + + for i := power.BitLen() - 1; i >= 0; i-- { + t.Square(sum) + if power.Bit(i) != 0 { + sum.Mul(t, a) + } else { + sum.Set(t) + } + } + + c.Set(sum) + return c +} + +func (e *gfP12) Square(a *gfP12) *gfP12 { + // Complex squaring algorithm + v0 := (&gfP6{}).Mul(&a.x, &a.y) + + t := (&gfP6{}).MulTau(&a.x) + t.Add(&a.y, t) + ty := (&gfP6{}).Add(&a.x, &a.y) + ty.Mul(ty, t).Sub(ty, v0) + t.MulTau(v0) + ty.Sub(ty, t) + + e.x.Add(v0, v0) + e.y.Set(ty) + return e +} + +func (e *gfP12) Invert(a *gfP12) *gfP12 { + // See "Implementing cryptographic pairings", M. Scott, section 3.2. + // ftp://136.206.11.249/pub/crypto/pairings.pdf + t1, t2 := &gfP6{}, &gfP6{} + + t1.Square(&a.x) + t2.Square(&a.y) + t1.MulTau(t1).Sub(t2, t1) + t2.Invert(t1) + + e.x.Neg(&a.x) + e.y.Set(&a.y) + e.MulScalar(e, t2) + return e +} diff --git a/crypto/bn256/cloudflare/gfp2.go b/crypto/bn256/cloudflare/gfp2.go new file mode 100644 index 000000000..90a89e8b4 --- /dev/null +++ b/crypto/bn256/cloudflare/gfp2.go @@ -0,0 +1,156 @@ +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. + +// gfP2 implements a field of size p² as a quadratic extension of the base field +// where i²=-1. +type gfP2 struct { + x, y gfP // value is xi+y. +} + +func gfP2Decode(in *gfP2) *gfP2 { + out := &gfP2{} + montDecode(&out.x, &in.x) + montDecode(&out.y, &in.y) + return out +} + +func (e *gfP2) String() string { + return "(" + e.x.String() + ", " + e.y.String() + ")" +} + +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 = gfP{0} + e.y = gfP{0} + return e +} + +func (e *gfP2) SetOne() *gfP2 { + e.x = gfP{0} + e.y = *newGFp(1) + return e +} + +func (e *gfP2) IsZero() bool { + zero := gfP{0} + return e.x == zero && e.y == zero +} + +func (e *gfP2) IsOne() bool { + zero, one := gfP{0}, *newGFp(1) + return e.x == zero && e.y == one +} + +func (e *gfP2) Conjugate(a *gfP2) *gfP2 { + e.y.Set(&a.y) + gfpNeg(&e.x, &a.x) + return e +} + +func (e *gfP2) Neg(a *gfP2) *gfP2 { + gfpNeg(&e.x, &a.x) + gfpNeg(&e.y, &a.y) + return e +} + +func (e *gfP2) Add(a, b *gfP2) *gfP2 { + gfpAdd(&e.x, &a.x, &b.x) + gfpAdd(&e.y, &a.y, &b.y) + return e +} + +func (e *gfP2) Sub(a, b *gfP2) *gfP2 { + gfpSub(&e.x, &a.x, &b.x) + gfpSub(&e.y, &a.y, &b.y) + return e +} + +// See "Multiplication and Squaring in Pairing-Friendly Fields", +// http://eprint.iacr.org/2006/471.pdf +func (e *gfP2) Mul(a, b *gfP2) *gfP2 { + tx, t := &gfP{}, &gfP{} + gfpMul(tx, &a.x, &b.y) + gfpMul(t, &b.x, &a.y) + gfpAdd(tx, tx, t) + + ty := &gfP{} + gfpMul(ty, &a.y, &b.y) + gfpMul(t, &a.x, &b.x) + gfpSub(ty, ty, t) + + e.x.Set(tx) + e.y.Set(ty) + return e +} + +func (e *gfP2) MulScalar(a *gfP2, b *gfP) *gfP2 { + gfpMul(&e.x, &a.x, b) + gfpMul(&e.y, &a.y, b) + return e +} + +// MulXi sets e=ξa where ξ=i+9 and then returns e. +func (e *gfP2) MulXi(a *gfP2) *gfP2 { + // (xi+y)(i+9) = (9x+y)i+(9y-x) + tx := &gfP{} + gfpAdd(tx, &a.x, &a.x) + gfpAdd(tx, tx, tx) + gfpAdd(tx, tx, tx) + gfpAdd(tx, tx, &a.x) + + gfpAdd(tx, tx, &a.y) + + ty := &gfP{} + gfpAdd(ty, &a.y, &a.y) + gfpAdd(ty, ty, ty) + gfpAdd(ty, ty, ty) + gfpAdd(ty, ty, &a.y) + + gfpSub(ty, ty, &a.x) + + e.x.Set(tx) + e.y.Set(ty) + return e +} + +func (e *gfP2) Square(a *gfP2) *gfP2 { + // Complex squaring algorithm: + // (xi+y)² = (x+y)(y-x) + 2*i*x*y + tx, ty := &gfP{}, &gfP{} + gfpSub(tx, &a.y, &a.x) + gfpAdd(ty, &a.x, &a.y) + gfpMul(ty, tx, ty) + + gfpMul(tx, &a.x, &a.y) + gfpAdd(tx, tx, tx) + + e.x.Set(tx) + e.y.Set(ty) + return e +} + +func (e *gfP2) Invert(a *gfP2) *gfP2 { + // See "Implementing cryptographic pairings", M. Scott, section 3.2. + // ftp://136.206.11.249/pub/crypto/pairings.pdf + t1, t2 := &gfP{}, &gfP{} + gfpMul(t1, &a.x, &a.x) + gfpMul(t2, &a.y, &a.y) + gfpAdd(t1, t1, t2) + + inv := &gfP{} + inv.Invert(t1) + + gfpNeg(t1, &a.x) + + gfpMul(&e.x, t1, inv) + gfpMul(&e.y, &a.y, inv) + return e +} diff --git a/crypto/bn256/cloudflare/gfp6.go b/crypto/bn256/cloudflare/gfp6.go new file mode 100644 index 000000000..83d61b781 --- /dev/null +++ b/crypto/bn256/cloudflare/gfp6.go @@ -0,0 +1,213 @@ +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. + +// 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 (e *gfP6) String() string { + return "(" + e.x.String() + ", " + e.y.String() + ", " + e.z.String() + ")" +} + +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) 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) Neg(a *gfP6) *gfP6 { + e.x.Neg(&a.x) + e.y.Neg(&a.y) + e.z.Neg(&a.z) + return e +} + +func (e *gfP6) Frobenius(a *gfP6) *gfP6 { + e.x.Conjugate(&a.x) + e.y.Conjugate(&a.y) + e.z.Conjugate(&a.z) + + e.x.Mul(&e.x, xiTo2PMinus2Over3) + e.y.Mul(&e.y, xiToPMinus1Over3) + 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) FrobeniusP4(a *gfP6) *gfP6 { + e.x.MulScalar(&a.x, xiToPSquaredMinus1Over3) + e.y.MulScalar(&a.y, xiTo2PSquaredMinus2Over3) + 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) Mul(a, b *gfP6) *gfP6 { + // "Multiplication and Squaring on Pairing-Friendly Fields" + // Section 4, Karatsuba method. + // http://eprint.iacr.org/2006/471.pdf + v0 := (&gfP2{}).Mul(&a.z, &b.z) + v1 := (&gfP2{}).Mul(&a.y, &b.y) + v2 := (&gfP2{}).Mul(&a.x, &b.x) + + t0 := (&gfP2{}).Add(&a.x, &a.y) + t1 := (&gfP2{}).Add(&b.x, &b.y) + tz := (&gfP2{}).Mul(t0, t1) + tz.Sub(tz, v1).Sub(tz, v2).MulXi(tz).Add(tz, v0) + + t0.Add(&a.y, &a.z) + t1.Add(&b.y, &b.z) + ty := (&gfP2{}).Mul(t0, t1) + t0.MulXi(v2) + ty.Sub(ty, v0).Sub(ty, v1).Add(ty, t0) + + t0.Add(&a.x, &a.z) + t1.Add(&b.x, &b.z) + tx := (&gfP2{}).Mul(t0, t1) + tx.Sub(tx, v0).Add(tx, v1).Sub(tx, v2) + + e.x.Set(tx) + e.y.Set(ty) + e.z.Set(tz) + return e +} + +func (e *gfP6) MulScalar(a *gfP6, b *gfP2) *gfP6 { + e.x.Mul(&a.x, b) + e.y.Mul(&a.y, b) + e.z.Mul(&a.z, b) + return e +} + +func (e *gfP6) MulGFP(a *gfP6, b *gfP) *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) *gfP6 { + tz := (&gfP2{}).MulXi(&a.x) + ty := (&gfP2{}).Set(&a.y) + + e.y.Set(&a.z) + e.x.Set(ty) + e.z.Set(tz) + return e +} + +func (e *gfP6) Square(a *gfP6) *gfP6 { + v0 := (&gfP2{}).Square(&a.z) + v1 := (&gfP2{}).Square(&a.y) + v2 := (&gfP2{}).Square(&a.x) + + c0 := (&gfP2{}).Add(&a.x, &a.y) + c0.Square(c0).Sub(c0, v1).Sub(c0, v2).MulXi(c0).Add(c0, v0) + + c1 := (&gfP2{}).Add(&a.y, &a.z) + c1.Square(c1).Sub(c1, v0).Sub(c1, v1) + xiV2 := (&gfP2{}).MulXi(v2) + c1.Add(c1, xiV2) + + c2 := (&gfP2{}).Add(&a.x, &a.z) + c2.Square(c2).Sub(c2, v0).Add(c2, v1).Sub(c2, v2) + + e.x.Set(c2) + e.y.Set(c1) + e.z.Set(c0) + return e +} + +func (e *gfP6) Invert(a *gfP6) *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 := (&gfP2{}).Mul(&a.x, &a.y) + t1.MulXi(t1) + + A := (&gfP2{}).Square(&a.z) + A.Sub(A, t1) + + B := (&gfP2{}).Square(&a.x) + B.MulXi(B) + t1.Mul(&a.y, &a.z) + B.Sub(B, t1) + + C := (&gfP2{}).Square(&a.y) + t1.Mul(&a.x, &a.z) + C.Sub(C, t1) + + F := (&gfP2{}).Mul(C, &a.y) + F.MulXi(F) + t1.Mul(A, &a.z) + F.Add(F, t1) + t1.Mul(B, &a.x).MulXi(t1) + F.Add(F, t1) + + F.Invert(F) + + e.x.Mul(C, F) + e.y.Mul(B, F) + e.z.Mul(A, F) + return e +} diff --git a/crypto/bn256/cloudflare/gfp_amd64.go b/crypto/bn256/cloudflare/gfp_amd64.go new file mode 100644 index 000000000..ac4f1a9c6 --- /dev/null +++ b/crypto/bn256/cloudflare/gfp_amd64.go @@ -0,0 +1,15 @@ +// +build amd64,!appengine,!gccgo + +package bn256 + +// go:noescape +func gfpNeg(c, a *gfP) + +//go:noescape +func gfpAdd(c, a, b *gfP) + +//go:noescape +func gfpSub(c, a, b *gfP) + +//go:noescape +func gfpMul(c, a, b *gfP) diff --git a/crypto/bn256/cloudflare/gfp_amd64.s b/crypto/bn256/cloudflare/gfp_amd64.s new file mode 100644 index 000000000..2d0176f2e --- /dev/null +++ b/crypto/bn256/cloudflare/gfp_amd64.s @@ -0,0 +1,97 @@ +// +build amd64,!appengine,!gccgo + +#include "gfp.h" +#include "mul.h" +#include "mul_bmi2.h" + +TEXT ·gfpNeg(SB),0,$0-16 + MOVQ ·p2+0(SB), R8 + MOVQ ·p2+8(SB), R9 + MOVQ ·p2+16(SB), R10 + MOVQ ·p2+24(SB), R11 + + MOVQ a+8(FP), DI + SUBQ 0(DI), R8 + SBBQ 8(DI), R9 + SBBQ 16(DI), R10 + SBBQ 24(DI), R11 + + MOVQ $0, AX + gfpCarry(R8,R9,R10,R11,AX, R12,R13,R14,R15,BX) + + MOVQ c+0(FP), DI + storeBlock(R8,R9,R10,R11, 0(DI)) + RET + +TEXT ·gfpAdd(SB),0,$0-24 + MOVQ a+8(FP), DI + MOVQ b+16(FP), SI + + loadBlock(0(DI), R8,R9,R10,R11) + MOVQ $0, R12 + + ADDQ 0(SI), R8 + ADCQ 8(SI), R9 + ADCQ 16(SI), R10 + ADCQ 24(SI), R11 + ADCQ $0, R12 + + gfpCarry(R8,R9,R10,R11,R12, R13,R14,R15,AX,BX) + + MOVQ c+0(FP), DI + storeBlock(R8,R9,R10,R11, 0(DI)) + RET + +TEXT ·gfpSub(SB),0,$0-24 + MOVQ a+8(FP), DI + MOVQ b+16(FP), SI + + loadBlock(0(DI), R8,R9,R10,R11) + + MOVQ ·p2+0(SB), R12 + MOVQ ·p2+8(SB), R13 + MOVQ ·p2+16(SB), R14 + MOVQ ·p2+24(SB), R15 + MOVQ $0, AX + + SUBQ 0(SI), R8 + SBBQ 8(SI), R9 + SBBQ 16(SI), R10 + SBBQ 24(SI), R11 + + CMOVQCC AX, R12 + CMOVQCC AX, R13 + CMOVQCC AX, R14 + CMOVQCC AX, R15 + + ADDQ R12, R8 + ADCQ R13, R9 + ADCQ R14, R10 + ADCQ R15, R11 + + MOVQ c+0(FP), DI + storeBlock(R8,R9,R10,R11, 0(DI)) + RET + +TEXT ·gfpMul(SB),0,$160-24 + MOVQ a+8(FP), DI + MOVQ b+16(FP), SI + + // Jump to a slightly different implementation if MULX isn't supported. + CMPB runtime·support_bmi2(SB), $0 + JE nobmi2Mul + + mulBMI2(0(DI),8(DI),16(DI),24(DI), 0(SI)) + storeBlock( R8, R9,R10,R11, 0(SP)) + storeBlock(R12,R13,R14,R15, 32(SP)) + gfpReduceBMI2() + JMP end + +nobmi2Mul: + mul(0(DI),8(DI),16(DI),24(DI), 0(SI), 0(SP)) + gfpReduce(0(SP)) + +end: + MOVQ c+0(FP), DI + storeBlock(R12,R13,R14,R15, 0(DI)) + RET diff --git a/crypto/bn256/cloudflare/gfp_pure.go b/crypto/bn256/cloudflare/gfp_pure.go new file mode 100644 index 000000000..8fa5d3053 --- /dev/null +++ b/crypto/bn256/cloudflare/gfp_pure.go @@ -0,0 +1,19 @@ +// +build !amd64 appengine gccgo + +package bn256 + +func gfpNeg(c, a *gfP) { + panic("unsupported architecture") +} + +func gfpAdd(c, a, b *gfP) { + panic("unsupported architecture") +} + +func gfpSub(c, a, b *gfP) { + panic("unsupported architecture") +} + +func gfpMul(c, a, b *gfP) { + panic("unsupported architecture") +} diff --git a/crypto/bn256/cloudflare/gfp_test.go b/crypto/bn256/cloudflare/gfp_test.go new file mode 100644 index 000000000..aff5e0531 --- /dev/null +++ b/crypto/bn256/cloudflare/gfp_test.go @@ -0,0 +1,62 @@ +// +build amd64,!appengine,!gccgo + +package bn256 + +import ( + "testing" +) + +// Tests that negation works the same way on both assembly-optimized and pure Go +// implementation. +func TestGFpNeg(t *testing.T) { + n := &gfP{0x0123456789abcdef, 0xfedcba9876543210, 0xdeadbeefdeadbeef, 0xfeebdaedfeebdaed} + w := &gfP{0xfedcba9876543211, 0x0123456789abcdef, 0x2152411021524110, 0x0114251201142512} + h := &gfP{} + + gfpNeg(h, n) + if *h != *w { + t.Errorf("negation mismatch: have %#x, want %#x", *h, *w) + } +} + +// Tests that addition works the same way on both assembly-optimized and pure Go +// implementation. +func TestGFpAdd(t *testing.T) { + a := &gfP{0x0123456789abcdef, 0xfedcba9876543210, 0xdeadbeefdeadbeef, 0xfeebdaedfeebdaed} + b := &gfP{0xfedcba9876543210, 0x0123456789abcdef, 0xfeebdaedfeebdaed, 0xdeadbeefdeadbeef} + w := &gfP{0xc3df73e9278302b8, 0x687e956e978e3572, 0x254954275c18417f, 0xad354b6afc67f9b4} + h := &gfP{} + + gfpAdd(h, a, b) + if *h != *w { + t.Errorf("addition mismatch: have %#x, want %#x", *h, *w) + } +} + +// Tests that subtraction works the same way on both assembly-optimized and pure Go +// implementation. +func TestGFpSub(t *testing.T) { + a := &gfP{0x0123456789abcdef, 0xfedcba9876543210, 0xdeadbeefdeadbeef, 0xfeebdaedfeebdaed} + b := &gfP{0xfedcba9876543210, 0x0123456789abcdef, 0xfeebdaedfeebdaed, 0xdeadbeefdeadbeef} + w := &gfP{0x02468acf13579bdf, 0xfdb97530eca86420, 0xdfc1e401dfc1e402, 0x203e1bfe203e1bfd} + h := &gfP{} + + gfpSub(h, a, b) + if *h != *w { + t.Errorf("subtraction mismatch: have %#x, want %#x", *h, *w) + } +} + +// Tests that multiplication works the same way on both assembly-optimized and pure Go +// implementation. +func TestGFpMul(t *testing.T) { + a := &gfP{0x0123456789abcdef, 0xfedcba9876543210, 0xdeadbeefdeadbeef, 0xfeebdaedfeebdaed} + b := &gfP{0xfedcba9876543210, 0x0123456789abcdef, 0xfeebdaedfeebdaed, 0xdeadbeefdeadbeef} + w := &gfP{0xcbcbd377f7ad22d3, 0x3b89ba5d849379bf, 0x87b61627bd38b6d2, 0xc44052a2a0e654b2} + h := &gfP{} + + gfpMul(h, a, b) + if *h != *w { + t.Errorf("multiplication mismatch: have %#x, want %#x", *h, *w) + } +} diff --git a/crypto/bn256/cloudflare/main_test.go b/crypto/bn256/cloudflare/main_test.go new file mode 100644 index 000000000..f0d59a404 --- /dev/null +++ b/crypto/bn256/cloudflare/main_test.go @@ -0,0 +1,73 @@ +// +build amd64,!appengine,!gccgo + +package bn256 + +import ( + "testing" + + "crypto/rand" +) + +func TestRandomG2Marshal(t *testing.T) { + for i := 0; i < 10; i++ { + n, g2, err := RandomG2(rand.Reader) + if err != nil { + t.Error(err) + continue + } + t.Logf("%d: %x\n", n, g2.Marshal()) + } +} + +func TestPairings(t *testing.T) { + a1 := new(G1).ScalarBaseMult(bigFromBase10("1")) + a2 := new(G1).ScalarBaseMult(bigFromBase10("2")) + a37 := new(G1).ScalarBaseMult(bigFromBase10("37")) + an1 := new(G1).ScalarBaseMult(bigFromBase10("21888242871839275222246405745257275088548364400416034343698204186575808495616")) + + b0 := new(G2).ScalarBaseMult(bigFromBase10("0")) + b1 := new(G2).ScalarBaseMult(bigFromBase10("1")) + b2 := new(G2).ScalarBaseMult(bigFromBase10("2")) + b27 := new(G2).ScalarBaseMult(bigFromBase10("27")) + b999 := new(G2).ScalarBaseMult(bigFromBase10("999")) + bn1 := new(G2).ScalarBaseMult(bigFromBase10("21888242871839275222246405745257275088548364400416034343698204186575808495616")) + + p1 := Pair(a1, b1) + pn1 := Pair(a1, bn1) + np1 := Pair(an1, b1) + if pn1.String() != np1.String() { + t.Error("Pairing mismatch: e(a, -b) != e(-a, b)") + } + if !PairingCheck([]*G1{a1, an1}, []*G2{b1, b1}) { + t.Error("MultiAte check gave false negative!") + } + p0 := new(GT).Add(p1, pn1) + p0_2 := Pair(a1, b0) + if p0.String() != p0_2.String() { + t.Error("Pairing mismatch: e(a, b) * e(a, -b) != 1") + } + p0_3 := new(GT).ScalarMult(p1, bigFromBase10("21888242871839275222246405745257275088548364400416034343698204186575808495617")) + if p0.String() != p0_3.String() { + t.Error("Pairing mismatch: e(a, b) has wrong order") + } + p2 := Pair(a2, b1) + p2_2 := Pair(a1, b2) + p2_3 := new(GT).ScalarMult(p1, bigFromBase10("2")) + if p2.String() != p2_2.String() { + t.Error("Pairing mismatch: e(a, b * 2) != e(a * 2, b)") + } + if p2.String() != p2_3.String() { + t.Error("Pairing mismatch: e(a, b * 2) != e(a, b) ** 2") + } + if p2.String() == p1.String() { + t.Error("Pairing is degenerate!") + } + if PairingCheck([]*G1{a1, a1}, []*G2{b1, b1}) { + t.Error("MultiAte check gave false positive!") + } + p999 := Pair(a37, b27) + p999_2 := Pair(a1, b999) + if p999.String() != p999_2.String() { + t.Error("Pairing mismatch: e(a * 37, b * 27) != e(a, b * 999)") + } +} diff --git a/crypto/bn256/cloudflare/mul.h b/crypto/bn256/cloudflare/mul.h new file mode 100644 index 000000000..bab5da831 --- /dev/null +++ b/crypto/bn256/cloudflare/mul.h @@ -0,0 +1,181 @@ +#define mul(a0,a1,a2,a3, rb, stack) \ + MOVQ a0, AX \ + MULQ 0+rb \ + MOVQ AX, R8 \ + MOVQ DX, R9 \ + MOVQ a0, AX \ + MULQ 8+rb \ + ADDQ AX, R9 \ + ADCQ $0, DX \ + MOVQ DX, R10 \ + MOVQ a0, AX \ + MULQ 16+rb \ + ADDQ AX, R10 \ + ADCQ $0, DX \ + MOVQ DX, R11 \ + MOVQ a0, AX \ + MULQ 24+rb \ + ADDQ AX, R11 \ + ADCQ $0, DX \ + MOVQ DX, R12 \ + \ + storeBlock(R8,R9,R10,R11, 0+stack) \ + MOVQ R12, 32+stack \ + \ + MOVQ a1, AX \ + MULQ 0+rb \ + MOVQ AX, R8 \ + MOVQ DX, R9 \ + MOVQ a1, AX \ + MULQ 8+rb \ + ADDQ AX, R9 \ + ADCQ $0, DX \ + MOVQ DX, R10 \ + MOVQ a1, AX \ + MULQ 16+rb \ + ADDQ AX, R10 \ + ADCQ $0, DX \ + MOVQ DX, R11 \ + MOVQ a1, AX \ + MULQ 24+rb \ + ADDQ AX, R11 \ + ADCQ $0, DX \ + MOVQ DX, R12 \ + \ + ADDQ 8+stack, R8 \ + ADCQ 16+stack, R9 \ + ADCQ 24+stack, R10 \ + ADCQ 32+stack, R11 \ + ADCQ $0, R12 \ + storeBlock(R8,R9,R10,R11, 8+stack) \ + MOVQ R12, 40+stack \ + \ + MOVQ a2, AX \ + MULQ 0+rb \ + MOVQ AX, R8 \ + MOVQ DX, R9 \ + MOVQ a2, AX \ + MULQ 8+rb \ + ADDQ AX, R9 \ + ADCQ $0, DX \ + MOVQ DX, R10 \ + MOVQ a2, AX \ + MULQ 16+rb \ + ADDQ AX, R10 \ + ADCQ $0, DX \ + MOVQ DX, R11 \ + MOVQ a2, AX \ + MULQ 24+rb \ + ADDQ AX, R11 \ + ADCQ $0, DX \ + MOVQ DX, R12 \ + \ + ADDQ 16+stack, R8 \ + ADCQ 24+stack, R9 \ + ADCQ 32+stack, R10 \ + ADCQ 40+stack, R11 \ + ADCQ $0, R12 \ + storeBlock(R8,R9,R10,R11, 16+stack) \ + MOVQ R12, 48+stack \ + \ + MOVQ a3, AX \ + MULQ 0+rb \ + MOVQ AX, R8 \ + MOVQ DX, R9 \ + MOVQ a3, AX \ + MULQ 8+rb \ + ADDQ AX, R9 \ + ADCQ $0, DX \ + MOVQ DX, R10 \ + MOVQ a3, AX \ + MULQ 16+rb \ + ADDQ AX, R10 \ + ADCQ $0, DX \ + MOVQ DX, R11 \ + MOVQ a3, AX \ + MULQ 24+rb \ + ADDQ AX, R11 \ + ADCQ $0, DX \ + MOVQ DX, R12 \ + \ + ADDQ 24+stack, R8 \ + ADCQ 32+stack, R9 \ + ADCQ 40+stack, R10 \ + ADCQ 48+stack, R11 \ + ADCQ $0, R12 \ + storeBlock(R8,R9,R10,R11, 24+stack) \ + MOVQ R12, 56+stack + +#define gfpReduce(stack) \ + \ // m = (T * N') mod R, store m in R8:R9:R10:R11 + MOVQ ·np+0(SB), AX \ + MULQ 0+stack \ + MOVQ AX, R8 \ + MOVQ DX, R9 \ + MOVQ ·np+0(SB), AX \ + MULQ 8+stack \ + ADDQ AX, R9 \ + ADCQ $0, DX \ + MOVQ DX, R10 \ + MOVQ ·np+0(SB), AX \ + MULQ 16+stack \ + ADDQ AX, R10 \ + ADCQ $0, DX \ + MOVQ DX, R11 \ + MOVQ ·np+0(SB), AX \ + MULQ 24+stack \ + ADDQ AX, R11 \ + \ + MOVQ ·np+8(SB), AX \ + MULQ 0+stack \ + MOVQ AX, R12 \ + MOVQ DX, R13 \ + MOVQ ·np+8(SB), AX \ + MULQ 8+stack \ + ADDQ AX, R13 \ + ADCQ $0, DX \ + MOVQ DX, R14 \ + MOVQ ·np+8(SB), AX \ + MULQ 16+stack \ + ADDQ AX, R14 \ + \ + ADDQ R12, R9 \ + ADCQ R13, R10 \ + ADCQ R14, R11 \ + \ + MOVQ ·np+16(SB), AX \ + MULQ 0+stack \ + MOVQ AX, R12 \ + MOVQ DX, R13 \ + MOVQ ·np+16(SB), AX \ + MULQ 8+stack \ + ADDQ AX, R13 \ + \ + ADDQ R12, R10 \ + ADCQ R13, R11 \ + \ + MOVQ ·np+24(SB), AX \ + MULQ 0+stack \ + ADDQ AX, R11 \ + \ + storeBlock(R8,R9,R10,R11, 64+stack) \ + \ + \ // m * N + mul(·p2+0(SB),·p2+8(SB),·p2+16(SB),·p2+24(SB), 64+stack, 96+stack) \ + \ + \ // Add the 512-bit intermediate to m*N + loadBlock(96+stack, R8,R9,R10,R11) \ + loadBlock(128+stack, R12,R13,R14,R15) \ + \ + MOVQ $0, AX \ + ADDQ 0+stack, R8 \ + ADCQ 8+stack, R9 \ + ADCQ 16+stack, R10 \ + ADCQ 24+stack, R11 \ + ADCQ 32+stack, R12 \ + ADCQ 40+stack, R13 \ + ADCQ 48+stack, R14 \ + ADCQ 56+stack, R15 \ + ADCQ $0, AX \ + \ + gfpCarry(R12,R13,R14,R15,AX, R8,R9,R10,R11,BX) diff --git a/crypto/bn256/cloudflare/mul_bmi2.h b/crypto/bn256/cloudflare/mul_bmi2.h new file mode 100644 index 000000000..71ad0499a --- /dev/null +++ b/crypto/bn256/cloudflare/mul_bmi2.h @@ -0,0 +1,112 @@ +#define mulBMI2(a0,a1,a2,a3, rb) \ + MOVQ a0, DX \ + MOVQ $0, R13 \ + MULXQ 0+rb, R8, R9 \ + MULXQ 8+rb, AX, R10 \ + ADDQ AX, R9 \ + MULXQ 16+rb, AX, R11 \ + ADCQ AX, R10 \ + MULXQ 24+rb, AX, R12 \ + ADCQ AX, R11 \ + ADCQ $0, R12 \ + ADCQ $0, R13 \ + \ + MOVQ a1, DX \ + MOVQ $0, R14 \ + MULXQ 0+rb, AX, BX \ + ADDQ AX, R9 \ + ADCQ BX, R10 \ + MULXQ 16+rb, AX, BX \ + ADCQ AX, R11 \ + ADCQ BX, R12 \ + ADCQ $0, R13 \ + MULXQ 8+rb, AX, BX \ + ADDQ AX, R10 \ + ADCQ BX, R11 \ + MULXQ 24+rb, AX, BX \ + ADCQ AX, R12 \ + ADCQ BX, R13 \ + ADCQ $0, R14 \ + \ + MOVQ a2, DX \ + MOVQ $0, R15 \ + MULXQ 0+rb, AX, BX \ + ADDQ AX, R10 \ + ADCQ BX, R11 \ + MULXQ 16+rb, AX, BX \ + ADCQ AX, R12 \ + ADCQ BX, R13 \ + ADCQ $0, R14 \ + MULXQ 8+rb, AX, BX \ + ADDQ AX, R11 \ + ADCQ BX, R12 \ + MULXQ 24+rb, AX, BX \ + ADCQ AX, R13 \ + ADCQ BX, R14 \ + ADCQ $0, R15 \ + \ + MOVQ a3, DX \ + MULXQ 0+rb, AX, BX \ + ADDQ AX, R11 \ + ADCQ BX, R12 \ + MULXQ 16+rb, AX, BX \ + ADCQ AX, R13 \ + ADCQ BX, R14 \ + ADCQ $0, R15 \ + MULXQ 8+rb, AX, BX \ + ADDQ AX, R12 \ + ADCQ BX, R13 \ + MULXQ 24+rb, AX, BX \ + ADCQ AX, R14 \ + ADCQ BX, R15 + +#define gfpReduceBMI2() \ + \ // m = (T * N') mod R, store m in R8:R9:R10:R11 + MOVQ ·np+0(SB), DX \ + MULXQ 0(SP), R8, R9 \ + MULXQ 8(SP), AX, R10 \ + ADDQ AX, R9 \ + MULXQ 16(SP), AX, R11 \ + ADCQ AX, R10 \ + MULXQ 24(SP), AX, BX \ + ADCQ AX, R11 \ + \ + MOVQ ·np+8(SB), DX \ + MULXQ 0(SP), AX, BX \ + ADDQ AX, R9 \ + ADCQ BX, R10 \ + MULXQ 16(SP), AX, BX \ + ADCQ AX, R11 \ + MULXQ 8(SP), AX, BX \ + ADDQ AX, R10 \ + ADCQ BX, R11 \ + \ + MOVQ ·np+16(SB), DX \ + MULXQ 0(SP), AX, BX \ + ADDQ AX, R10 \ + ADCQ BX, R11 \ + MULXQ 8(SP), AX, BX \ + ADDQ AX, R11 \ + \ + MOVQ ·np+24(SB), DX \ + MULXQ 0(SP), AX, BX \ + ADDQ AX, R11 \ + \ + storeBlock(R8,R9,R10,R11, 64(SP)) \ + \ + \ // m * N + mulBMI2(·p2+0(SB),·p2+8(SB),·p2+16(SB),·p2+24(SB), 64(SP)) \ + \ + \ // Add the 512-bit intermediate to m*N + MOVQ $0, AX \ + ADDQ 0(SP), R8 \ + ADCQ 8(SP), R9 \ + ADCQ 16(SP), R10 \ + ADCQ 24(SP), R11 \ + ADCQ 32(SP), R12 \ + ADCQ 40(SP), R13 \ + ADCQ 48(SP), R14 \ + ADCQ 56(SP), R15 \ + ADCQ $0, AX \ + \ + gfpCarry(R12,R13,R14,R15,AX, R8,R9,R10,R11,BX) diff --git a/crypto/bn256/cloudflare/optate.go b/crypto/bn256/cloudflare/optate.go new file mode 100644 index 000000000..b71e50e3a --- /dev/null +++ b/crypto/bn256/cloudflare/optate.go @@ -0,0 +1,271 @@ +package bn256 + +func lineFunctionAdd(r, p *twistPoint, q *curvePoint, r2 *gfP2) (a, b, c *gfP2, rOut *twistPoint) { + // See the mixed addition algorithm from "Faster Computation of the + // Tate Pairing", http://arxiv.org/pdf/0904.0854v3.pdf + B := (&gfP2{}).Mul(&p.x, &r.t) + + D := (&gfP2{}).Add(&p.y, &r.z) + D.Square(D).Sub(D, r2).Sub(D, &r.t).Mul(D, &r.t) + + H := (&gfP2{}).Sub(B, &r.x) + I := (&gfP2{}).Square(H) + + E := (&gfP2{}).Add(I, I) + E.Add(E, E) + + J := (&gfP2{}).Mul(H, E) + + L1 := (&gfP2{}).Sub(D, &r.y) + L1.Sub(L1, &r.y) + + V := (&gfP2{}).Mul(&r.x, E) + + rOut = &twistPoint{} + rOut.x.Square(L1).Sub(&rOut.x, J).Sub(&rOut.x, V).Sub(&rOut.x, V) + + rOut.z.Add(&r.z, H).Square(&rOut.z).Sub(&rOut.z, &r.t).Sub(&rOut.z, I) + + t := (&gfP2{}).Sub(V, &rOut.x) + t.Mul(t, L1) + t2 := (&gfP2{}).Mul(&r.y, J) + t2.Add(t2, t2) + rOut.y.Sub(t, t2) + + rOut.t.Square(&rOut.z) + + t.Add(&p.y, &rOut.z).Square(t).Sub(t, r2).Sub(t, &rOut.t) + + t2.Mul(L1, &p.x) + t2.Add(t2, t2) + a = (&gfP2{}).Sub(t2, t) + + c = (&gfP2{}).MulScalar(&rOut.z, &q.y) + c.Add(c, c) + + b = (&gfP2{}).Neg(L1) + b.MulScalar(b, &q.x).Add(b, b) + + return +} + +func lineFunctionDouble(r *twistPoint, q *curvePoint) (a, b, c *gfP2, rOut *twistPoint) { + // See the doubling algorithm for a=0 from "Faster Computation of the + // Tate Pairing", http://arxiv.org/pdf/0904.0854v3.pdf + A := (&gfP2{}).Square(&r.x) + B := (&gfP2{}).Square(&r.y) + C := (&gfP2{}).Square(B) + + D := (&gfP2{}).Add(&r.x, B) + D.Square(D).Sub(D, A).Sub(D, C).Add(D, D) + + E := (&gfP2{}).Add(A, A) + E.Add(E, A) + + G := (&gfP2{}).Square(E) + + rOut = &twistPoint{} + rOut.x.Sub(G, D).Sub(&rOut.x, D) + + rOut.z.Add(&r.y, &r.z).Square(&rOut.z).Sub(&rOut.z, B).Sub(&rOut.z, &r.t) + + rOut.y.Sub(D, &rOut.x).Mul(&rOut.y, E) + t := (&gfP2{}).Add(C, C) + t.Add(t, t).Add(t, t) + rOut.y.Sub(&rOut.y, t) + + rOut.t.Square(&rOut.z) + + t.Mul(E, &r.t).Add(t, t) + b = (&gfP2{}).Neg(t) + b.MulScalar(b, &q.x) + + a = (&gfP2{}).Add(&r.x, E) + a.Square(a).Sub(a, A).Sub(a, G) + t.Add(B, B).Add(t, t) + a.Sub(a, t) + + c = (&gfP2{}).Mul(&rOut.z, &r.t) + c.Add(c, c).MulScalar(c, &q.y) + + return +} + +func mulLine(ret *gfP12, a, b, c *gfP2) { + a2 := &gfP6{} + a2.y.Set(a) + a2.z.Set(b) + a2.Mul(a2, &ret.x) + t3 := (&gfP6{}).MulScalar(&ret.y, c) + + t := (&gfP2{}).Add(b, c) + t2 := &gfP6{} + t2.y.Set(a) + t2.z.Set(t) + ret.x.Add(&ret.x, &ret.y) + + ret.y.Set(t3) + + ret.x.Mul(&ret.x, t2).Sub(&ret.x, a2).Sub(&ret.x, &ret.y) + a2.MulTau(a2) + ret.y.Add(&ret.y, a2) +} + +// sixuPlus2NAF is 6u+2 in non-adjacent form. +var sixuPlus2NAF = []int8{0, 0, 0, 1, 0, 1, 0, -1, 0, 0, 1, -1, 0, 0, 1, 0, + 0, 1, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 1, 1, + 1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1, + 1, 0, 0, -1, 0, 0, 0, 1, 1, 0, -1, 0, 0, 1, 0, 1, 1} + +// miller implements the Miller loop for calculating the Optimal Ate pairing. +// See algorithm 1 from http://cryptojedi.org/papers/dclxvi-20100714.pdf +func miller(q *twistPoint, p *curvePoint) *gfP12 { + ret := (&gfP12{}).SetOne() + + aAffine := &twistPoint{} + aAffine.Set(q) + aAffine.MakeAffine() + + bAffine := &curvePoint{} + bAffine.Set(p) + bAffine.MakeAffine() + + minusA := &twistPoint{} + minusA.Neg(aAffine) + + r := &twistPoint{} + r.Set(aAffine) + + r2 := (&gfP2{}).Square(&aAffine.y) + + for i := len(sixuPlus2NAF) - 1; i > 0; i-- { + a, b, c, newR := lineFunctionDouble(r, bAffine) + if i != len(sixuPlus2NAF)-1 { + ret.Square(ret) + } + + mulLine(ret, a, b, c) + r = newR + + switch sixuPlus2NAF[i-1] { + case 1: + a, b, c, newR = lineFunctionAdd(r, aAffine, bAffine, r2) + case -1: + a, b, c, newR = lineFunctionAdd(r, minusA, bAffine, r2) + default: + continue + } + + mulLine(ret, a, b, c) + r = newR + } + + // In order to calculate Q1 we have to convert q from the sextic twist + // to the full GF(p^12) group, apply the Frobenius there, and convert + // back. + // + // The twist isomorphism is (x', y') -> (xω², yω³). If we consider just + // x for a moment, then after applying the Frobenius, we have x̄ω^(2p) + // where x̄ is the conjugate of x. If we are going to apply the inverse + // isomorphism we need a value with a single coefficient of ω² so we + // rewrite this as x̄ω^(2p-2)ω². ξ⁶ = ω and, due to the construction of + // p, 2p-2 is a multiple of six. Therefore we can rewrite as + // x̄ξ^((p-1)/3)ω² and applying the inverse isomorphism eliminates the + // ω². + // + // A similar argument can be made for the y value. + + q1 := &twistPoint{} + q1.x.Conjugate(&aAffine.x).Mul(&q1.x, xiToPMinus1Over3) + q1.y.Conjugate(&aAffine.y).Mul(&q1.y, xiToPMinus1Over2) + q1.z.SetOne() + q1.t.SetOne() + + // For Q2 we are applying the p² Frobenius. The two conjugations cancel + // out and we are left only with the factors from the isomorphism. In + // the case of x, we end up with a pure number which is why + // xiToPSquaredMinus1Over3 is ∈ GF(p). With y we get a factor of -1. We + // ignore this to end up with -Q2. + + minusQ2 := &twistPoint{} + minusQ2.x.MulScalar(&aAffine.x, xiToPSquaredMinus1Over3) + minusQ2.y.Set(&aAffine.y) + minusQ2.z.SetOne() + minusQ2.t.SetOne() + + r2.Square(&q1.y) + a, b, c, newR := lineFunctionAdd(r, q1, bAffine, r2) + mulLine(ret, a, b, c) + r = newR + + r2.Square(&minusQ2.y) + a, b, c, newR = lineFunctionAdd(r, minusQ2, bAffine, r2) + mulLine(ret, a, b, c) + r = newR + + return ret +} + +// finalExponentiation computes the (p¹²-1)/Order-th power of an element of +// GF(p¹²) to obtain an element of GT (steps 13-15 of algorithm 1 from +// http://cryptojedi.org/papers/dclxvi-20100714.pdf) +func finalExponentiation(in *gfP12) *gfP12 { + t1 := &gfP12{} + + // This is the p^6-Frobenius + t1.x.Neg(&in.x) + t1.y.Set(&in.y) + + inv := &gfP12{} + inv.Invert(in) + t1.Mul(t1, inv) + + t2 := (&gfP12{}).FrobeniusP2(t1) + t1.Mul(t1, t2) + + fp := (&gfP12{}).Frobenius(t1) + fp2 := (&gfP12{}).FrobeniusP2(t1) + fp3 := (&gfP12{}).Frobenius(fp2) + + fu := (&gfP12{}).Exp(t1, u) + fu2 := (&gfP12{}).Exp(fu, u) + fu3 := (&gfP12{}).Exp(fu2, u) + + y3 := (&gfP12{}).Frobenius(fu) + fu2p := (&gfP12{}).Frobenius(fu2) + fu3p := (&gfP12{}).Frobenius(fu3) + y2 := (&gfP12{}).FrobeniusP2(fu2) + + y0 := &gfP12{} + y0.Mul(fp, fp2).Mul(y0, fp3) + + y1 := (&gfP12{}).Conjugate(t1) + y5 := (&gfP12{}).Conjugate(fu2) + y3.Conjugate(y3) + y4 := (&gfP12{}).Mul(fu, fu2p) + y4.Conjugate(y4) + + y6 := (&gfP12{}).Mul(fu3, fu3p) + y6.Conjugate(y6) + + t0 := (&gfP12{}).Square(y6) + t0.Mul(t0, y4).Mul(t0, y5) + t1.Mul(y3, y5).Mul(t1, t0) + t0.Mul(t0, y2) + t1.Square(t1).Mul(t1, t0).Square(t1) + t0.Mul(t1, y1) + t1.Mul(t1, y0) + t0.Square(t0).Mul(t0, t1) + + return t0 +} + +func optimalAte(a *twistPoint, b *curvePoint) *gfP12 { + e := miller(a, b) + ret := finalExponentiation(e) + + if a.IsInfinity() || b.IsInfinity() { + ret.SetOne() + } + return ret +} diff --git a/crypto/bn256/cloudflare/twist.go b/crypto/bn256/cloudflare/twist.go new file mode 100644 index 000000000..0c2f80d4e --- /dev/null +++ b/crypto/bn256/cloudflare/twist.go @@ -0,0 +1,204 @@ +package bn256 + +import ( + "math/big" +) + +// twistPoint implements the elliptic curve y²=x³+3/ξ over GF(p²). Points are +// kept in Jacobian form and t=z² when valid. The group G₂ is the set of +// n-torsion points of this curve over GF(p²) (where n = Order) +type twistPoint struct { + x, y, z, t gfP2 +} + +var twistB = &gfP2{ + gfP{0x38e7ecccd1dcff67, 0x65f0b37d93ce0d3e, 0xd749d0dd22ac00aa, 0x0141b9ce4a688d4d}, + gfP{0x3bf938e377b802a8, 0x020b1b273633535d, 0x26b7edf049755260, 0x2514c6324384a86d}, +} + +// twistGen is the generator of group G₂. +var twistGen = &twistPoint{ + gfP2{ + gfP{0xafb4737da84c6140, 0x6043dd5a5802d8c4, 0x09e950fc52a02f86, 0x14fef0833aea7b6b}, + gfP{0x8e83b5d102bc2026, 0xdceb1935497b0172, 0xfbb8264797811adf, 0x19573841af96503b}, + }, + gfP2{ + gfP{0x64095b56c71856ee, 0xdc57f922327d3cbb, 0x55f935be33351076, 0x0da4a0e693fd6482}, + gfP{0x619dfa9d886be9f6, 0xfe7fd297f59e9b78, 0xff9e1a62231b7dfe, 0x28fd7eebae9e4206}, + }, + gfP2{*newGFp(0), *newGFp(1)}, + gfP2{*newGFp(0), *newGFp(1)}, +} + +func (c *twistPoint) String() string { + c.MakeAffine() + x, y := gfP2Decode(&c.x), gfP2Decode(&c.y) + return "(" + x.String() + ", " + y.String() + ")" +} + +func (c *twistPoint) Set(a *twistPoint) { + c.x.Set(&a.x) + c.y.Set(&a.y) + c.z.Set(&a.z) + c.t.Set(&a.t) +} + +// IsOnCurve returns true iff c is on the curve. +func (c *twistPoint) IsOnCurve() bool { + c.MakeAffine() + if c.IsInfinity() { + return true + } + + y2, x3 := &gfP2{}, &gfP2{} + y2.Square(&c.y) + x3.Square(&c.x).Mul(x3, &c.x).Add(x3, twistB) + + if *y2 != *x3 { + return false + } + cneg := &twistPoint{} + cneg.Mul(c, Order) + return cneg.z.IsZero() +} + +func (c *twistPoint) SetInfinity() { + c.x.SetZero() + c.y.SetOne() + c.z.SetZero() + c.t.SetZero() +} + +func (c *twistPoint) IsInfinity() bool { + return c.z.IsZero() +} + +func (c *twistPoint) Add(a, b *twistPoint) { + // For additional comments, see the same function in curve.go. + + if a.IsInfinity() { + c.Set(b) + return + } + if b.IsInfinity() { + c.Set(a) + return + } + + // See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/addition/add-2007-bl.op3 + z12 := (&gfP2{}).Square(&a.z) + z22 := (&gfP2{}).Square(&b.z) + u1 := (&gfP2{}).Mul(&a.x, z22) + u2 := (&gfP2{}).Mul(&b.x, z12) + + t := (&gfP2{}).Mul(&b.z, z22) + s1 := (&gfP2{}).Mul(&a.y, t) + + t.Mul(&a.z, z12) + s2 := (&gfP2{}).Mul(&b.y, t) + + h := (&gfP2{}).Sub(u2, u1) + xEqual := h.IsZero() + + t.Add(h, h) + i := (&gfP2{}).Square(t) + j := (&gfP2{}).Mul(h, i) + + t.Sub(s2, s1) + yEqual := t.IsZero() + if xEqual && yEqual { + c.Double(a) + return + } + r := (&gfP2{}).Add(t, t) + + v := (&gfP2{}).Mul(u1, i) + + t4 := (&gfP2{}).Square(r) + t.Add(v, v) + t6 := (&gfP2{}).Sub(t4, j) + c.x.Sub(t6, t) + + t.Sub(v, &c.x) // t7 + t4.Mul(s1, j) // t8 + t6.Add(t4, t4) // t9 + t4.Mul(r, t) // t10 + c.y.Sub(t4, t6) + + t.Add(&a.z, &b.z) // t11 + t4.Square(t) // t12 + t.Sub(t4, z12) // t13 + t4.Sub(t, z22) // t14 + c.z.Mul(t4, h) +} + +func (c *twistPoint) Double(a *twistPoint) { + // See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/doubling/dbl-2009-l.op3 + A := (&gfP2{}).Square(&a.x) + B := (&gfP2{}).Square(&a.y) + C := (&gfP2{}).Square(B) + + t := (&gfP2{}).Add(&a.x, B) + t2 := (&gfP2{}).Square(t) + t.Sub(t2, A) + t2.Sub(t, C) + d := (&gfP2{}).Add(t2, t2) + t.Add(A, A) + e := (&gfP2{}).Add(t, A) + f := (&gfP2{}).Square(e) + + t.Add(d, d) + c.x.Sub(f, t) + + t.Add(C, C) + t2.Add(t, t) + t.Add(t2, t2) + c.y.Sub(d, &c.x) + t2.Mul(e, &c.y) + c.y.Sub(t2, t) + + t.Mul(&a.y, &a.z) + c.z.Add(t, t) +} + +func (c *twistPoint) Mul(a *twistPoint, scalar *big.Int) { + sum, t := &twistPoint{}, &twistPoint{} + + for i := scalar.BitLen(); i >= 0; i-- { + t.Double(sum) + if scalar.Bit(i) != 0 { + sum.Add(t, a) + } else { + sum.Set(t) + } + } + + c.Set(sum) +} + +func (c *twistPoint) MakeAffine() { + if c.z.IsOne() { + return + } else if c.z.IsZero() { + c.x.SetZero() + c.y.SetOne() + c.t.SetZero() + return + } + + zInv := (&gfP2{}).Invert(&c.z) + t := (&gfP2{}).Mul(&c.y, zInv) + zInv2 := (&gfP2{}).Square(zInv) + c.y.Mul(t, zInv2) + t.Mul(&c.x, zInv2) + c.x.Set(t) + c.z.SetOne() + c.t.SetOne() +} + +func (c *twistPoint) Neg(a *twistPoint) { + c.x.Set(&a.x) + c.y.Neg(&a.y) + c.z.Set(&a.z) + c.t.SetZero() +} diff --git a/crypto/bn256/constants.go b/crypto/bn256/constants.go deleted file mode 100644 index ab649d7f3..000000000 --- a/crypto/bn256/constants.go +++ /dev/null @@ -1,44 +0,0 @@ -// 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 - -import ( - "math/big" -) - -func bigFromBase10(s string) *big.Int { - n, _ := new(big.Int).SetString(s, 10) - return n -} - -// u is the BN parameter that determines the prime: 1868033³. -var u = bigFromBase10("4965661367192848881") - -// p is a prime over which we form a basic field: 36u⁴+36u³+24u²+6u+1. -var P = bigFromBase10("21888242871839275222246405745257275088696311157297823662689037894645226208583") - -// Order is the number of elements in both G₁ and G₂: 36u⁴+36u³+18u²+6u+1. -var Order = bigFromBase10("21888242871839275222246405745257275088548364400416034343698204186575808495617") - -// xiToPMinus1Over6 is ξ^((p-1)/6) where ξ = i+9. -var xiToPMinus1Over6 = &gfP2{bigFromBase10("16469823323077808223889137241176536799009286646108169935659301613961712198316"), bigFromBase10("8376118865763821496583973867626364092589906065868298776909617916018768340080")} - -// xiToPMinus1Over3 is ξ^((p-1)/3) where ξ = i+9. -var xiToPMinus1Over3 = &gfP2{bigFromBase10("10307601595873709700152284273816112264069230130616436755625194854815875713954"), bigFromBase10("21575463638280843010398324269430826099269044274347216827212613867836435027261")} - -// xiToPMinus1Over2 is ξ^((p-1)/2) where ξ = i+9. -var xiToPMinus1Over2 = &gfP2{bigFromBase10("3505843767911556378687030309984248845540243509899259641013678093033130930403"), bigFromBase10("2821565182194536844548159561693502659359617185244120367078079554186484126554")} - -// xiToPSquaredMinus1Over3 is ξ^((p²-1)/3) where ξ = i+9. -var xiToPSquaredMinus1Over3 = bigFromBase10("21888242871839275220042445260109153167277707414472061641714758635765020556616") - -// xiTo2PSquaredMinus2Over3 is ξ^((2p²-2)/3) where ξ = i+9 (a cubic root of unity, mod p). -var xiTo2PSquaredMinus2Over3 = bigFromBase10("2203960485148121921418603742825762020974279258880205651966") - -// xiToPSquaredMinus1Over6 is ξ^((1p²-1)/6) where ξ = i+9 (a cubic root of -1, mod p). -var xiToPSquaredMinus1Over6 = bigFromBase10("21888242871839275220042445260109153167277707414472061641714758635765020556617") - -// xiTo2PMinus2Over3 is ξ^((2p-2)/3) where ξ = i+9. -var xiTo2PMinus2Over3 = &gfP2{bigFromBase10("19937756971775647987995932169929341994314640652964949448313374472400716661030"), bigFromBase10("2581911344467009335267311115468803099551665605076196740867805258568234346338")} diff --git a/crypto/bn256/curve.go b/crypto/bn256/curve.go deleted file mode 100644 index 3e679fdc7..000000000 --- a/crypto/bn256/curve.go +++ /dev/null @@ -1,278 +0,0 @@ -// 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 - -import ( - "math/big" -) - -// curvePoint implements the elliptic curve y²=x³+3. Points are kept in -// Jacobian form and t=z² when valid. G₁ is the set of points of this curve on -// GF(p). -type curvePoint struct { - x, y, z, t *big.Int -} - -var curveB = new(big.Int).SetInt64(3) - -// curveGen is the generator of G₁. -var curveGen = &curvePoint{ - new(big.Int).SetInt64(1), - new(big.Int).SetInt64(2), - new(big.Int).SetInt64(1), - new(big.Int).SetInt64(1), -} - -func newCurvePoint(pool *bnPool) *curvePoint { - return &curvePoint{ - pool.Get(), - pool.Get(), - pool.Get(), - pool.Get(), - } -} - -func (c *curvePoint) String() string { - c.MakeAffine(new(bnPool)) - return "(" + c.x.String() + ", " + c.y.String() + ")" -} - -func (c *curvePoint) Put(pool *bnPool) { - pool.Put(c.x) - pool.Put(c.y) - pool.Put(c.z) - pool.Put(c.t) -} - -func (c *curvePoint) Set(a *curvePoint) { - c.x.Set(a.x) - c.y.Set(a.y) - c.z.Set(a.z) - c.t.Set(a.t) -} - -// IsOnCurve returns true iff c is on the curve where c must be in affine form. -func (c *curvePoint) IsOnCurve() bool { - yy := new(big.Int).Mul(c.y, c.y) - xxx := new(big.Int).Mul(c.x, c.x) - xxx.Mul(xxx, c.x) - yy.Sub(yy, xxx) - yy.Sub(yy, curveB) - if yy.Sign() < 0 || yy.Cmp(P) >= 0 { - yy.Mod(yy, P) - } - return yy.Sign() == 0 -} - -func (c *curvePoint) SetInfinity() { - c.z.SetInt64(0) -} - -func (c *curvePoint) IsInfinity() bool { - return c.z.Sign() == 0 -} - -func (c *curvePoint) Add(a, b *curvePoint, pool *bnPool) { - if a.IsInfinity() { - c.Set(b) - return - } - if b.IsInfinity() { - c.Set(a) - return - } - - // See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/addition/add-2007-bl.op3 - - // Normalize the points by replacing a = [x1:y1:z1] and b = [x2:y2:z2] - // by [u1:s1:z1·z2] and [u2:s2:z1·z2] - // where u1 = x1·z2², s1 = y1·z2³ and u1 = x2·z1², s2 = y2·z1³ - z1z1 := pool.Get().Mul(a.z, a.z) - z1z1.Mod(z1z1, P) - z2z2 := pool.Get().Mul(b.z, b.z) - z2z2.Mod(z2z2, P) - u1 := pool.Get().Mul(a.x, z2z2) - u1.Mod(u1, P) - u2 := pool.Get().Mul(b.x, z1z1) - u2.Mod(u2, P) - - t := pool.Get().Mul(b.z, z2z2) - t.Mod(t, P) - s1 := pool.Get().Mul(a.y, t) - s1.Mod(s1, P) - - t.Mul(a.z, z1z1) - t.Mod(t, P) - s2 := pool.Get().Mul(b.y, t) - s2.Mod(s2, P) - - // Compute x = (2h)²(s²-u1-u2) - // where s = (s2-s1)/(u2-u1) is the slope of the line through - // (u1,s1) and (u2,s2). The extra factor 2h = 2(u2-u1) comes from the value of z below. - // This is also: - // 4(s2-s1)² - 4h²(u1+u2) = 4(s2-s1)² - 4h³ - 4h²(2u1) - // = r² - j - 2v - // with the notations below. - h := pool.Get().Sub(u2, u1) - xEqual := h.Sign() == 0 - - t.Add(h, h) - // i = 4h² - i := pool.Get().Mul(t, t) - i.Mod(i, P) - // j = 4h³ - j := pool.Get().Mul(h, i) - j.Mod(j, P) - - t.Sub(s2, s1) - yEqual := t.Sign() == 0 - if xEqual && yEqual { - c.Double(a, pool) - return - } - r := pool.Get().Add(t, t) - - v := pool.Get().Mul(u1, i) - v.Mod(v, P) - - // t4 = 4(s2-s1)² - t4 := pool.Get().Mul(r, r) - t4.Mod(t4, P) - t.Add(v, v) - t6 := pool.Get().Sub(t4, j) - c.x.Sub(t6, t) - - // Set y = -(2h)³(s1 + s*(x/4h²-u1)) - // This is also - // y = - 2·s1·j - (s2-s1)(2x - 2i·u1) = r(v-x) - 2·s1·j - t.Sub(v, c.x) // t7 - t4.Mul(s1, j) // t8 - t4.Mod(t4, P) - t6.Add(t4, t4) // t9 - t4.Mul(r, t) // t10 - t4.Mod(t4, P) - c.y.Sub(t4, t6) - - // Set z = 2(u2-u1)·z1·z2 = 2h·z1·z2 - t.Add(a.z, b.z) // t11 - t4.Mul(t, t) // t12 - t4.Mod(t4, P) - t.Sub(t4, z1z1) // t13 - t4.Sub(t, z2z2) // t14 - c.z.Mul(t4, h) - c.z.Mod(c.z, P) - - pool.Put(z1z1) - pool.Put(z2z2) - pool.Put(u1) - pool.Put(u2) - pool.Put(t) - pool.Put(s1) - pool.Put(s2) - pool.Put(h) - pool.Put(i) - pool.Put(j) - pool.Put(r) - pool.Put(v) - pool.Put(t4) - pool.Put(t6) -} - -func (c *curvePoint) Double(a *curvePoint, pool *bnPool) { - // See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/doubling/dbl-2009-l.op3 - A := pool.Get().Mul(a.x, a.x) - A.Mod(A, P) - B := pool.Get().Mul(a.y, a.y) - B.Mod(B, P) - C_ := pool.Get().Mul(B, B) - C_.Mod(C_, P) - - t := pool.Get().Add(a.x, B) - t2 := pool.Get().Mul(t, t) - t2.Mod(t2, P) - t.Sub(t2, A) - t2.Sub(t, C_) - d := pool.Get().Add(t2, t2) - t.Add(A, A) - e := pool.Get().Add(t, A) - f := pool.Get().Mul(e, e) - f.Mod(f, P) - - t.Add(d, d) - c.x.Sub(f, t) - - t.Add(C_, C_) - t2.Add(t, t) - t.Add(t2, t2) - c.y.Sub(d, c.x) - t2.Mul(e, c.y) - t2.Mod(t2, P) - c.y.Sub(t2, t) - - t.Mul(a.y, a.z) - t.Mod(t, P) - c.z.Add(t, t) - - pool.Put(A) - pool.Put(B) - pool.Put(C_) - pool.Put(t) - pool.Put(t2) - pool.Put(d) - pool.Put(e) - pool.Put(f) -} - -func (c *curvePoint) Mul(a *curvePoint, scalar *big.Int, pool *bnPool) *curvePoint { - sum := newCurvePoint(pool) - sum.SetInfinity() - t := newCurvePoint(pool) - - for i := scalar.BitLen(); i >= 0; i-- { - t.Double(sum, pool) - if scalar.Bit(i) != 0 { - sum.Add(t, a, pool) - } else { - sum.Set(t) - } - } - - c.Set(sum) - sum.Put(pool) - t.Put(pool) - return