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crypto/bn256: full switchover to cloudflare's code

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This commit is contained in:
Péter Szilágyi 2018-03-12 11:43:03 +02:00
parent 6a2d2869f6
commit aa48242399
No known key found for this signature in database
GPG key ID: E9AE538CEDF8293D
36 changed files with 473 additions and 2773 deletions
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0
crypto/bn256/cloudflare/bn256.go → crypto/bn256/bn256.go
View file

63
crypto/bn256/bn256_amd64.go
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@ -1,63 +0,0 @@
// 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 <http://www.gnu.org/licenses/>.
// +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)
}

63
crypto/bn256/bn256_other.go
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@ -1,63 +0,0 @@
// 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 <http://www.gnu.org/licenses/>.
// +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)
}

2
crypto/bn256/cloudflare/bn256_test.go → crypto/bn256/bn256_test.go
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@ -1,5 +1,3 @@
// +build amd64,!appengine,!gccgo
package bn256
import (

45
crypto/bn256/cloudflare/example_test.go
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@ -1,45 +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.
// +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.
}

32
crypto/bn256/cloudflare/gfp.h
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@ -1,32 +0,0 @@
#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

15
crypto/bn256/cloudflare/gfp_amd64.go
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@ -1,15 +0,0 @@
// +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)

19
crypto/bn256/cloudflare/gfp_pure.go
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@ -1,19 +0,0 @@
// +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")
}

73
crypto/bn256/cloudflare/main_test.go
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@ -1,73 +0,0 @@
// +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)")
}
}

0
crypto/bn256/cloudflare/constants.go → crypto/bn256/constants.go
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0
crypto/bn256/cloudflare/curve.go → crypto/bn256/curve.go
View file

0
crypto/bn256/google/example_test.go → crypto/bn256/example_test.go
View file

0
crypto/bn256/cloudflare/gfp.go → crypto/bn256/gfp.go
View file

0
crypto/bn256/cloudflare/gfp12.go → crypto/bn256/gfp12.go
View file

0
crypto/bn256/cloudflare/gfp2.go → crypto/bn256/gfp2.go
View file

0
crypto/bn256/cloudflare/gfp6.go → crypto/bn256/gfp6.go
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40
crypto/bn256/cloudflare/gfp_amd64.s → crypto/bn256/gfp_amd64.s
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@ -1,8 +1,40 @@
// +build amd64,!appengine,!gccgo
// +build amd64,!generic
#include "gfp.h"
#include "mul.h"
#include "mul_bmi2.h"
#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
#include "mul_amd64.h"
#include "mul_bmi2_amd64.h"
TEXT ·gfpNeg(SB),0,$0-16
MOVQ ·p2+0(SB), R8

113
crypto/bn256/gfp_arm64.s Normal file
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@ -0,0 +1,113 @@
// +build arm64,!generic
#define storeBlock(a0,a1,a2,a3, r) \
MOVD a0, 0+r \
MOVD a1, 8+r \
MOVD a2, 16+r \
MOVD a3, 24+r
#define loadBlock(r, a0,a1,a2,a3) \
MOVD 0+r, a0 \
MOVD 8+r, a1 \
MOVD 16+r, a2 \
MOVD 24+r, a3
#define loadModulus(p0,p1,p2,p3) \
MOVD ·p2+0(SB), p0 \
MOVD ·p2+8(SB), p1 \
MOVD ·p2+16(SB), p2 \
MOVD ·p2+24(SB), p3
#include "mul_arm64.h"
TEXT ·gfpNeg(SB),0,$0-16
MOVD a+8(FP), R0
loadBlock(0(R0), R1,R2,R3,R4)
loadModulus(R5,R6,R7,R8)
SUBS R1, R5, R1
SBCS R2, R6, R2
SBCS R3, R7, R3
SBCS R4, R8, R4
SUBS R5, R1, R5
SBCS R6, R2, R6
SBCS R7, R3, R7
SBCS R8, R4, R8
CSEL CS, R5, R1, R1
CSEL CS, R6, R2, R2
CSEL CS, R7, R3, R3
CSEL CS, R8, R4, R4
MOVD c+0(FP), R0
storeBlock(R1,R2,R3,R4, 0(R0))
RET
TEXT ·gfpAdd(SB),0,$0-24
MOVD a+8(FP), R0
loadBlock(0(R0), R1,R2,R3,R4)
MOVD b+16(FP), R0
loadBlock(0(R0), R5,R6,R7,R8)
loadModulus(R9,R10,R11,R12)
MOVD ZR, R0
ADDS R5, R1
ADCS R6, R2
ADCS R7, R3
ADCS R8, R4
ADCS ZR, R0
SUBS R9, R1, R5
SBCS R10, R2, R6
SBCS R11, R3, R7
SBCS R12, R4, R8
SBCS ZR, R0, R0
CSEL CS, R5, R1, R1
CSEL CS, R6, R2, R2
CSEL CS, R7, R3, R3
CSEL CS, R8, R4, R4
MOVD c+0(FP), R0
storeBlock(R1,R2,R3,R4, 0(R0))
RET
TEXT ·gfpSub(SB),0,$0-24
MOVD a+8(FP), R0
loadBlock(0(R0), R1,R2,R3,R4)
MOVD b+16(FP), R0
loadBlock(0(R0), R5,R6,R7,R8)
loadModulus(R9,R10,R11,R12)
SUBS R5, R1
SBCS R6, R2
SBCS R7, R3
SBCS R8, R4
CSEL CS, ZR, R9, R9
CSEL CS, ZR, R10, R10
CSEL CS, ZR, R11, R11
CSEL CS, ZR, R12, R12
ADDS R9, R1
ADCS R10, R2
ADCS R11, R3
ADCS R12, R4
MOVD c+0(FP), R0
storeBlock(R1,R2,R3,R4, 0(R0))
RET
TEXT ·gfpMul(SB),0,$0-24
MOVD a+8(FP), R0
loadBlock(0(R0), R1,R2,R3,R4)
MOVD b+16(FP), R0
loadBlock(0(R0), R5,R6,R7,R8)
mul(R9,R10,R11,R12,R13,R14,R15,R16)
gfpReduce()
MOVD c+0(FP), R0
storeBlock(R1,R2,R3,R4, 0(R0))
RET

18
crypto/bn256/gfp_decl.go Normal file
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@ -0,0 +1,18 @@
// +build amd64,!generic arm64,!generic
package bn256
// This file contains forward declarations for the architecture-specific
// assembly implementations of these functions, provided that they exist.
// 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)