c -} - -func (c *curvePoint) MakeAffine(pool *bnPool) *curvePoint { - if words := c.z.Bits(); len(words) == 1 && words[0] == 1 { - return c - } - - zInv := pool.Get().ModInverse(c.z, P) - t := pool.Get().Mul(c.y, zInv) - t.Mod(t, P) - zInv2 := pool.Get().Mul(zInv, zInv) - zInv2.Mod(zInv2, P) - c.y.Mul(t, zInv2) - c.y.Mod(c.y, P) - t.Mul(c.x, zInv2) - t.Mod(t, P) - c.x.Set(t) - c.z.SetInt64(1) - c.t.SetInt64(1) - - pool.Put(zInv) - pool.Put(t) - pool.Put(zInv2) - - return c -} - -func (c *curvePoint) Negative(a *curvePoint) { - c.x.Set(a.x) - c.y.Neg(a.y) - c.z.Set(a.z) - c.t.SetInt64(0) -} diff --git a/crypto/bn256/example_test.go b/crypto/bn256/example_test.go deleted file mode 100644 index b2d19807a..000000000 --- a/crypto/bn256/example_test.go +++ /dev/null @@ -1,43 +0,0 @@ -// 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 - -import ( - "crypto/rand" -) - -func ExamplePair() { - // This implements the tripartite Diffie-Hellman algorithm from "A One - // Round Protocol for Tripartite Diffie-Hellman", A. Joux. - // http://www.springerlink.com/content/cddc57yyva0hburb/fulltext.pdf - - // Each of three parties, a, b and c, generate a private value. - a, _ := rand.Int(rand.Reader, Order) - b, _ := rand.Int(rand.Reader, Order) - c, _ := rand.Int(rand.Reader, Order) - - // Then each party calculates g₁ and g₂ times their private value. - pa := new(G1).ScalarBaseMult(a) - qa := new(G2).ScalarBaseMult(a) - - pb := new(G1).ScalarBaseMult(b) - qb := new(G2).ScalarBaseMult(b) - - pc := new(G1).ScalarBaseMult(c) - qc := new(G2).ScalarBaseMult(c) - - // Now each party exchanges its public values with the other two and - // all parties can calculate the shared key. - k1 := Pair(pb, qc) - k1.ScalarMult(k1, a) - - k2 := Pair(pc, qa) - k2.ScalarMult(k2, b) - - k3 := Pair(pa, qb) - k3.ScalarMult(k3, c) - - // k1, k2 and k3 will all be equal. -} diff --git a/crypto/bn256/gfp12.go b/crypto/bn256/gfp12.go deleted file mode 100644 index f084eddf2..000000000 --- a/crypto/bn256/gfp12.go +++ /dev/null @@ -1,200 +0,0 @@ -// 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(e.x, b, pool) - e.y.Mul(e.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 -} diff --git a/crypto/bn256/gfp2.go b/crypto/bn256/gfp2.go deleted file mode 100644 index 3981f6cb4..000000000 --- a/crypto/bn256/gfp2.go +++ /dev/null @@ -1,227 +0,0 @@ -// 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+9 and then returns e. -func (e *gfP2) MulXi(a *gfP2, pool *bnPool) *gfP2 { - // (xi+y)(i+3) = (9x+y)i+(9y-x) - tx := pool.Get().Lsh(a.x, 3) - tx.Add(tx, a.x) - tx.Add(tx, a.y) - - ty := pool.Get().Lsh(a.y, 3) - 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 -} - -func (e *gfP2) Real() *big.Int { - return e.x -} - -func (e *gfP2) Imag() *big.Int { - return e.y -} diff --git a/crypto/bn256/gfp6.go b/crypto/bn256/gfp6.go deleted file mode 100644 index 218856617..000000000 --- a/crypto/bn256/gfp6.go +++ /dev/null @@ -1,296 +0,0 @@ -// 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+9. -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 -} diff --git a/crypto/bn256/google/bn256.go b/crypto/bn256/google/bn256.go new file mode 100644 index 000000000..5da83e033 --- /dev/null +++ b/crypto/bn256/google/bn256.go @@ -0,0 +1,447 @@ +// 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 implements a particular bilinear group at the 128-bit security level. +// +// Bilinear groups are the basis of many of the new cryptographic protocols +// that have been proposed over the past decade. They consist of a triplet of +// groups (G₁, G₂ and GT) such that there exists a function e(g₁ˣ,g₂ʸ)=gTˣʸ +// (where gₓ is a generator of the respective group). That function is called +// a pairing function. +// +// This package specifically implements the Optimal Ate pairing over a 256-bit +// Barreto-Naehrig curve as described in +// http://cryptojedi.org/papers/dclxvi-20100714.pdf. Its output is compatible +// with the implementation described in that paper. +package bn256 + +import ( + "crypto/rand" + "errors" + "io" + "math/big" +) + +// BUG(agl): this implementation is not constant time. +// TODO(agl): keep GF(p²) elements in Mongomery form. + +// G1 is an abstract cyclic group. The zero value is suitable for use as the +// output of an operation, but cannot be used as an input. +type G1 struct { + p *curvePoint +} + +// RandomG1 returns x and g₁ˣ where x is a random, non-zero number read from r. +func RandomG1(r io.Reader) (*big.Int, *G1, error) { + var k *big.Int + var err error + + for { + k, err = rand.Int(r, Order) + if err != nil { + return nil, nil, err + } + if k.Sign() > 0 { + break + } + } + + return k, new(G1).ScalarBaseMult(k), nil +} + +func (g *G1) String() string { + return "bn256.G1" + g.p.String() +} + +// CurvePoints returns p's curve points in big integer +func (e *G1) CurvePoints() (*big.Int, *big.Int, *big.Int, *big.Int) { + return e.p.x, e.p.y, e.p.z, e.p.t +} + +// ScalarBaseMult sets e to g*k where g is the generator of the group and +// then returns e. +func (e *G1) ScalarBaseMult(k *big.Int) *G1 { + if e.p == nil { + e.p = newCurvePoint(nil) + } + e.p.Mul(curveGen, k, new(bnPool)) + return e +} + +// ScalarMult sets e to a*k and then returns e. +func (e *G1) ScalarMult(a *G1, k *big.Int) *G1 { + if e.p == nil { + e.p = newCurvePoint(nil) + } + e.p.Mul(a.p, k, new(bnPool)) + return e +} + +// Add sets e to a+b and then returns e. +// BUG(agl): this function is not complete: a==b fails. +func (e *G1) Add(a, b *G1) *G1 { + if e.p == nil { + e.p = newCurvePoint(nil) + } + e.p.Add(a.p, b.p, new(bnPool)) + return e +} + +// Neg sets e to -a and then returns e. +func (e *G1) Neg(a *G1) *G1 { + if e.p == nil { + e.p = newCurvePoint(nil) + } + e.p.Negative(a.p) + return e +} + +// Marshal converts n to a byte slice. +func (n *G1) Marshal() []byte { + n.p.MakeAffine(nil) + + xBytes := new(big.Int).Mod(n.p.x, P).Bytes() + yBytes := new(big.Int).Mod(n.p.y, P).Bytes() + + // Each value is a 256-bit number. + const numBytes = 256 / 8 + + ret := make([]byte, numBytes*2) + copy(ret[1*numBytes-len(xBytes):], xBytes) + copy(ret[2*numBytes-len(yBytes):], yBytes) + + return ret +} + +// Unmarshal sets e to the result of converting the output of Marshal back into +// a group element and then returns e. +func (e *G1) Unmarshal(m []byte) ([]byte, error) { + // Each value is a 256-bit number. + const numBytes = 256 / 8 + if len(m) != 2*numBytes { + return nil, errors.New("bn256: not enough data") + } + // Unmarshal the points and check their caps + if e.p == nil { + e.p = newCurvePoint(nil) + } + e.p.x.SetBytes(m[0*numBytes : 1*numBytes]) + if e.p.x.Cmp(P) >= 0 { + return nil, errors.New("bn256: coordinate exceeds modulus") + } + e.p.y.SetBytes(m[1*numBytes : 2*numBytes]) + if e.p.y.Cmp(P) >= 0 { + return nil, errors.New("bn256: coordinate exceeds modulus") + } + // Ensure the point is on the curve + if e.p.x.Sign() == 0 && e.p.y.Sign() == 0 { + // This is the point at infinity. + e.p.y.SetInt64(1) + e.p.z.SetInt64(0) + e.p.t.SetInt64(0) + } else { + e.p.z.SetInt64(1) + e.p.t.SetInt64(1) + + if !e.p.IsOnCurve() { + return nil, errors.New("bn256: malformed point") + } + } + return m[2*numBytes:], nil +} + +// G2 is an abstract cyclic group. The zero value is suitable for use as the +// output of an operation, but cannot be used as an input. +type G2 struct { + p *twistPoint +} + +// RandomG1 returns x and g₂ˣ where x is a random, non-zero number read from r. +func RandomG2(r io.Reader) (*big.Int, *G2, error) { + var k *big.Int + var err error + + for { + k, err = rand.Int(r, Order) + if err != nil { + return nil, nil, err + } + if k.Sign() > 0 { + break + } + } + + return k, new(G2).ScalarBaseMult(k), nil +} + +func (g *G2) String() string { + return "bn256.G2" + g.p.String() +} + +// CurvePoints returns the curve points of p which includes the real +// and imaginary parts of the curve point. +func (e *G2) CurvePoints() (*gfP2, *gfP2, *gfP2, *gfP2) { + return e.p.x, e.p.y, e.p.z, e.p.t +} + +// ScalarBaseMult sets e to g*k where g is the generator of the group and +// then returns out. +func (e *G2) ScalarBaseMult(k *big.Int) *G2 { + if e.p == nil { + e.p = newTwistPoint(nil) + } + e.p.Mul(twistGen, k, new(bnPool)) + return e +} + +// ScalarMult sets e to a*k and then returns e. +func (e *G2) ScalarMult(a *G2, k *big.Int) *G2 { + if e.p == nil { + e.p = newTwistPoint(nil) + } + e.p.Mul(a.p, k, new(bnPool)) + return e +} + +// Add sets e to a+b and then returns e. +// BUG(agl): this function is not complete: a==b fails. +func (e *G2) Add(a, b *G2) *G2 { + if e.p == nil { + e.p = newTwistPoint(nil) + } + e.p.Add(a.p, b.p, new(bnPool)) + return e +} + +// Marshal converts n into a byte slice. +func (n *G2) Marshal() []byte { + n.p.MakeAffine(nil) + + xxBytes := new(big.Int).Mod(n.p.x.x, P).Bytes() + xyBytes := new(big.Int).Mod(n.p.x.y, P).Bytes() + yxBytes := new(big.Int).Mod(n.p.y.x, P).Bytes() + yyBytes := new(big.Int).Mod(n.p.y.y, P).Bytes() + + // Each value is a 256-bit number. + const numBytes = 256 / 8 + + ret := make([]byte, numBytes*4) + copy(ret[1*numBytes-len(xxBytes):], xxBytes) + copy(ret[2*numBytes-len(xyBytes):], xyBytes) + copy(ret[3*numBytes-len(yxBytes):], yxBytes) + copy(ret[4*numBytes-len(yyBytes):], yyBytes) + + return ret +} + +// Unmarshal sets e to the result of converting the output of Marshal back into +// a group element and then returns e. +func (e *G2) Unmarshal(m []byte) ([]byte, error) { + // Each value is a 256-bit number. + const numBytes = 256 / 8 + if len(m) != 4*numBytes { + return nil, errors.New("bn256: not enough data") + } + // Unmarshal the points and check their caps + if e.p == nil { + e.p = newTwistPoint(nil) + } + e.p.x.x.SetBytes(m[0*numBytes : 1*numBytes]) + if e.p.x.x.Cmp(P) >= 0 { + return nil, errors.New("bn256: coordinate exceeds modulus") + } + e.p.x.y.SetBytes(m[1*numBytes : 2*numBytes]) + if e.p.x.y.Cmp(P) >= 0 { + return nil, errors.New("bn256: coordinate exceeds modulus") + } + e.p.y.x.SetBytes(m[2*numBytes : 3*numBytes]) + if e.p.y.x.Cmp(P) >= 0 { + return nil, errors.New("bn256: coordinate exceeds modulus") + } + e.p.y.y.SetBytes(m[3*numBytes : 4*numBytes]) + if e.p.y.y.Cmp(P) >= 0 { + return nil, errors.New("bn256: coordinate exceeds modulus") + } + // Ensure the point is on the curve + if e.p.x.x.Sign() == 0 && + e.p.x.y.Sign() == 0 && + e.p.y.x.Sign() == 0 && + e.p.y.y.Sign() == 0 { + // This is the point at infinity. + e.p.y.SetOne() + e.p.z.SetZero() + e.p.t.SetZero() + } else { + e.p.z.SetOne() + e.p.t.SetOne() + + if !e.p.IsOnCurve() { + return nil, errors.New("bn256: malformed point") + } + } + return m[4*numBytes:], nil +} + +// GT is an abstract cyclic group. The zero value is suitable for use as the +// output of an operation, but cannot be used as an input. +type GT struct { + p *gfP12 +} + +func (g *GT) String() string { + return "bn256.GT" + g.p.String() +} + +// ScalarMult sets e to a*k and then returns e. +func (e *GT) ScalarMult(a *GT, k *big.Int) *GT { + if e.p == nil { + e.p = newGFp12(nil) + } + e.p.Exp(a.p, k, new(bnPool)) + return e +} + +// Add sets e to a+b and then returns e. +func (e *GT) Add(a, b *GT) *GT { + if e.p == nil { + e.p = newGFp12(nil) + } + e.p.Mul(a.p, b.p, new(bnPool)) + return e +} + +// Neg sets e to -a and then returns e. +func (e *GT) Neg(a *GT) *GT { + if e.p == nil { + e.p = newGFp12(nil) + } + e.p.Invert(a.p, new(bnPool)) + return e +} + +// Marshal converts n into a byte slice. +func (n *GT) Marshal() []byte { + n.p.Minimal() + + xxxBytes := n.p.x.x.x.Bytes() + xxyBytes := n.p.x.x.y.Bytes() + xyxBytes := n.p.x.y.x.Bytes() + xyyBytes := n.p.x.y.y.Bytes() + xzxBytes := n.p.x.z.x.Bytes() + xzyBytes := n.p.x.z.y.Bytes() + yxxBytes := n.p.y.x.x.Bytes() + yxyBytes := n.p.y.x.y.Bytes() + yyxBytes := n.p.y.y.x.Bytes() + yyyBytes := n.p.y.y.y.Bytes() + yzxBytes := n.p.y.z.x.Bytes() + yzyBytes := n.p.y.z.y.Bytes() + + // Each value is a 256-bit number. + const numBytes = 256 / 8 + + ret := make([]byte, numBytes*12) + copy(ret[1*numBytes-len(xxxBytes):], xxxBytes) + copy(ret[2*numBytes-len(xxyBytes):], xxyBytes) + copy(ret[3*numBytes-len(xyxBytes):], xyxBytes) + copy(ret[4*numBytes-len(xyyBytes):], xyyBytes) + copy(ret[5*numBytes-len(xzxBytes):], xzxBytes) + copy(ret[6*numBytes-len(xzyBytes):], xzyBytes) + copy(ret[7*numBytes-len(yxxBytes):], yxxBytes) + copy(ret[8*numBytes-len(yxyBytes):], yxyBytes) + copy(ret[9*numBytes-len(yyxBytes):], yyxBytes) + copy(ret[10*numBytes-len(yyyBytes):], yyyBytes) + copy(ret[11*numBytes-len(yzxBytes):], yzxBytes) + copy(ret[12*numBytes-len(yzyBytes):], yzyBytes) + + return ret +} + +// Unmarshal sets e to the result of converting the output of Marshal back into +// a group element and then returns e. +func (e *GT) Unmarshal(m []byte) (*GT, bool) { + // Each value is a 256-bit number. + const numBytes = 256 / 8 + + if len(m) != 12*numBytes { + return nil, false + } + + if e.p == nil { + e.p = newGFp12(nil) + } + + e.p.x.x.x.SetBytes(m[0*numBytes : 1*numBytes]) + e.p.x.x.y.SetBytes(m[1*numBytes : 2*numBytes]) + e.p.x.y.x.SetBytes(m[2*numBytes : 3*numBytes]) + e.p.x.y.y.SetBytes(m[3*numBytes : 4*numBytes]) + e.p.x.z.x.SetBytes(m[4*numBytes : 5*numBytes]) + e.p.x.z.y.SetBytes(m[5*numBytes : 6*numBytes]) + e.p.y.x.x.SetBytes(m[6*numBytes : 7*numBytes]) + e.p.y.x.y.SetBytes(m[7*numBytes : 8*numBytes]) + e.p.y.y.x.SetBytes(m[8*numBytes : 9*numBytes]) + e.p.y.y.y.SetBytes(m[9*numBytes : 10*numBytes]) + e.p.y.z.x.SetBytes(m[10*numBytes : 11*numBytes]) + e.p.y.z.y.SetBytes(m[11*numBytes : 12*numBytes]) + + return e, true +} + +// Pair calculates an Optimal Ate pairing. +func Pair(g1 *G1, g2 *G2) *GT { + return >{optimalAte(g2.p, g1.p, new(bnPool))} +} + +// PairingCheck calculates the Optimal Ate pairing for a set of points. +func PairingCheck(a []*G1, b []*G2) bool { + pool := new(bnPool) + + acc := newGFp12(pool) + acc.SetOne() + + for i := 0; i < len(a); i++ { + if a[i].p.IsInfinity() || b[i].p.IsInfinity() { + continue + } + 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 +} diff --git a/crypto/bn256/google/bn256_test.go b/crypto/bn256/google/bn256_test.go new file mode 100644 index 000000000..a4497ada9 --- /dev/null +++ b/crypto/bn256/google/bn256_test.go @@ -0,0 +1,311 @@ +// 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 + +import ( + "bytes" + "crypto/rand" + "math/big" + "testing" +) + +func TestGFp2Invert(t *testing.T) { + pool := new(bnPool) + + a := newGFp2(pool) + a.x.SetString("23423492374", 10) + a.y.SetString("12934872398472394827398470", 10) + + inv := newGFp2(pool) + inv.Invert(a, pool) + + b := newGFp2(pool).Mul(inv, a, pool) + if b.x.Int64() != 0 || b.y.Int64() != 1 { + t.Fatalf("bad result for a^-1*a: %s %s", b.x, b.y) + } + + a.Put(pool) + b.Put(pool) + inv.Put(pool) + + if c := pool.Count(); c > 0 { + t.Errorf("Pool count non-zero: %d\n", c) + } +} + +func isZero(n *big.Int) bool { + return new(big.Int).Mod(n, P).Int64() == 0 +} + +func isOne(n *big.Int) bool { + return new(big.Int).Mod(n, P).Int64() == 1 +} + +func TestGFp6Invert(t *testing.T) { + pool := new(bnPool) + + a := newGFp6(pool) + a.x.x.SetString("239487238491", 10) + a.x.y.SetString("2356249827341", 10) + a.y.x.SetString("082659782", 10) + a.y.y.SetString("182703523765", 10) + a.z.x.SetString("978236549263", 10) + a.z.y.SetString("64893242", 10) + + inv := newGFp6(pool) + inv.Invert(a, pool) + + b := newGFp6(pool).Mul(inv, a, pool) + if !isZero(b.x.x) || + !isZero(b.x.y) || + !isZero(b.y.x) || + !isZero(b.y.y) || + !isZero(b.z.x) || + !isOne(b.z.y) { + t.Fatalf("bad result for a^-1*a: %s", b) + } + + a.Put(pool) + b.Put(pool) + inv.Put(pool) + + if c := pool.Count(); c > 0 { + t.Errorf("Pool count non-zero: %d\n", c) + } +} + +func TestGFp12Invert(t *testing.T) { + pool := new(bnPool) + + a := newGFp12(pool) + a.x.x.x.SetString("239846234862342323958623", 10) + a.x.x.y.SetString("2359862352529835623", 10) + a.x.y.x.SetString("928836523", 10) + a.x.y.y.SetString("9856234", 10) + a.x.z.x.SetString("235635286", 10) + a.x.z.y.SetString("5628392833", 10) + a.y.x.x.SetString("252936598265329856238956532167968", 10) + a.y.x.y.SetString("23596239865236954178968", 10) + a.y.y.x.SetString("95421692834", 10) + a.y.y.y.SetString("236548", 10) + a.y.z.x.SetString("924523", 10) + a.y.z.y.SetString("12954623", 10) + + inv := newGFp12(pool) + inv.Invert(a, pool) + + b := newGFp12(pool).Mul(inv, a, pool) + if !isZero(b.x.x.x) || + !isZero(b.x.x.y) || + !isZero(b.x.y.x) || + !isZero(b.x.y.y) || + !isZero(b.x.z.x) || + !isZero(b.x.z.y) || + !isZero(b.y.x.x) || + !isZero(b.y.x.y) || + !isZero(b.y.y.x) || + !isZero(b.y.y.y) || + !isZero(b.y.z.x) || + !isOne(b.y.z.y) { + t.Fatalf("bad result for a^-1*a: %s", b) + } + + a.Put(pool) + b.Put(pool) + inv.Put(pool) + + if c := pool.Count(); c > 0 { + t.Errorf("Pool count non-zero: %d\n", c) + } +} + +func TestCurveImpl(t *testing.T) { + pool := new(bnPool) + + g := &curvePoint{ + pool.Get().SetInt64(1), + pool.Get().SetInt64(-2), + pool.Get().SetInt64(1), + pool.Get().SetInt64(0), + } + + x := pool.Get().SetInt64(32498273234) + X := newCurvePoint(pool).Mul(g, x, pool) + + y := pool.Get().SetInt64(98732423523) + Y := newCurvePoint(pool).Mul(g, y, pool) + + s1 := newCurvePoint(pool).Mul(X, y, pool).MakeAffine(pool) + s2 := newCurvePoint(pool).Mul(Y, x, pool).MakeAffine(pool) + + if s1.x.Cmp(s2.x) != 0 || + s2.x.Cmp(s1.x) != 0 { + t.Errorf("DH points don't match: (%s, %s) (%s, %s)", s1.x, s1.y, s2.x, s2.y) + } + + pool.Put(x) + X.Put(pool) + pool.Put(y) + Y.Put(pool) + s1.Put(pool) + s2.Put(pool) + g.Put(pool) + + if c := pool.Count(); c > 0 { + t.Errorf("Pool count non-zero: %d\n", c) + } +} + +func TestOrderG1(t *testing.T) { + g := new(G1).ScalarBaseMult(Order) + if !g.p.IsInfinity() { + t.Error("G1 has incorrect order") + } + + one := new(G1).ScalarBaseMult(new(big.Int).SetInt64(1)) + g.Add(g, one) + g.p.MakeAffine(nil) + if g.p.x.Cmp(one.p.x) != 0 || g.p.y.Cmp(one.p.y) != 0 { + t.Errorf("1+0 != 1 in G1") + } +} + +func TestOrderG2(t *testing.T) { + g := new(G2).ScalarBaseMult(Order) + if !g.p.IsInfinity() { + t.Error("G2 has incorrect order") + } + + one := new(G2).ScalarBaseMult(new(big.Int).SetInt64(1)) + g.Add(g, one) + g.p.MakeAffine(nil) + if g.p.x.x.Cmp(one.p.x.x) != 0 || + g.p.x.y.Cmp(one.p.x.y) != 0 || + g.p.y.x.Cmp(one.p.y.x) != 0 || + g.p.y.y.Cmp(one.p.y.y) != 0 { + t.Errorf("1+0 != 1 in G2") + } +} + +func TestOrderGT(t *testing.T) { + gt := Pair(&G1{curveGen}, &G2{twistGen}) + g := new(GT).ScalarMult(gt, Order) + if !g.p.IsOne() { + t.Error("GT has incorrect order") + } +} + +func TestBilinearity(t *testing.T) { + for i := 0; i < 2; i++ { + a, p1, _ := RandomG1(rand.Reader) + b, p2, _ := RandomG2(rand.Reader) + e1 := Pair(p1, p2) + + e2 := Pair(&G1{curveGen}, &G2{twistGen}) + e2.ScalarMult(e2, a) + e2.ScalarMult(e2, b) + + minusE2 := new(GT).Neg(e2) + e1.Add(e1, minusE2) + + if !e1.p.IsOne() { + t.Fatalf("bad pairing result: %s", e1) + } + } +} + +func TestG1Marshal(t *testing.T) { + g := new(G1).ScalarBaseMult(new(big.Int).SetInt64(1)) + form := g.Marshal() + _, err := new(G1).Unmarshal(form) + if err != nil { + t.Fatalf("failed to unmarshal") + } + + g.ScalarBaseMult(Order) + form = g.Marshal() + + g2 := new(G1) + if _, err = g2.Unmarshal(form); err != nil { + t.Fatalf("failed to unmarshal ∞") + } + if !g2.p.IsInfinity() { + t.Fatalf("∞ unmarshaled incorrectly") + } +} + +func TestG2Marshal(t *testing.T) { + g := new(G2).ScalarBaseMult(new(big.Int).SetInt64(1)) + form := g.Marshal() + _, err := new(G2).Unmarshal(form) + if err != nil { + t.Fatalf("failed to unmarshal") + } + + g.ScalarBaseMult(Order) + form = g.Marshal() + g2 := new(G2) + if _, err = g2.Unmarshal(form); err != nil { + t.Fatalf("failed to unmarshal ∞") + } + if !g2.p.IsInfinity() { + t.Fatalf("∞ unmarshaled incorrectly") + } +} + +func TestG1Identity(t *testing.T) { + g := new(G1).ScalarBaseMult(new(big.Int).SetInt64(0)) + if !g.p.IsInfinity() { + t.Error("failure") + } +} + +func TestG2Identity(t *testing.T) { + g := new(G2).ScalarBaseMult(new(big.Int).SetInt64(0)) + if !g.p.IsInfinity() { + t.Error("failure") + } +} + +func TestTripartiteDiffieHellman(t *testing.T) { + a, _ := rand.Int(rand.Reader, Order) + b, _ := rand.Int(rand.Reader, Order) + c, _ := rand.Int(rand.Reader, Order) + + pa := new(G1) + pa.Unmarshal(new(G1).ScalarBaseMult(a).Marshal()) + qa := new(G2) + qa.Unmarshal(new(G2).ScalarBaseMult(a).Marshal()) + pb := new(G1) + pb.Unmarshal(new(G1).ScalarBaseMult(b).Marshal()) + qb := new(G2) + qb.Unmarshal(new(G2).ScalarBaseMult(b).Marshal()) + pc := new(G1) + pc.Unmarshal(new(G1).ScalarBaseMult(c).Marshal()) + qc := new(G2) + qc.Unmarshal(new(G2).ScalarBaseMult(c).Marshal()) + + k1 := Pair(pb, qc) + k1.ScalarMult(k1, a) + k1Bytes := k1.Marshal() + + k2 := Pair(pc, qa) + k2.ScalarMult(k2, b) + k2Bytes := k2.Marshal() + + k3 := Pair(pa, qb) + k3.ScalarMult(k3, c) + k3Bytes := k3.Marshal() + + if !bytes.Equal(k1Bytes, k2Bytes) || !bytes.Equal(k2Bytes, k3Bytes) { + t.Errorf("keys didn't agree") + } +} + +func BenchmarkPairing(b *testing.B) { + for i := 0; i < b.N; i++ { + Pair(&G1{curveGen}, &G2{twistGen}) + } +} diff --git a/crypto/bn256/google/constants.go b/crypto/bn256/google/constants.go new file mode 100644 index 000000000..ab649d7f3 --- /dev/null +++ b/crypto/bn256/google/constants.go @@ -0,0 +1,44 @@ +// 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 + +import ( + "math/big" +) + +func bigFromBase10(s string) *big.Int { + n, _ := new(big.Int).SetString(s, 10) + return n +} + +// u is the BN parameter that determines the prime: 1868033³. +var u = bigFromBase10("4965661367192848881") + +// p is a prime over which we form a basic field: 36u⁴+36u³+24u²+6u+1. +var P = bigFromBase10("21888242871839275222246405745257275088696311157297823662689037894645226208583") + +// Order is the number of elements in both G₁ and G₂: 36u⁴+36u³+18u²+6u+1. +var Order = bigFromBase10("21888242871839275222246405745257275088548364400416034343698204186575808495617") + +// xiToPMinus1Over6 is ξ^((p-1)/6) where ξ = i+9. +var xiToPMinus1Over6 = &gfP2{bigFromBase10("16469823323077808223889137241176536799009286646108169935659301613961712198316"), bigFromBase10("8376118865763821496583973867626364092589906065868298776909617916018768340080")} + +// xiToPMinus1Over3 is ξ^((p-1)/3) where ξ = i+9. +var xiToPMinus1Over3 = &gfP2{bigFromBase10("10307601595873709700152284273816112264069230130616436755625194854815875713954"), bigFromBase10("21575463638280843010398324269430826099269044274347216827212613867836435027261")} + +// xiToPMinus1Over2 is ξ^((p-1)/2) where ξ = i+9. +var xiToPMinus1Over2 = &gfP2{bigFromBase10("3505843767911556378687030309984248845540243509899259641013678093033130930403"), bigFromBase10("2821565182194536844548159561693502659359617185244120367078079554186484126554")} + +// xiToPSquaredMinus1Over3 is ξ^((p²-1)/3) where ξ = i+9. +var xiToPSquaredMinus1Over3 = bigFromBase10("21888242871839275220042445260109153167277707414472061641714758635765020556616") + +// xiTo2PSquaredMinus2Over3 is ξ^((2p²-2)/3) where ξ = i+9 (a cubic root of unity, mod p). +var xiTo2PSquaredMinus2Over3 = bigFromBase10("2203960485148121921418603742825762020974279258880205651966") + +// xiToPSquaredMinus1Over6 is ξ^((1p²-1)/6) where ξ = i+9 (a cubic root of -1, mod p). +var xiToPSquaredMinus1Over6 = bigFromBase10("21888242871839275220042445260109153167277707414472061641714758635765020556617") + +// xiTo2PMinus2Over3 is ξ^((2p-2)/3) where ξ = i+9. +var xiTo2PMinus2Over3 = &gfP2{bigFromBase10("19937756971775647987995932169929341994314640652964949448313374472400716661030"), bigFromBase10("2581911344467009335267311115468803099551665605076196740867805258568234346338")} diff --git a/crypto/bn256/google/curve.go b/crypto/bn256/google/curve.go new file mode 100644 index 000000000..3e679fdc7 --- /dev/null +++ b/crypto/bn256/google/curve.go @@ -0,0 +1,278 @@ +// 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 + +import ( + "math/big" +) + +// curvePoint implements the elliptic curve y²=x³+3. Points are kept in +// Jacobian form and t=z² when valid. G₁ is the set of points of this curve on +// GF(p). +type curvePoint struct { + x, y, z, t *big.Int +} + +var curveB = new(big.Int).SetInt64(3) + +// curveGen is the generator of G₁. +var curveGen = &curvePoint{ + new(big.Int).SetInt64(1), + new(big.Int).SetInt64(2), + new(big.Int).SetInt64(1), + new(big.Int).SetInt64(1), +} + +func newCurvePoint(pool *bnPool) *curvePoint { + return &curvePoint{ + pool.Get(), + pool.Get(), + pool.Get(), + pool.Get(), + } +} + +func (c *curvePoint) String() string { + c.MakeAffine(new(bnPool)) + return "(" + c.x.String() + ", " + c.y.String() + ")" +} + +func (c *curvePoint) Put(pool *bnPool) { + pool.Put(c.x) + pool.Put(c.y) + pool.Put(c.z) + pool.Put(c.t) +} + +func (c *curvePoint) Set(a *curvePoint) { + c.x.Set(a.x) + c.y.Set(a.y) + c.z.Set(a.z) + c.t.Set(a.t) +} + +// IsOnCurve returns true iff c is on the curve where c must be in affine form. +func (c *curvePoint) IsOnCurve() bool { + yy := new(big.Int).Mul(c.y, c.y) + xxx := new(big.Int).Mul(c.x, c.x) + xxx.Mul(xxx, c.x) + yy.Sub(yy, xxx) + yy.Sub(yy, curveB) + if yy.Sign() < 0 || yy.Cmp(P) >= 0 { + yy.Mod(yy, P) + } + return yy.Sign() == 0 +} + +func (c *curvePoint) SetInfinity() { + c.z.SetInt64(0) +} + +func (c *curvePoint) IsInfinity() bool { + return c.z.Sign() == 0 +} + +func (c *curvePoint) Add(a, b *curvePoint, pool *bnPool) { + if a.IsInfinity() { + c.Set(b) + return + } + if b.IsInfinity() { + c.Set(a) + return + } + + // See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/addition/add-2007-bl.op3 + + // Normalize the points by replacing a = [x1:y1:z1] and b = [x2:y2:z2] + // by [u1:s1:z1·z2] and [u2:s2:z1·z2] + // where u1 = x1·z2², s1 = y1·z2³ and u1 = x2·z1², s2 = y2·z1³ + z1z1 := pool.Get().Mul(a.z, a.z) + z1z1.Mod(z1z1, P) + z2z2 := pool.Get().Mul(b.z, b.z) + z2z2.Mod(z2z2, P) + u1 := pool.Get().Mul(a.x, z2z2) + u1.Mod(u1, P) + u2 := pool.Get().Mul(b.x, z1z1) + u2.Mod(u2, P) + + t := pool.Get().Mul(b.z, z2z2) + t.Mod(t, P) + s1 := pool.Get().Mul(a.y, t) + s1.Mod(s1, P) + + t.Mul(a.z, z1z1) + t.Mod(t, P) + s2 := pool.Get().Mul(b.y, t) + s2.Mod(s2, P) + + // Compute x = (2h)²(s²-u1-u2) + // where s = (s2-s1)/(u2-u1) is the slope of the line through + // (u1,s1) and (u2,s2). The extra factor 2h = 2(u2-u1) comes from the value of z below. + // This is also: + // 4(s2-s1)² - 4h²(u1+u2) = 4(s2-s1)² - 4h³ - 4h²(2u1) + // = r² - j - 2v + // with the notations below. + h := pool.Get().Sub(u2, u1) + xEqual := h.Sign() == 0 + + t.Add(h, h) + // i = 4h² + i := pool.Get().Mul(t, t) + i.Mod(i, P) + // j = 4h³ + j := pool.Get().Mul(h, i) + j.Mod(j, P) + + t.Sub(s2, s1) + yEqual := t.Sign() == 0 + if xEqual && yEqual { + c.Double(a, pool) + return + } + r := pool.Get().Add(t, t) + + v := pool.Get().Mul(u1, i) + v.Mod(v, P) + + // t4 = 4(s2-s1)² + t4 := pool.Get().Mul(r, r) + t4.Mod(t4, P) + t.Add(v, v) + t6 := pool.Get().Sub(t4, j) + c.x.Sub(t6, t) + + // Set y = -(2h)³(s1 + s*(x/4h²-u1)) + // This is also + // y = - 2·s1·j - (s2-s1)(2x - 2i·u1) = r(v-x) - 2·s1·j + t.Sub(v, c.x) // t7 + t4.Mul(s1, j) // t8 + t4.Mod(t4, P) + t6.Add(t4, t4) // t9 + t4.Mul(r, t) // t10 + t4.Mod(t4, P) + c.y.Sub(t4, t6) + + // Set z = 2(u2-u1)·z1·z2 = 2h·z1·z2 + t.Add(a.z, b.z) // t11 + t4.Mul(t, t) // t12 + t4.Mod(t4, P) + t.Sub(t4, z1z1) // t13 + t4.Sub(t, z2z2) // t14 + c.z.Mul(t4, h) + c.z.Mod(c.z, P) + + pool.Put(z1z1) + pool.Put(z2z2) + pool.Put(u1) + pool.Put(u2) + pool.Put(t) + pool.Put(s1) + pool.Put(s2) + pool.Put(h) + pool.Put(i) + pool.Put(j) + pool.Put(r) + pool.Put(v) + pool.Put(t4) + pool.Put(t6) +} + +func (c *curvePoint) Double(a *curvePoint, pool *bnPool) { + // See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/doubling/dbl-2009-l.op3 + A := pool.Get().Mul(a.x, a.x) + A.Mod(A, P) + B := pool.Get().Mul(a.y, a.y) + B.Mod(B, P) + C_ := pool.Get().Mul(B, B) + C_.Mod(C_, P) + + t := pool.Get().Add(a.x, B) + t2 := pool.Get().Mul(t, t) + t2.Mod(t2, P) + t.Sub(t2, A) + t2.Sub(t, C_) + d := pool.Get().Add(t2, t2) + t.Add(A, A) + e := pool.Get().Add(t, A) + f := pool.Get().Mul(e, e) + f.Mod(f, P) + + t.Add(d, d) + c.x.Sub(f, t) + + t.Add(C_, C_) + t2.Add(t, t) + t.Add(t2, t2) + c.y.Sub(d, c.x) + t2.Mul(e, c.y) + t2.Mod(t2, P) + c.y.Sub(t2, t) + + t.Mul(a.y, a.z) + t.Mod(t, P) + c.z.Add(t, t) + + pool.Put(A) + pool.Put(B) + pool.Put(C_) + pool.Put(t) + pool.Put(t2) + pool.Put(d) + pool.Put(e) + pool.Put(f) +} + +func (c *curvePoint) Mul(a *curvePoint, scalar *big.Int, pool *bnPool) *curvePoint { + sum := newCurvePoint(pool) + sum.SetInfinity() + t := newCurvePoint(pool) + + for i := scalar.BitLen(); i >= 0; i-- { + t.Double(sum, pool) + if scalar.Bit(i) != 0 { + sum.Add(t, a, pool) + } else { + sum.Set(t) + } + } + + c.Set(sum) + sum.Put(pool) + t.Put(pool) + return c +} + +func (c *curvePoint) MakeAffine(pool *bnPool) *curvePoint { + if words := c.z.Bits(); len(words) == 1 && words[0] == 1 { + return c + } + + zInv := pool.Get().ModInverse(c.z, P) + t := pool.Get().Mul(c.y, zInv) + t.Mod(t, P) + zInv2 := pool.Get().Mul(zInv, zInv) + zInv2.Mod(zInv2, P) + c.y.Mul(t, zInv2) + c.y.Mod(c.y, P) + t.Mul(c.x, zInv2) + t.Mod(t, P) + c.x.Set(t) + c.z.SetInt64(1) + c.t.SetInt64(1) + + pool.Put(zInv) + pool.Put(t) + pool.Put(zInv2) + + return c +} + +func (c *curvePoint) Negative(a *curvePoint) { + c.x.Set(a.x) + c.y.Neg(a.y) + c.z.Set(a.z) + c.t.SetInt64(0) +} diff --git a/crypto/bn256/google/example_test.go b/crypto/bn256/google/example_test.go new file mode 100644 index 000000000..b2d19807a --- /dev/null +++ b/crypto/bn256/google/example_test.go @@ -0,0 +1,43 @@ +// 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 + +import ( + "crypto/rand" +) + +func ExamplePair() { + // This implements the tripartite Diffie-Hellman algorithm from "A One + // Round Protocol for Tripartite Diffie-Hellman", A. Joux. + // http://www.springerlink.com/content/cddc57yyva0hburb/fulltext.pdf + + // Each of three parties, a, b and c, generate a private value. + a, _ := rand.Int(rand.Reader, Order) + b, _ := rand.Int(rand.Reader, Order) + c, _ := rand.Int(rand.Reader, Order) + + // Then each party calculates g₁ and g₂ times their private value. + pa := new(G1).ScalarBaseMult(a) + qa := new(G2).ScalarBaseMult(a) + + pb := new(G1).ScalarBaseMult(b) + qb := new(G2).ScalarBaseMult(b) + + pc := new(G1).ScalarBaseMult(c) + qc := new(G2).ScalarBaseMult(c) + + // Now each party exchanges its public values with the other two and + // all parties can calculate the shared key. + k1 := Pair(pb, qc) + k1.ScalarMult(k1, a) + + k2 := Pair(pc, qa) + k2.ScalarMult(k2, b) + + k3 := Pair(pa, qb) + k3.ScalarMult(k3, c) + + // k1, k2 and k3 will all be equal. +} diff --git a/crypto/bn256/google/gfp12.go b/crypto/bn256/google/gfp12.go new file mode 100644 index 000000000..f084eddf2 --- /dev/null +++ b/crypto/bn256/google/gfp12.go @@ -0,0 +1,200 @@ +// 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(e.x, b, pool) + e.y.Mul(e.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 +} diff --git a/crypto/bn256/google/gfp2.go b/crypto/bn256/google/gfp2.go new file mode 100644 index 000000000..3981f6cb4 --- /dev/null +++ b/crypto/bn256/google/gfp2.go @@ -0,0 +1,227 @@ +// 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+9 and then returns e. +func (e *gfP2) MulXi(a *gfP2, pool *bnPool) *gfP2 { + // (xi+y)(i+3) = (9x+y)i+(9y-x) + tx := pool.Get().Lsh(a.x, 3) + tx.Add(tx, a.x) + tx.Add(tx, a.y) + + ty := pool.Get().Lsh(a.y, 3) + 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 +} + +func (e *gfP2) Real() *big.Int { + return e.x +} + +func (e *gfP2) Imag() *big.Int { + return e.y +} diff --git a/crypto/bn256/google/gfp6.go b/crypto/bn256/google/gfp6.go new file mode 100644 index 000000000..218856617 --- /dev/null +++ b/crypto/bn256/google/gfp6.go @@ -0,0 +1,296 @@ +// 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+9. +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 +} diff --git a/crypto/bn256/google/main_test.go b/crypto/bn256/google/main_test.go new file mode 100644 index 000000000..0230f1b19 --- /dev/null +++ b/crypto/bn256/google/main_test.go @@ -0,0 +1,71 @@ +package bn256 + +import ( + "testing" + + "crypto/rand" +) + +func TestRandomG2Marshal(t *testing.T) { + for i := 0; i < 10; i++ { + n, g2, err := RandomG2(rand.Reader) + if err != nil { + t.Error(err) + continue + } + t.Logf("%d: %x\n", n, g2.Marshal()) + } +} + +func TestPairings(t *testing.T) { + a1 := new(G1).ScalarBaseMult(bigFromBase10("1")) + a2 := new(G1).ScalarBaseMult(bigFromBase10("2")) + a37 := new(G1).ScalarBaseMult(bigFromBase10("37")) + an1 := new(G1).ScalarBaseMult(bigFromBase10("21888242871839275222246405745257275088548364400416034343698204186575808495616")) + + b0 := new(G2).ScalarBaseMult(bigFromBase10("0")) + b1 := new(G2).ScalarBaseMult(bigFromBase10("1")) + b2 := new(G2).ScalarBaseMult(bigFromBase10("2")) + b27 := new(G2).ScalarBaseMult(bigFromBase10("27")) + b999 := new(G2).ScalarBaseMult(bigFromBase10("999")) + bn1 := new(G2).ScalarBaseMult(bigFromBase10("21888242871839275222246405745257275088548364400416034343698204186575808495616")) + + p1 := Pair(a1, b1) + pn1 := Pair(a1, bn1) + np1 := Pair(an1, b1) + if pn1.String() != np1.String() { + t.Error("Pairing mismatch: e(a, -b) != e(-a, b)") + } + if !PairingCheck([]*G1{a1, an1}, []*G2{b1, b1}) { + t.Error("MultiAte check gave false negative!") + } + p0 := new(GT).Add(p1, pn1) + p0_2 := Pair(a1, b0) + if p0.String() != p0_2.String() { + t.Error("Pairing mismatch: e(a, b) * e(a, -b) != 1") + } + p0_3 := new(GT).ScalarMult(p1, bigFromBase10("21888242871839275222246405745257275088548364400416034343698204186575808495617")) + if p0.String() != p0_3.String() { + t.Error("Pairing mismatch: e(a, b) has wrong order") + } + p2 := Pair(a2, b1) + p2_2 := Pair(a1, b2) + p2_3 := new(GT).ScalarMult(p1, bigFromBase10("2")) + if p2.String() != p2_2.String() { + t.Error("Pairing mismatch: e(a, b * 2) != e(a * 2, b)") + } + if p2.String() != p2_3.String() { + t.Error("Pairing mismatch: e(a, b * 2) != e(a, b) ** 2") + } + if p2.String() == p1.String() { + t.Error("Pairing is degenerate!") + } + if PairingCheck([]*G1{a1, a1}, []*G2{b1, b1}) { + t.Error("MultiAte check gave false positive!") + } + p999 := Pair(a37, b27) + p999_2 := Pair(a1, b999) + if p999.String() != p999_2.String() { + t.Error("Pairing mismatch: e(a * 37, b * 27) != e(a, b * 999)") + } +} diff --git a/crypto/bn256/google/optate.go b/crypto/bn256/google/optate.go new file mode 100644 index 000000000..9d6957062 --- /dev/null +++ b/crypto/bn256/google/optate.go @@ -0,0 +1,397 @@ +// 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 + +func lineFunctionAdd(r, p *twistPoint, q *curvePoint, r2 *gfP2, pool *bnPool) (a, b, c *gfP2, rOut *twistPoint) { + // See the mixed addition algorithm from "Faster Computation of the + // Tate Pairing", http://arxiv.org/pdf/0904.0854v3.pdf + + B := newGFp2(pool).Mul(p.x, r.t, pool) + + D := newGFp2(pool).Add(p.y, r.z) + D.Square(D, pool) + D.Sub(D, r2) + D.Sub(D, r.t) + D.Mul(D, r.t, pool) + + H := newGFp2(pool).Sub(B, r.x) + I := newGFp2(pool).Square(H, pool) + + E := newGFp2(pool).Add(I, I) + E.Add(E, E) + + J := newGFp2(pool).Mul(H, E, pool) + + L1 := newGFp2(pool).Sub(D, r.y) + L1.Sub(L1, r.y) + + V := newGFp2(pool).Mul(r.x, E, pool) + + rOut = newTwistPoint(pool) + rOut.x.Square(L1, pool) + rOut.x.Sub(rOut.x, J) + rOut.x.Sub(rOut.x, V) + rOut.x.Sub(rOut.x, V) + + rOut.z.Add(r.z, H) + rOut.z.Square(rOut.z, pool) + rOut.z.Sub(rOut.z, r.t) + rOut.z.Sub(rOut.z, I) + + t := newGFp2(pool).Sub(V, rOut.x) + t.Mul(t, L1, pool) + t2 := newGFp2(pool).Mul(r.y, J, pool) + t2.Add(t2, t2) + rOut.y.Sub(t, t2) + + rOut.t.Square(rOut.z, pool) + + t.Add(p.y, rOut.z) + t.Square(t, pool) + t.Sub(t, r2) + t.Sub(t, rOut.t) + + t2.Mul(L1, p.x, pool) + t2.Add(t2, t2) + a = newGFp2(pool) + a.Sub(t2, t) + + c = newGFp2(pool) + c.MulScalar(rOut.z, q.y) + c.Add(c, c) + + b = newGFp2(pool) + b.SetZero() + b.Sub(b, L1) + b.MulScalar(b, q.x) + b.Add(b, b) + + B.Put(pool) + D.Put(pool) + H.Put(pool) + I.Put(pool) + E.Put(pool) + J.Put(pool) + L1.Put(pool) + V.Put(pool) + t.Put(pool) + t2.Put(pool) + + return +} + +func lineFunctionDouble(r *twistPoint, q *curvePoint, pool *bnPool) (a, b, c *gfP2, rOut *twistPoint) { + // See the doubling algorithm for a=0 from "Faster Computation of the + // Tate Pairing", http://arxiv.org/pdf/0904.0854v3.pdf + + A := newGFp2(pool).Square(r.x, pool) + B := newGFp2(pool).Square(r.y, pool) + C_ := newGFp2(pool).Square(B, pool) + + D := newGFp2(pool).Add(r.x, B) + D.Square(D, pool) + D.Sub(D, A) + D.Sub(D, C_) + D.Add(D, D) + + E := newGFp2(pool).Add(A, A) + E.Add(E, A) + + G := newGFp2(pool).Square(E, pool) + + rOut = newTwistPoint(pool) + rOut.x.Sub(G, D) + rOut.x.Sub(rOut.x, D) + + rOut.z.Add(r.y, r.z) + rOut.z.Square(rOut.z, pool) + rOut.z.Sub(rOut.z, B) + rOut.z.Sub(rOut.z, r.t) + + rOut.y.Sub(D, rOut.x) + rOut.y.Mul(rOut.y, E, pool) + t := newGFp2(pool).Add(C_, C_) + t.Add(t, t) + t.Add(t, t) + rOut.y.Sub(rOut.y, t) + + rOut.t.Square(rOut.z, pool) + + t.Mul(E, r.t, pool) + t.Add(t, t) + b = newGFp2(pool) + b.SetZero() + b.Sub(b, t) + b.MulScalar(b, q.x) + + a = newGFp2(pool) + a.Add(r.x, E) + a.Square(a, pool) + a.Sub(a, A) + a.Sub(a, G) + t.Add(B, B) + t.Add(t, t) + a.Sub(a, t) + + c = newGFp2(pool) + c.Mul(rOut.z, r.t, pool) + c.Add(c, c) + c.MulScalar(c, q.y) + + A.Put(pool) + B.Put(pool) + C_.Put(pool) + D.Put(pool) + E.Put(pool) + G.Put(pool) + t.Put(pool) + + return +} + +func mulLine(ret *gfP12, a, b, c *gfP2, pool *bnPool) { + a2 := newGFp6(pool) + a2.x.SetZero() + a2.y.Set(a) + a2.z.Set(b) + a2.Mul(a2, ret.x, pool) + t3 := newGFp6(pool).MulScalar(ret.y, c, pool) + + t := newGFp2(pool) + t.Add(b, c) + t2 := newGFp6(pool) + t2.x.SetZero() + t2.y.Set(a) + t2.z.Set(t) + ret.x.Add(ret.x, ret.y) + + ret.y.Set(t3) + + ret.x.Mul(ret.x, t2, pool) + ret.x.Sub(ret.x, a2) + ret.x.Sub(ret.x, ret.y) + a2.MulTau(a2, pool) + ret.y.Add(ret.y, a2) + + a2.Put(pool) + t3.Put(pool) + t2.Put(pool) + t.Put(pool) +} + +// sixuPlus2NAF is 6u+2 in non-adjacent form. +var sixuPlus2NAF = []int8{0, 0, 0, 1, 0, 1, 0, -1, 0, 0, 1, -1, 0, 0, 1, 0, + 0, 1, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 1, 1, + 1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1, + 1, 0, 0, -1, 0, 0, 0, 1, 1, 0, -1, 0, 0, 1, 0, 1, 1} + +// miller implements the Miller loop for calculating the Optimal Ate pairing. +// See algorithm 1 from http://cryptojedi.org/papers/dclxvi-20100714.pdf +func miller(q *twistPoint, p *curvePoint, pool *bnPool) *gfP12 { + ret := newGFp12(pool) + ret.SetOne() + + aAffine := newTwistPoint(pool) + aAffine.Set(q) + aAffine.MakeAffine(pool) + + bAffine := newCurvePoint(pool) + bAffine.Set(p) + bAffine.MakeAffine(pool) + + minusA := newTwistPoint(pool) + minusA.Negative(aAffine, pool) + + r := newTwistPoint(pool) + r.Set(aAffine) + + r2 := newGFp2(pool) + r2.Square(aAffine.y, pool) + + for i := len(sixuPlus2NAF) - 1; i > 0; i-- { + a, b, c, newR := lineFunctionDouble(r, bAffine, pool) + if i != len(sixuPlus2NAF)-1 { + ret.Square(ret, pool) + } + + mulLine(ret, a, b, c, pool) + a.Put(pool) + b.Put(pool) + c.Put(pool) + r.Put(pool) + r = newR + + switch sixuPlus2NAF[i-1] { + case 1: + a, b, c, newR = lineFunctionAdd(r, aAffine, bAffine, r2, pool) + case -1: + a, b, c, newR = lineFunctionAdd(r, minusA, bAffine, r2, pool) + default: + continue + } + + mulLine(ret, a, b, c, pool) + a.Put(pool) + b.Put(pool) + c.Put(pool) + r.Put(pool) + r = newR + } + + // In order to calculate Q1 we have to convert q from the sextic twist + // to the full GF(p^12) group, apply the Frobenius there, and convert + // back. + // + // The twist isomorphism is (x', y') -> (xω², yω³). If we consider just + // x for a