173
crypto/bn256/gfp_generic.go Normal file
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@ -0,0 +1,173 @@
// +build !amd64,!arm64 generic
package bn256
func gfpCarry(a *gfP, head uint64) {
b := &gfP{}
var carry uint64
for i, pi := range p2 {
ai := a[i]
bi := ai - pi - carry
b[i] = bi
carry = (pi&^ai | (pi|^ai)&bi) >> 63
}
carry = carry &^ head
// If b is negative, then return a.
// Else return b.
carry = -carry
ncarry := ^carry
for i := 0; i < 4; i++ {
a[i] = (a[i] & carry) | (b[i] & ncarry)
}
}
func gfpNeg(c, a *gfP) {
var carry uint64
for i, pi := range p2 {
ai := a[i]
ci := pi - ai - carry
c[i] = ci
carry = (ai&^pi | (ai|^pi)&ci) >> 63
}
gfpCarry(c, 0)
}
func gfpAdd(c, a, b *gfP) {
var carry uint64
for i, ai := range a {
bi := b[i]
ci := ai + bi + carry
c[i] = ci
carry = (ai&bi | (ai|bi)&^ci) >> 63
}
gfpCarry(c, carry)
}
func gfpSub(c, a, b *gfP) {
t := &gfP{}
var carry uint64
for i, pi := range p2 {
bi := b[i]
ti := pi - bi - carry
t[i] = ti
carry = (bi&^pi | (bi|^pi)&ti) >> 63
}
carry = 0
for i, ai := range a {
ti := t[i]
ci := ai + ti + carry
c[i] = ci
carry = (ai&ti | (ai|ti)&^ci) >> 63
}
gfpCarry(c, carry)
}
func mul(a, b [4]uint64) [8]uint64 {
const (
mask16 uint64 = 0x0000ffff
mask32 uint64 = 0xffffffff
)
var buff [32]uint64
for i, ai := range a {
a0, a1, a2, a3 := ai&mask16, (ai>>16)&mask16, (ai>>32)&mask16, ai>>48
for j, bj := range b {
b0, b2 := bj&mask32, bj>>32
off := 4 * (i + j)
buff[off+0] += a0 * b0
buff[off+1] += a1 * b0
buff[off+2] += a2*b0 + a0*b2
buff[off+3] += a3*b0 + a1*b2
buff[off+4] += a2 * b2
buff[off+5] += a3 * b2
}
}
for i := uint(1); i < 4; i++ {
shift := 16 * i
var head, carry uint64
for j := uint(0); j < 8; j++ {
block := 4 * j
xi := buff[block]
yi := (buff[block+i] << shift) + head
zi := xi + yi + carry
buff[block] = zi
carry = (xi&yi | (xi|yi)&^zi) >> 63
head = buff[block+i] >> (64 - shift)
}
}
return [8]uint64{buff[0], buff[4], buff[8], buff[12], buff[16], buff[20], buff[24], buff[28]}
}
func halfMul(a, b [4]uint64) [4]uint64 {
const (
mask16 uint64 = 0x0000ffff
mask32 uint64 = 0xffffffff
)
var buff [18]uint64
for i, ai := range a {
a0, a1, a2, a3 := ai&mask16, (ai>>16)&mask16, (ai>>32)&mask16, ai>>48
for j, bj := range b {
if i+j > 3 {
break
}
b0, b2 := bj&mask32, bj>>32
off := 4 * (i + j)
buff[off+0] += a0 * b0
buff[off+1] += a1 * b0
buff[off+2] += a2*b0 + a0*b2
buff[off+3] += a3*b0 + a1*b2
buff[off+4] += a2 * b2
buff[off+5] += a3 * b2
}
}
for i := uint(1); i < 4; i++ {
shift := 16 * i
var head, carry uint64
for j := uint(0); j < 4; j++ {
block := 4 * j
xi := buff[block]
yi := (buff[block+i] << shift) + head
zi := xi + yi + carry
buff[block] = zi
carry = (xi&yi | (xi|yi)&^zi) >> 63
head = buff[block+i] >> (64 - shift)
}
}
return [4]uint64{buff[0], buff[4], buff[8], buff[12]}
}
func gfpMul(c, a, b *gfP) {
T := mul(*a, *b)
m := halfMul([4]uint64{T[0], T[1], T[2], T[3]}, np)
t := mul([4]uint64{m[0], m[1], m[2], m[3]}, p2)
var carry uint64
for i, Ti := range T {
ti := t[i]
zi := Ti + ti + carry
T[i] = zi
carry = (Ti&ti | (Ti|ti)&^zi) >> 63
}
*c = gfP{T[4], T[5], T[6], T[7]}
gfpCarry(c, carry)
}

2
crypto/bn256/cloudflare/gfp_test.go → crypto/bn256/gfp_test.go
View file

@ -1,5 +1,3 @@
// +build amd64,!appengine,!gccgo
package bn256
import (

447
crypto/bn256/google/bn256.go
View file

@ -1,447 +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"
"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 &GT{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
}

311
crypto/bn256/google/bn256_test.go
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@ -1,311 +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()
_, 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})
}
}

44
crypto/bn256/google/constants.go
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@ -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")}

278
crypto/bn256/google/curve.go
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// Copyright 2012 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package bn256
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)
}