moment, then after applying the Frobenius, we have x̄ω^(2p) + // where x̄ is the conjugate of x. If we are going to apply the inverse + // isomorphism we need a value with a single coefficient of ω² so we + // rewrite this as x̄ω^(2p-2)ω². ξ⁶ = ω and, due to the construction of + // p, 2p-2 is a multiple of six. Therefore we can rewrite as + // x̄ξ^((p-1)/3)ω² and applying the inverse isomorphism eliminates the + // ω². + // + // A similar argument can be made for the y value. + + q1 := newTwistPoint(pool) + q1.x.Conjugate(aAffine.x) + q1.x.Mul(q1.x, xiToPMinus1Over3, pool) + q1.y.Conjugate(aAffine.y) + q1.y.Mul(q1.y, xiToPMinus1Over2, pool) + q1.z.SetOne() + q1.t.SetOne() + + // For Q2 we are applying the p² Frobenius. The two conjugations cancel + // out and we are left only with the factors from the isomorphism. In + // the case of x, we end up with a pure number which is why + // xiToPSquaredMinus1Over3 is ∈ GF(p). With y we get a factor of -1. We + // ignore this to end up with -Q2. + + minusQ2 := newTwistPoint(pool) + minusQ2.x.MulScalar(aAffine.x, xiToPSquaredMinus1Over3) + minusQ2.y.Set(aAffine.y) + minusQ2.z.SetOne() + minusQ2.t.SetOne() + + r2.Square(q1.y, pool) + a, b, c, newR := lineFunctionAdd(r, q1, bAffine, r2, pool) + mulLine(ret, a, b, c, pool) + a.Put(pool) + b.Put(pool) + c.Put(pool) + r.Put(pool) + r = newR + + r2.Square(minusQ2.y, pool) + a, b, c, newR = lineFunctionAdd(r, minusQ2, bAffine, r2, pool) + mulLine(ret, a, b, c, pool) + a.Put(pool) + b.Put(pool) + c.Put(pool) + r.Put(pool) + r = newR + + aAffine.Put(pool) + bAffine.Put(pool) + minusA.Put(pool) + r.Put(pool) + r2.Put(pool) + + return ret +} + +// finalExponentiation computes the (p¹²-1)/Order-th power of an element of +// GF(p¹²) to obtain an element of GT (steps 13-15 of algorithm 1 from +// http://cryptojedi.org/papers/dclxvi-20100714.pdf) +func finalExponentiation(in *gfP12, pool *bnPool) *gfP12 { + t1 := newGFp12(pool) + + // This is the p^6-Frobenius + t1.x.Negative(in.x) + t1.y.Set(in.y) + + inv := newGFp12(pool) + inv.Invert(in, pool) + t1.Mul(t1, inv, pool) + + t2 := newGFp12(pool).FrobeniusP2(t1, pool) + t1.Mul(t1, t2, pool) + + fp := newGFp12(pool).Frobenius(t1, pool) + fp2 := newGFp12(pool).FrobeniusP2(t1, pool) + fp3 := newGFp12(pool).Frobenius(fp2, pool) + + fu, fu2, fu3 := newGFp12(pool), newGFp12(pool), newGFp12(pool) + fu.Exp(t1, u, pool) + fu2.Exp(fu, u, pool) + fu3.Exp(fu2, u, pool) + + y3 := newGFp12(pool).Frobenius(fu, pool) + fu2p := newGFp12(pool).Frobenius(fu2, pool) + fu3p := newGFp12(pool).Frobenius(fu3, pool) + y2 := newGFp12(pool).FrobeniusP2(fu2, pool) + + y0 := newGFp12(pool) + y0.Mul(fp, fp2, pool) + y0.Mul(y0, fp3, pool) + + y1, y4, y5 := newGFp12(pool), newGFp12(pool), newGFp12(pool) + y1.Conjugate(t1) + y5.Conjugate(fu2) + y3.Conjugate(y3) + y4.Mul(fu, fu2p, pool) + y4.Conjugate(y4) + + y6 := newGFp12(pool) + y6.Mul(fu3, fu3p, pool) + y6.Conjugate(y6) + + t0 := newGFp12(pool) + t0.Square(y6, pool) + t0.Mul(t0, y4, pool) + t0.Mul(t0, y5, pool) + t1.Mul(y3, y5, pool) + t1.Mul(t1, t0, pool) + t0.Mul(t0, y2, pool) + t1.Square(t1, pool) + t1.Mul(t1, t0, pool) + t1.Square(t1, pool) + t0.Mul(t1, y1, pool) + t1.Mul(t1, y0, pool) + t0.Square(t0, pool) + t0.Mul(t0, t1, pool) + + inv.Put(pool) + t1.Put(pool) + t2.Put(pool) + fp.Put(pool) + fp2.Put(pool) + fp3.Put(pool) + fu.Put(pool) + fu2.Put(pool) + fu3.Put(pool) + fu2p.Put(pool) + fu3p.Put(pool) + y0.Put(pool) + y1.Put(pool) + y2.Put(pool) + y3.Put(pool) + y4.Put(pool) + y5.Put(pool) + y6.Put(pool) + + return t0 +} + +func optimalAte(a *twistPoint, b *curvePoint, pool *bnPool) *gfP12 { + e := miller(a, b, pool) + ret := finalExponentiation(e, pool) + e.Put(pool) + + if a.IsInfinity() || b.IsInfinity() { + ret.SetOne() + } + return ret +} diff --git a/crypto/bn256/google/twist.go b/crypto/bn256/google/twist.go new file mode 100644 index 000000000..1f5a4d9de --- /dev/null +++ b/crypto/bn256/google/twist.go @@ -0,0 +1,255 @@ +// 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 + +import ( + "math/big" +) + +// twistPoint implements the elliptic curve y²=x³+3/ξ over GF(p²). Points are +// kept in Jacobian form and t=z² when valid. The group G₂ is the set of +// n-torsion points of this curve over GF(p²) (where n = Order) +type twistPoint struct { + x, y, z, t *gfP2 +} + +var twistB = &gfP2{ + bigFromBase10("266929791119991161246907387137283842545076965332900288569378510910307636690"), + bigFromBase10("19485874751759354771024239261021720505790618469301721065564631296452457478373"), +} + +// twistGen is the generator of group G₂. +var twistGen = &twistPoint{ + &gfP2{ + bigFromBase10("11559732032986387107991004021392285783925812861821192530917403151452391805634"), + bigFromBase10("10857046999023057135944570762232829481370756359578518086990519993285655852781"), + }, + &gfP2{ + bigFromBase10("4082367875863433681332203403145435568316851327593401208105741076214120093531"), + bigFromBase10("8495653923123431417604973247489272438418190587263600148770280649306958101930"), + }, + &gfP2{ + bigFromBase10("0"), + bigFromBase10("1"), + }, + &gfP2{ + bigFromBase10("0"), + bigFromBase10("1"), + }, +} + +func newTwistPoint(pool *bnPool) *twistPoint { + return &twistPoint{ + newGFp2(pool), + newGFp2(pool), + newGFp2(pool), + newGFp2(pool), + } +} + +func (c *twistPoint) String() string { + return "(" + c.x.String() + ", " + c.y.String() + ", " + c.z.String() + ")" +} + +func (c *twistPoint) Put(pool *bnPool) { + c.x.Put(pool) + c.y.Put(pool) + c.z.Put(pool) + c.t.Put(pool) +} + +func (c *twistPoint) Set(a *twistPoint) { + c.x.Set(a.x) + c.y.Set(a.y) + c.z.Set(a.z) + c.t.Set(a.t) +} + +// IsOnCurve returns true iff c is on the curve where c must be in affine form. +func (c *twistPoint) IsOnCurve() bool { + pool := new(bnPool) + yy := newGFp2(pool).Square(c.y, pool) + xxx := newGFp2(pool).Square(c.x, pool) + xxx.Mul(xxx, c.x, pool) + yy.Sub(yy, xxx) + yy.Sub(yy, twistB) + yy.Minimal() + + if yy.x.Sign() != 0 || yy.y.Sign() != 0 { + return false + } + cneg := newTwistPoint(pool) + cneg.Mul(c, Order, pool) + return cneg.z.IsZero() +} + +func (c *twistPoint) SetInfinity() { + c.z.SetZero() +} + +func (c *twistPoint) IsInfinity() bool { + return c.z.IsZero() +} + +func (c *twistPoint) Add(a, b *twistPoint, pool *bnPool) { + // For additional comments, see the same function in curve.go. + + if a.IsInfinity() { + c.Set(b) + return + } + if b.IsInfinity() { + c.Set(a) + return + } + + // See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/addition/add-2007-bl.op3 + z1z1 := newGFp2(pool).Square(a.z, pool) + z2z2 := newGFp2(pool).Square(b.z, pool) + u1 := newGFp2(pool).Mul(a.x, z2z2, pool) + u2 := newGFp2(pool).Mul(b.x, z1z1, pool) + + t := newGFp2(pool).Mul(b.z, z2z2, pool) + s1 := newGFp2(pool).Mul(a.y, t, pool) + + t.Mul(a.z, z1z1, pool) + s2 := newGFp2(pool).Mul(b.y, t, pool) + + h := newGFp2(pool).Sub(u2, u1) + xEqual := h.IsZero() + + t.Add(h, h) + i := newGFp2(pool).Square(t, pool) + j := newGFp2(pool).Mul(h, i, pool) + + t.Sub(s2, s1) + yEqual := t.IsZero() + if xEqual && yEqual { + c.Double(a, pool) + return + } + r := newGFp2(pool).Add(t, t) + + v := newGFp2(pool).Mul(u1, i, pool) + + t4 := newGFp2(pool).Square(r, pool) + t.Add(v, v) + t6 := newGFp2(pool).Sub(t4, j) + c.x.Sub(t6, t) + + t.Sub(v, c.x) // t7 + t4.Mul(s1, j, pool) // t8 + t6.Add(t4, t4) // t9 + t4.Mul(r, t, pool) // t10 + c.y.Sub(t4, t6) + + t.Add(a.z, b.z) // t11 + t4.Square(t, pool) // t12 + t.Sub(t4, z1z1) // t13 + t4.Sub(t, z2z2) // t14 + c.z.Mul(t4, h, pool) + + z1z1.Put(pool) + z2z2.Put(pool) + u1.Put(pool) + u2.Put(pool) + t.Put(pool) + s1.Put(pool) + s2.Put(pool) + h.Put(pool) + i.Put(pool) + j.Put(pool) + r.Put(pool) + v.Put(pool) + t4.Put(pool) + t6.Put(pool) +} + +func (c *twistPoint) Double(a *twistPoint, pool *bnPool) { + // See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/doubling/dbl-2009-l.op3 + A := newGFp2(pool).Square(a.x, pool) + B := newGFp2(pool).Square(a.y, pool) + C_ := newGFp2(pool).Square(B, pool) + + t := newGFp2(pool).Add(a.x, B) + t2 := newGFp2(pool).Square(t, pool) + t.Sub(t2, A) + t2.Sub(t, C_) + d := newGFp2(pool).Add(t2, t2) + t.Add(A, A) + e := newGFp2(pool).Add(t, A) + f := newGFp2(pool).Square(e, pool) + + t.Add(d, d) + c.x.Sub(f, t) + + t.Add(C_, C_) + t2.Add(t, t) + t.Add(t2, t2) + c.y.Sub(d, c.x) + t2.Mul(e, c.y, pool) + c.y.Sub(t2, t) + + t.Mul(a.y, a.z, pool) + c.z.Add(t, t) + + A.Put(pool) + B.Put(pool) + C_.Put(pool) + t.Put(pool) + t2.Put(pool) + d.Put(pool) + e.Put(pool) + f.Put(pool) +} + +func (c *twistPoint) Mul(a *twistPoint, scalar *big.Int, pool *bnPool) *twistPoint { + sum := newTwistPoint(pool) + sum.SetInfinity() + t := newTwistPoint(pool) + + for i := scalar.BitLen(); i >= 0; i-- { + t.Double(sum, pool) + if scalar.Bit(i) != 0 { + sum.Add(t, a, pool) + } else { + sum.Set(t) + } + } + + c.Set(sum) + sum.Put(pool) + t.Put(pool) + return c +} + +func (c *twistPoint) MakeAffine(pool *bnPool) *twistPoint { + if c.z.IsOne() { + return c + } + + zInv := newGFp2(pool).Invert(c.z, pool) + t := newGFp2(pool).Mul(c.y, zInv, pool) + zInv2 := newGFp2(pool).Square(zInv, pool) + c.y.Mul(t, zInv2, pool) + t.Mul(c.x, zInv2, pool) + c.x.Set(t) + c.z.SetOne() + c.t.SetOne() + + zInv.Put(pool) + t.Put(pool) + zInv2.Put(pool) + + return c +} + +func (c *twistPoint) Negative(a *twistPoint, pool *bnPool) { + c.x.Set(a.x) + c.y.SetZero() + c.y.Sub(c.y, a.y) + c.z.Set(a.z) + c.t.SetZero() +} diff --git a/crypto/bn256/main_test.go b/crypto/bn256/main_test.go deleted file mode 100644 index 0230f1b19..000000000 --- a/crypto/bn256/main_test.go +++ /dev/null @@ -1,71 +0,0 @@ -package bn256 - -import ( - "testing" - - "crypto/rand" -) - -func TestRandomG2Marshal(t *testing.T) { - for i := 0; i < 10; i++ { - n, g2, err := RandomG2(rand.Reader) - if err != nil { - t.Error(err) - continue - } - t.Logf("%d: %x\n", n, g2.Marshal()) - } -} - -func TestPairings(t *testing.T) { - a1 := new(G1).ScalarBaseMult(bigFromBase10("1")) - a2 := new(G1).ScalarBaseMult(bigFromBase10("2")) - a37 := new(G1).ScalarBaseMult(bigFromBase10("37")) - an1 := new(G1).ScalarBaseMult(bigFromBase10("21888242871839275222246405745257275088548364400416034343698204186575808495616")) - - b0 := new(G2).ScalarBaseMult(bigFromBase10("0")) - b1 := new(G2).ScalarBaseMult(bigFromBase10("1")) - b2 := new(G2).ScalarBaseMult(bigFromBase10("2")) - b27 := new(G2).ScalarBaseMult(bigFromBase10("27")) - b999 := new(G2).ScalarBaseMult(bigFromBase10("999")) - bn1 := new(G2).ScalarBaseMult(bigFromBase10("21888242871839275222246405745257275088548364400416034343698204186575808495616")) - - p1 := Pair(a1, b1) - pn1 := Pair(a1, bn1) - np1 := Pair(an1, b1) - if pn1.String() != np1.String() { - t.Error("Pairing mismatch: e(a, -b) != e(-a, b)") - } - if !PairingCheck([]*G1{a1, an1}, []*G2{b1, b1}) { - t.Error("MultiAte check gave false negative!") - } - p0 := new(GT).Add(p1, pn1) - p0_2 := Pair(a1, b0) - if p0.String() != p0_2.String() { - t.Error("Pairing mismatch: e(a, b) * e(a, -b) != 1") - } - p0_3 := new(GT).ScalarMult(p1, bigFromBase10("21888242871839275222246405745257275088548364400416034343698204186575808495617")) - if p0.String() != p0_3.String() { - t.Error("Pairing mismatch: e(a, b) has wrong order") - } - p2 := Pair(a2, b1) - p2_2 := Pair(a1, b2) - p2_3 := new(GT).ScalarMult(p1, bigFromBase10("2")) - if p2.String() != p2_2.String() { - t.Error("Pairing mismatch: e(a, b * 2) != e(a * 2, b)") - } - if p2.String() != p2_3.String() { - t.Error("Pairing mismatch: e(a, b * 2) != e(a, b) ** 2") - } - if p2.String() == p1.String() { - t.Error("Pairing is degenerate!") - } - if PairingCheck([]*G1{a1, a1}, []*G2{b1, b1}) { - t.Error("MultiAte check gave false positive!") - } - p999 := Pair(a37, b27) - p999_2 := Pair(a1, b999) - if p999.String() != p999_2.String() { - t.Error("Pairing mismatch: e(a * 37, b * 27) != e(a, b * 999)") - } -} diff --git a/crypto/bn256/optate.go b/crypto/bn256/optate.go deleted file mode 100644 index 9d6957062..000000000 --- a/crypto/bn256/optate.go +++ /dev/null @@ -1,397 +0,0 @@ -// 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 - -func lineFunctionAdd(r, p *twistPoint, q *curvePoint, r2 *gfP2, pool *bnPool) (a, b, c *gfP2, rOut *twistPoint) { - // See the mixed addition algorithm from "Faster Computation of the - // Tate Pairing", http://arxiv.org/pdf/0904.0854v3.pdf - - B := newGFp2(pool).Mul(p.x, r.t, pool) - - D := newGFp2(pool).Add(p.y, r.z) - D.Square(D, pool) - D.Sub(D, r2) - D.Sub(D, r.t) - D.Mul(D, r.t, pool) - - H := newGFp2(pool).Sub(B, r.x) - I := newGFp2(pool).Square(H, pool) - - E := newGFp2(pool).Add(I, I) - E.Add(E, E) - - J := newGFp2(pool).Mul(H, E, pool) - - L1 := newGFp2(pool).Sub(D, r.y) - L1.Sub(L1, r.y) - - V := newGFp2(pool).Mul(r.x, E, pool) - - rOut = newTwistPoint(pool) - rOut.x.Square(L1, pool) - rOut.x.Sub(rOut.x, J) - rOut.x.Sub(rOut.x, V) - rOut.x.Sub(rOut.x, V) - - rOut.z.Add(r.z, H) - rOut.z.Square(rOut.z, pool) - rOut.z.Sub(rOut.z, r.t) - rOut.z.Sub(rOut.z, I) - - t := newGFp2(pool).Sub(V, rOut.x) - t.Mul(t, L1, pool) - t2 := newGFp2(pool).Mul(r.y, J, pool) - t2.Add(t2, t2) - rOut.y.Sub(t, t2) - - rOut.t.Square(rOut.z, pool) - - t.Add(p.y, rOut.z) - t.Square(t, pool) - t.Sub(t, r2) - t.Sub(t, rOut.t) - - t2.Mul(L1, p.x, pool) - t2.Add(t2, t2) - a = newGFp2(pool) - a.Sub(t2, t) - - c = newGFp2(pool) - c.MulScalar(rOut.z, q.y) - c.Add(c, c) - - b = newGFp2(pool) - b.SetZero() - b.Sub(b, L1) - b.MulScalar(b, q.x) - b.Add(b, b) - - B.Put(pool) - D.Put(pool) - H.Put(pool) - I.Put(pool) - E.Put(pool) - J.Put(pool) - L1.Put(pool) - V.Put(pool) - t.Put(pool) - t2.Put(pool) - - return -} - -func lineFunctionDouble(r *twistPoint, q *curvePoint, pool *bnPool) (a, b, c *gfP2, rOut *twistPoint) { - // See the doubling algorithm for a=0 from "Faster Computation of the - // Tate Pairing", http://arxiv.org/pdf/0904.0854v3.pdf - - A := newGFp2(pool).Square(r.x, pool) - B := newGFp2(pool).Square(r.y, pool) - C_ := newGFp2(pool).Square(B, pool) - - D := newGFp2(pool).Add(r.x, B) - D.Square(D, pool) - D.Sub(D, A) - D.Sub(D, C_) - D.Add(D, D) - - E := newGFp2(pool).Add(A, A) - E.Add(E, A) - - G := newGFp2(pool).Square(E, pool) - - rOut = newTwistPoint(pool) - rOut.x.Sub(G, D) - rOut.x.Sub(rOut.x, D) - - rOut.z.Add(r.y, r.z) - rOut.z.Square(rOut.z, pool) - rOut.z.Sub(rOut.z, B) - rOut.z.Sub(rOut.z, r.t) - - rOut.y.Sub(D, rOut.x) - rOut.y.Mul(rOut.y, E, pool) - t := newGFp2(pool).Add(C_, C_) - t.Add(t, t) - t.Add(t, t) - rOut.y.Sub(rOut.y, t) - - rOut.t.Square(rOut.z, pool) - - t.Mul(E, r.t, pool) - t.Add(t, t) - b = newGFp2(pool) - b.SetZero() - b.Sub(b, t) - b.MulScalar(b, q.x) - - a = newGFp2(pool) - a.Add(r.x, E) - a.Square(a, pool) - a.Sub(a, A) - a.Sub(a, G) - t.Add(B, B) - t.Add(t, t) - a.Sub(a, t) - - c = newGFp2(pool) - c.Mul(rOut.z, r.t, pool) - c.Add(c, c) - c.MulScalar(c, q.y) - - A.Put(pool) - B.Put(pool) - C_.Put(pool) - D.Put(pool) - E.Put(pool) - G.Put(pool) - t.Put(pool) - - return -} - -func mulLine(ret *gfP12, a, b, c *gfP2, pool *bnPool) { - a2 := newGFp6(pool) - a2.x.SetZero() - a2.y.Set(a) - a2.z.Set(b) - a2.Mul(a2, ret.x, pool) - t3 := newGFp6(pool).MulScalar(ret.y, c, pool) - - t := newGFp2(pool) - t.Add(b, c) - t2 := newGFp6(pool) - t2.x.SetZero() - t2.y.Set(a) - t2.z.Set(t) - ret.x.Add(ret.x, ret.y) - - ret.y.Set(t3) - - ret.x.Mul(ret.x, t2, pool) - ret.x.Sub(ret.x, a2) - ret.x.Sub(ret.x, ret.y) - a2.MulTau(a2, pool) - ret.y.Add(ret.y, a2) - - a2.Put(pool) - t3.Put(pool) - t2.Put(pool) - t.Put(pool) -} - -// sixuPlus2NAF is 6u+2 in non-adjacent form. -var sixuPlus2NAF = []int8{0, 0, 0, 1, 0, 1, 0, -1, 0, 0, 1, -1, 0, 0, 1, 0, - 0, 1, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 1, 1, - 1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1, - 1, 0, 0, -1, 0, 0, 0, 1, 1, 0, -1, 0, 0, 1, 0, 1, 1} - -// miller implements the Miller loop for calculating the Optimal Ate pairing. -// See algorithm 1 from http://cryptojedi.org/papers/dclxvi-20100714.pdf -func miller(q *twistPoint, p *curvePoint, pool *bnPool) *gfP12 { - ret := newGFp12(pool) - ret.SetOne() - - aAffine := newTwistPoint(pool) - aAffine.Set(q) - aAffine.MakeAffine(pool) - - bAffine := newCurvePoint(pool) - bAffine.Set(p) - bAffine.MakeAffine(pool) - - minusA := newTwistPoint(pool) - minusA.Negative(aAffine, pool) - - r := newTwistPoint(pool) - r.Set(aAffine) - - r2 := newGFp2(pool) - r2.Square(aAffine.y, pool) - - for i := len(sixuPlus2NAF) - 1; i > 0; i-- { - a, b, c, newR := lineFunctionDouble(r, bAffine, pool) - if i != len(sixuPlus2NAF)-1 { - ret.Square(ret, pool) - } - - mulLine(ret, a, b, c, pool) - a.Put(pool) - b.Put(pool) - c.Put(pool) - r.Put(pool) - r = newR - - switch sixuPlus2NAF[i-1] { - case 1: - a, b, c, newR = lineFunctionAdd(r, aAffine, bAffine, r2, pool) - case -1: - a, b, c, newR = lineFunctionAdd(r, minusA, bAffine, r2, pool) - default: - continue - } - - mulLine(ret, a, b, c, pool) - a.Put(pool) - b.Put(pool) - c.Put(pool) - r.Put(pool) - r = newR - } - - // In order to calculate Q1 we have to convert q from the sextic twist - // to the full GF(p^12) group, apply the Frobenius there, and convert - // back. - // - // The twist isomorphism is (x', y') -> (xω², yω³). If we consider just - // x for a moment, then after applying the Frobenius, we have x̄ω^(2p) - // where x̄ is the conjugate of x. If we are going to apply the inverse - // isomorphism we need a value with a single coefficient of ω² so we - // rewrite this as x̄ω^(2p-2)ω². ξ⁶ = ω and, due to the construction of - // p, 2p-2 is a multiple of six. Therefore we can rewrite as - // x̄ξ^((p-1)/3)ω² and applying the inverse isomorphism eliminates the - // ω². - // - // A similar argument can be made for the y value. - - q1 := newTwistPoint(pool) - q1.x.Conjugate(aAffine.x) - q1.x.Mul(q1.x, xiToPMinus1Over3, pool) - q1.y.Conjugate(aAffine.y) - q1.y.Mul(q1.y, xiToPMinus1Over2, pool) - q1.z.SetOne() - q1.t.SetOne() - - // For Q2 we are applying the p² Frobenius. The two conjugations cancel - // out and we are left only with the factors from the isomorphism. In - // the case of x, we end up with a pure number which is why - // xiToPSquaredMinus1Over3 is ∈ GF(p). With y we get a factor of -1. We - // ignore this to end up with -Q2. - - minusQ2 := newTwistPoint(pool) - minusQ2.x.MulScalar(aAffine.x, xiToPSquaredMinus1Over3) - minusQ2.y.Set(aAffine.y) - minusQ2.z.SetOne() - minusQ2.t.SetOne() - - r2.Square(q1.y, pool) - a, b, c, newR := lineFunctionAdd(r, q1, bAffine, r2, pool) - mulLine(ret, a, b, c, pool) - a.Put(pool) - b.Put(pool) - c.Put(pool) - r.Put(pool) - r = newR - - r2.Square(minusQ2.y, pool) - a, b, c, newR = lineFunctionAdd(r, minusQ2, bAffine, r2, pool) - mulLine(ret, a, b, c, pool) - a.Put(pool) - b.Put(pool) - c.Put(pool) - r.Put(pool) - r = newR - - aAffine.Put(pool) - bAffine.Put(pool) - minusA.Put(pool) - r.Put(pool) - r2.Put(pool) - - return ret -} - -// finalExponentiation computes the (p¹²-1)/Order-th power of an element of -// GF(p¹²) to obtain an element of GT (steps 13-15 of algorithm 1 from -// http://cryptojedi.org/papers/dclxvi-20100714.pdf) -func finalExponentiation(in *gfP12, pool *bnPool) *gfP12 { - t1 := newGFp12(pool) - - // This is the p^6-Frobenius - t1.x.Negative(in.x) - t1.y.Set(in.y) - - inv := newGFp12(pool) - inv.Invert(in, pool) - t1.Mul(t1, inv, pool) - - t2 := newGFp12(pool).FrobeniusP2(t1, pool) - t1.Mul(t1, t2, pool) - - fp := newGFp12(pool).Frobenius(t1, pool) - fp2 := newGFp12(pool).FrobeniusP2(t1, pool) - fp3 := newGFp12(pool).Frobenius(fp2, pool) - - fu, fu2, fu3 := newGFp12(pool), newGFp12(pool), newGFp12(pool) - fu.Exp(t1, u, pool) - fu2.Exp(fu, u, pool) - fu3.Exp(fu2, u, pool) - - y3 := newGFp12(pool).Frobenius(fu, pool) - fu2p := newGFp12(pool).Frobenius(fu2, pool) - fu3p := newGFp12(pool).Frobenius(fu3, pool) - y2 := newGFp12(pool).FrobeniusP2(fu2, pool) - - y0 := newGFp12(pool) - y0.Mul(fp, fp2, pool) - y0.Mul(y0, fp3, pool) - - y1, y4, y5 := newGFp12(pool), newGFp12(pool), newGFp12(pool) - y1.Conjugate(t1) - y5.Conjugate(fu2) - y3.Conjugate(y3) - y4.Mul(fu, fu2p, pool) - y4.Conjugate(y4) - - y6 := newGFp12(pool) - y6.Mul(fu3, fu3p, pool) - y6.Conjugate(y6) - - t0 := newGFp12(pool) - t0.Square(y6, pool) - t0.Mul(t0, y4, pool) - t0.Mul(t0, y5, pool) - t1.Mul(y3, y5, pool) - t1.Mul(t1, t0, pool) - t0.Mul(t0, y2, pool) - t1.Square(t1, pool) - t1.Mul(t1, t0, pool) - t1.Square(t1, pool) - t0.Mul(t1, y1, pool) - t1.Mul(t1, y0, pool) - t0.Square(t0, pool) - t0.Mul(t0, t1, pool) - - inv.Put(pool) - t1.Put(pool) - t2.Put(pool) - fp.Put(pool) - fp2.Put(pool) - fp3.Put(pool) - fu.Put(pool) - fu2.Put(pool) - fu3.Put(pool) - fu2p.Put(pool) - fu3p.Put(pool) - y0.Put(pool) - y1.Put(pool) - y2.Put(pool) - y3.Put(pool) - y4.Put(pool) - y5.Put(pool) - y6.Put(pool) - - return t0 -} - -func optimalAte(a *twistPoint, b *curvePoint, pool *bnPool) *gfP12 { - e := miller(a, b, pool) - ret := finalExponentiation(e, pool) - e.Put(pool) - - if a.IsInfinity() || b.IsInfinity() { - ret.SetOne() - } - return ret -} diff --git a/crypto/bn256/twist.go b/crypto/bn256/twist.go deleted file mode 100644 index 95b966e2e..000000000 --- a/crypto/bn256/twist.go +++ /dev/null @@ -1,249 +0,0 @@ -// 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 - -import ( - "math/big" -) - -// twistPoint implements the elliptic curve y²=x³+3/ξ over GF(p²). Points are -// kept in Jacobian form and t=z² when valid. The group G₂ is the set of -// n-torsion points of this curve over GF(p²) (where n = Order) -type twistPoint struct { - x, y, z, t *gfP2 -} - -var twistB = &gfP2{ - bigFromBase10("266929791119991161246907387137283842545076965332900288569378510910307636690"), - bigFromBase10("19485874751759354771024239261021720505790618469301721065564631296452457478373"), -} - -// twistGen is the generator of group G₂. -var twistGen = &twistPoint{ - &gfP2{ - bigFromBase10("11559732032986387107991004021392285783925812861821192530917403151452391805634"), - bigFromBase10("10857046999023057135944570762232829481370756359578518086990519993285655852781"), - }, - &gfP2{ - bigFromBase10("4082367875863433681332203403145435568316851327593401208105741076214120093531"), - bigFromBase10("8495653923123431417604973247489272438418190587263600148770280649306958101930"), - }, - &gfP2{ - bigFromBase10("0"), - bigFromBase10("1"), - }, - &gfP2{ - bigFromBase10("0"), - bigFromBase10("1"), - }, -} - -func newTwistPoint(pool *bnPool) *twistPoint { - return &twistPoint{ - newGFp2(pool), - newGFp2(pool), - newGFp2(pool), - newGFp2(pool), - } -} - -func (c *twistPoint) String() string { - return "(" + c.x.String() + ", " + c.y.String() + ", " + c.z.String() + ")" -} - -func (c *twistPoint) Put(pool *bnPool) { - c.x.Put(pool) - c.y.Put(pool) - c.z.Put(pool) - c.t.Put(pool) -} - -func (c *twistPoint) Set(a *twistPoint) { - c.x.Set(a.x) - c.y.Set(a.y) - c.z.Set(a.z) - c.t.Set(a.t) -} - -// IsOnCurve returns true iff c is on the curve where c must be in affine form. -func (c *twistPoint) IsOnCurve() bool { - pool := new(bnPool) - yy := newGFp2(pool).Square(c.y, pool) - xxx := newGFp2(pool).Square(c.x, pool) - xxx.Mul(xxx, c.x, pool) - yy.Sub(yy, xxx) - yy.Sub(yy, twistB) - yy.Minimal() - return yy.x.Sign() == 0 && yy.y.Sign() == 0 -} - -func (c *twistPoint) SetInfinity() { - c.z.SetZero() -} - -func (c *twistPoint) IsInfinity() bool { - return c.z.IsZero() -} - -func (c *twistPoint) Add(a, b *twistPoint, pool *bnPool) { - // For additional comments, see the same function in curve.go. - - if a.IsInfinity() { - c.Set(b) - return - } - if b.IsInfinity() { - c.Set(a) - return - } - - // See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/addition/add-2007-bl.op3 - z1z1 := newGFp2(pool).Square(a.z, pool) - z2z2 := newGFp2(pool).Square(b.z, pool) - u1 := newGFp2(pool).Mul(a.x, z2z2, pool) - u2 := newGFp2(pool).Mul(b.x, z1z1, pool) - - t := newGFp2(pool).Mul(b.z, z2z2, pool) - s1 := newGFp2(pool).Mul(a.y, t, pool) - - t.Mul(a.z, z1z1, pool) - s2 := newGFp2(pool).Mul(b.y, t, pool) - - h := newGFp2(pool).Sub(u2, u1) - xEqual := h.IsZero() - - t.Add(h, h) - i := newGFp2(pool).Square(t, pool) - j := newGFp2(pool).Mul(h, i, pool) - - t.Sub(s2, s1) - yEqual := t.IsZero() - if xEqual && yEqual { - c.Double(a, pool) - return - } - r := newGFp2(pool).Add(t, t) - - v := newGFp2(pool).Mul(u1, i, pool) - - t4 := newGFp2(pool).Square(r, pool) - t.Add(v, v) - t6 := newGFp2(pool).Sub(t4, j) - c.x.Sub(t6, t) - - t.Sub(v, c.x) // t7 - t4.Mul(s1, j, pool) // t8 - t6.Add(t4, t4) // t9 - t4.Mul(r, t, pool) // t10 - c.y.Sub(t4, t6) - - t.Add(a.z, b.z) // t11 - t4.Square(t, pool) // t12 - t.Sub(t4, z1z1) // t13 - t4.Sub(t, z2z2) // t14 - c.z.Mul(t4, h, pool) - - z1z1.Put(pool) - z2z2.Put(pool) - u1.Put(pool) - u2.Put(pool) - t.Put(pool) - s1.Put(pool) - s2.Put(pool) - h.Put(pool) - i.Put(pool) - j.Put(pool) - r.Put(pool) - v.Put(pool) - t4.Put(pool) - t6.Put(pool) -} - -func (c *twistPoint) Double(a *twistPoint, pool *bnPool) { - // See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/doubling/dbl-2009-l.op3 - A := newGFp2(pool).Square(a.x, pool) - B := newGFp2(pool).Square(a.y, pool) - C_ := newGFp2(pool).Square(B, pool) - - t := newGFp2(pool).Add(a.x, B) - t2 := newGFp2(pool).Square(t, pool) - t.Sub(t2, A) - t2.Sub(t, C_) - d := newGFp2(pool).Add(t2, t2) - t.Add(A, A) - e := newGFp2(pool).Add(t, A) - f := newGFp2(pool).Square(e, pool) - - t.Add(d, d) - c.x.Sub(f, t) - - t.Add(C_, C_) - t2.Add(t, t) - t.Add(t2, t2) - c.y.Sub(d, c.x) - t2.Mul(e, c.y, pool) - c.y.Sub(t2, t) - - t.Mul(a.y, a.z, pool) - c.z.Add(t, t) - - A.Put(pool) - B.Put(pool) - C_.Put(pool) - t.Put(pool) - t2.Put(pool) - d.Put(pool) - e.Put(pool) - f.Put(pool) -} - -func (c *twistPoint) Mul(a *twistPoint, scalar *big.Int, pool *bnPool) *twistPoint { - sum := newTwistPoint(pool) - sum.SetInfinity() - t := newTwistPoint(pool) - - for i := scalar.BitLen(); i >= 0; i-- { - t.Double(sum, pool) - if scalar.Bit(i) != 0 { - sum.Add(t, a, pool) - } else { - sum.Set(t) - } - } - - c.Set(sum) - sum.Put(pool) - t.Put(pool) - return c -} - -func (c *twistPoint) MakeAffine(pool *bnPool) *twistPoint { - if c.z.IsOne() { - return c - } - - zInv := newGFp2(pool).Invert(c.z, pool) - t := newGFp2(pool).Mul(c.y, zInv, pool) - zInv2 := newGFp2(pool).Square(zInv, pool) - c.y.Mul(t, zInv2, pool) - t.Mul(c.x, zInv2, pool) - c.x.Set(t) - c.z.SetOne() - c.t.SetOne() - - zInv.Put(pool) - t.Put(pool) - zInv2.Put(pool) - - return c -} - -func (c *twistPoint) Negative(a *twistPoint, pool *bnPool) { - c.x.Set(a.x) - c.y.SetZero() - c.y.Sub(c.y, a.y) - c.z.Set(a.z) - c.t.SetZero() -} -- cgit v1.2.3