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crypto/bn256/google/gfp12.go
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// Copyright 2012 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package bn256
// For details of the algorithms used, see "Multiplication and Squaring on
// Pairing-Friendly Fields, Devegili et al.
// http://eprint.iacr.org/2006/471.pdf.
import (
"math/big"
)
// gfP12 implements the field of size p¹² as a quadratic extension of gfP6
// where ω²=τ.
type gfP12 struct {
x, y *gfP6 // value is xω + y
}
func newGFp12(pool *bnPool) *gfP12 {
return &gfP12{newGFp6(pool), newGFp6(pool)}
}
func (e *gfP12) String() string {
return "(" + e.x.String() + "," + e.y.String() + ")"
}
func (e *gfP12) Put(pool *bnPool) {
e.x.Put(pool)
e.y.Put(pool)
}
func (e *gfP12) Set(a *gfP12) *gfP12 {
e.x.Set(a.x)
e.y.Set(a.y)
return e
}
func (e *gfP12) SetZero() *gfP12 {
e.x.SetZero()
e.y.SetZero()
return e
}
func (e *gfP12) SetOne() *gfP12 {
e.x.SetZero()
e.y.SetOne()
return e
}
func (e *gfP12) Minimal() {
e.x.Minimal()
e.y.Minimal()
}
func (e *gfP12) IsZero() bool {
e.Minimal()
return e.x.IsZero() && e.y.IsZero()
}
func (e *gfP12) IsOne() bool {
e.Minimal()
return e.x.IsZero() && e.y.IsOne()
}
func (e *gfP12) Conjugate(a *gfP12) *gfP12 {
e.x.Negative(a.x)
e.y.Set(a.y)
return a
}
func (e *gfP12) Negative(a *gfP12) *gfP12 {
e.x.Negative(a.x)
e.y.Negative(a.y)
return e
}
// Frobenius computes (xω+y)^p = x^p ω·ξ^((p-1)/6) + y^p
func (e *gfP12) Frobenius(a *gfP12, pool *bnPool) *gfP12 {
e.x.Frobenius(a.x, pool)
e.y.Frobenius(a.y, pool)
e.x.MulScalar(e.x, xiToPMinus1Over6, pool)
return e
}
// FrobeniusP2 computes (xω+y)^p² = x^p² ω·ξ^((p²-1)/6) + y^p²
func (e *gfP12) FrobeniusP2(a *gfP12, pool *bnPool) *gfP12 {
e.x.FrobeniusP2(a.x)
e.x.MulGFP(e.x, xiToPSquaredMinus1Over6)
e.y.FrobeniusP2(a.y)
return e
}
func (e *gfP12) Add(a, b *gfP12) *gfP12 {
e.x.Add(a.x, b.x)
e.y.Add(a.y, b.y)
return e
}
func (e *gfP12) Sub(a, b *gfP12) *gfP12 {
e.x.Sub(a.x, b.x)
e.y.Sub(a.y, b.y)
return e
}
func (e *gfP12) Mul(a, b *gfP12, pool *bnPool) *gfP12 {
tx := newGFp6(pool)
tx.Mul(a.x, b.y, pool)
t := newGFp6(pool)
t.Mul(b.x, a.y, pool)
tx.Add(tx, t)
ty := newGFp6(pool)
ty.Mul(a.y, b.y, pool)
t.Mul(a.x, b.x, pool)
t.MulTau(t, pool)
e.y.Add(ty, t)
e.x.Set(tx)
tx.Put(pool)
ty.Put(pool)
t.Put(pool)
return e
}
func (e *gfP12) MulScalar(a *gfP12, b *gfP6, pool *bnPool) *gfP12 {
e.x.Mul(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
}

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crypto/bn256/google/gfp2.go
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@ -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
}

296
crypto/bn256/google/gfp6.go
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@ -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
}

397
crypto/bn256/google/optate.go
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@ -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
}

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crypto/bn256/google/twist.go
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@ -1,255 +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()
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()
}

0
crypto/bn256/google/main_test.go → crypto/bn256/main_test.go
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0
crypto/bn256/cloudflare/mul.h → crypto/bn256/mul_amd64.h
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133
crypto/bn256/mul_arm64.h Normal file
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@ -0,0 +1,133 @@
#define mul(c0,c1,c2,c3,c4,c5,c6,c7) \
MUL R1, R5, c0 \
UMULH R1, R5, c1 \
MUL R1, R6, R0 \
ADDS R0, c1 \
UMULH R1, R6, c2 \
MUL R1, R7, R0 \
ADCS R0, c2 \
UMULH R1, R7, c3 \
MUL R1, R8, R0 \
ADCS R0, c3 \
UMULH R1, R8, c4 \
ADCS ZR, c4 \
\
MUL R2, R5, R25 \
UMULH R2, R5, R26 \
MUL R2, R6, R0 \
ADDS R0, R26 \
UMULH R2, R6, R27 \
MUL R2, R7, R0 \
ADCS R0, R27 \
UMULH R2, R7, R29 \
MUL R2, R8, R0 \
ADCS R0, R29 \
UMULH R2, R8, c5 \
ADCS ZR, c5 \
ADDS R25, c1 \
ADCS R26, c2 \
ADCS R27, c3 \
ADCS R29, c4 \
ADCS ZR, c5 \
\
MUL R3, R5, R25 \
UMULH R3, R5, R26 \
MUL R3, R6, R0 \
ADDS R0, R26 \
UMULH R3, R6, R27 \
MUL R3, R7, R0 \
ADCS R0, R27 \
UMULH R3, R7, R29 \
MUL R3, R8, R0 \
ADCS R0, R29 \
UMULH R3, R8, c6 \
ADCS ZR, c6 \
ADDS R25, c2 \
ADCS R26, c3 \
ADCS R27, c4 \
ADCS R29, c5 \
ADCS ZR, c6 \
\
MUL R4, R5, R25 \
UMULH R4, R5, R26 \
MUL R4, R6, R0 \
ADDS R0, R26 \
UMULH R4, R6, R27 \
MUL R4, R7, R0 \
ADCS R0, R27 \
UMULH R4, R7, R29 \
MUL R4, R8, R0 \
ADCS R0, R29 \
UMULH R4, R8, c7 \
ADCS ZR, c7 \
ADDS R25, c3 \
ADCS R26, c4 \
ADCS R27, c5 \
ADCS R29, c6 \
ADCS ZR, c7
#define gfpReduce() \
\ // m = (T * N') mod R, store m in R1:R2:R3:R4
MOVD ·np+0(SB), R17 \
MOVD ·np+8(SB), R18 \
MOVD ·np+16(SB), R19 \
MOVD ·np+24(SB), R20 \
\
MUL R9, R17, R1 \
UMULH R9, R17, R2 \
MUL R9, R18, R0 \
ADDS R0, R2 \
UMULH R9, R18, R3 \
MUL R9, R19, R0 \
ADCS R0, R3 \
UMULH R9, R19, R4 \
MUL R9, R20, R0 \
ADCS R0, R4 \
\
MUL R10, R17, R21 \
UMULH R10, R17, R22 \
MUL R10, R18, R0 \
ADDS R0, R22 \
UMULH R10, R18, R23 \
MUL R10, R19, R0 \
ADCS R0, R23 \
ADDS R21, R2 \
ADCS R22, R3 \
ADCS R23, R4 \
\
MUL R11, R17, R21 \
UMULH R11, R17, R22 \
MUL R11, R18, R0 \
ADDS R0, R22 \
ADDS R21, R3 \
ADCS R22, R4 \
\
MUL R12, R17, R21 \
ADDS R21, R4 \
\
\ // m * N
loadModulus(R5,R6,R7,R8) \
mul(R17,R18,R19,R20,R21,R22,R23,R24) \
\
\ // Add the 512-bit intermediate to m*N
MOVD ZR, R25 \
ADDS R9, R17 \
ADCS R10, R18 \
ADCS R11, R19 \
ADCS R12, R20 \
ADCS R13, R21 \
ADCS R14, R22 \
ADCS R15, R23 \
ADCS R16, R24 \
ADCS ZR, R25 \
\
\ // Our output is R21:R22:R23:R24. Reduce mod p if necessary.
SUBS R5, R21, R10 \
SBCS R6, R22, R11 \
SBCS R7, R23, R12 \
SBCS R8, R24, R13 \
\
CSEL CS, R10, R21, R1 \
CSEL CS, R11, R22, R2 \
CSEL CS, R12, R23, R3 \
CSEL CS, R13, R24, R4

0
crypto/bn256/cloudflare/mul_bmi2.h → crypto/bn256/mul_bmi2_amd64.h
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0
crypto/bn256/cloudflare/optate.go → crypto/bn256/optate.go
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0
crypto/bn256/cloudflare/twist.go → crypto/bn256/twist.go
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