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bn128 pairing implemented

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arnaucube
2018-11-24 01:11:28 +01:00
parent 3dfb1e5875
commit d55d875d7a
15 changed files with 745 additions and 176 deletions
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45
README.md
View File

@@ -419,26 +419,32 @@ if verified {
## Bn128
**[not finished]**
This is implemented followng the implementations and info from:
- https://github.com/iden3/zksnark
- https://github.com/zcash/zcash/tree/master/src/snark
- https://github.com/ethereum/py_ecc/tree/master/py_ecc/bn128
This is implemented followng the info and the implementations from:
- `Multiplication and Squaring on Pairing-Friendly
Fields`, Augusto Jun Devegili, Colm Ó hÉigeartaigh, Michael Scott, and Ricardo Dahab https://pdfs.semanticscholar.org/3e01/de88d7428076b2547b60072088507d881bf1.pdf
- `Optimal Pairings`, Frederik Vercauteren https://www.cosic.esat.kuleuven.be/bcrypt/optimal.pdf
- `Optimal Pairings`, Frederik Vercauteren https://www.cosic.esat.kuleuven.be/bcrypt/optimal.pdf , https://eprint.iacr.org/2008/096.pdf
- `Double-and-Add with Relative Jacobian
Coordinates`, Björn Fay https://eprint.iacr.org/2014/1014.pdf
- `Fast and Regular Algorithms for Scalar Multiplication
over Elliptic Curves`, Matthieu Rivain https://eprint.iacr.org/2011/338.pdf
- `High-Speed Software Implementation of the Optimal Ate Pairing over Barreto–Naehrig Curves`, Jean-Luc Beuchat, Jorge E. González-Díaz, Shigeo Mitsunari, Eiji Okamoto, Francisco Rodríguez-Henríquez, and Tadanori Teruya https://eprint.iacr.org/2010/354.pdf
- `New software speed records for cryptographic pairings`, Michael Naehrig, Ruben Niederhagen, Peter Schwabe https://cryptojedi.org/papers/dclxvi-20100714.pdf
- https://github.com/zcash/zcash/tree/master/src/snark
- https://github.com/iden3/snarkjs
- https://github.com/ethereum/py_ecc/tree/master/py_ecc/bn128
- [x] Fq, Fq2, Fq6, Fq12 operations
- [x] G1, G2 operations
- [x] preparePairing
- [x] PreComupteG1, PreComupteG2
- [x] DoubleStep, AddStep
- [x] MillerLoop
- [x] Pairing
#### Usage
First let's define three basic functions to convert integer compositions to big integer compositions:
First let's assume that we have these three basic functions to convert integer compositions to big integer compositions:
```go
func iToBig(a int) *big.Int {
return big.NewInt(int64(a))
@@ -453,6 +459,28 @@ func iiiToBig(a, b int) [2]*big.Int {
}
```
- Pairing
```go
bn128, err := NewBn128()
assert.Nil(t, err)
big25 := big.NewInt(int64(25))
big30 := big.NewInt(int64(30))
g1a := bn128.G1.MulScalar(bn128.G1.G, big25)
g2a := bn128.G2.MulScalar(bn128.G2.G, big30)
g1b := bn128.G1.MulScalar(bn128.G1.G, big30)
g2b := bn128.G2.MulScalar(bn128.G2.G, big25)
pA, err := bn128.Pairing(g1a, g2a)
assert.Nil(t, err)
pB, err := bn128.Pairing(g1b, g2b)
assert.Nil(t, err)
assert.True(t, bn128.Fq12.Equal(pA, pB))
```
- Finite Fields (1, 2, 6, 12) operations
```go
// new finite field of order 1
@@ -468,7 +496,7 @@ res = fq1.Inverse(iToBig(4))
res = fq1.Square(iToBig(5))
// new finite field of order 2
nonResidueFq2str := "-1" // i / Beta
nonResidueFq2str := "-1" // i/j
nonResidueFq2, ok := new(big.Int).SetString(nonResidueFq2str, 10)
fq2 := Fq2{fq1, nonResidueFq2}
nonResidueFq6 := iiToBig(9, 1)
@@ -564,6 +592,7 @@ b := bn128.G2.Affine(grsum2)
assert.Equal(t, a, b)
```
---
To run all tests:

44
bn128/README.md
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@@ -1,24 +1,30 @@
## Bn128
**[not finished]**
This is implemented followng the implementations and info from:
- https://github.com/zcash/zcash/tree/master/src/snark
- https://github.com/iden3/snarkjs
- https://github.com/ethereum/py_ecc/tree/master/py_ecc/bn128
This is implemented followng the info and the implementations from:
- `Multiplication and Squaring on Pairing-Friendly
Fields`, Augusto Jun Devegili, Colm Ó hÉigeartaigh, Michael Scott, and Ricardo Dahab https://pdfs.semanticscholar.org/3e01/de88d7428076b2547b60072088507d881bf1.pdf
- `Optimal Pairings`, Frederik Vercauteren https://www.cosic.esat.kuleuven.be/bcrypt/optimal.pdf
- `Optimal Pairings`, Frederik Vercauteren https://www.cosic.esat.kuleuven.be/bcrypt/optimal.pdf , https://eprint.iacr.org/2008/096.pdf
- `Double-and-Add with Relative Jacobian
Coordinates`, Björn Fay https://eprint.iacr.org/2014/1014.pdf
- `Fast and Regular Algorithms for Scalar Multiplication
over Elliptic Curves`, Matthieu Rivain https://eprint.iacr.org/2011/338.pdf
- `High-Speed Software Implementation of the Optimal Ate Pairing over Barreto–Naehrig Curves`, Jean-Luc Beuchat, Jorge E. González-Díaz, Shigeo Mitsunari, Eiji Okamoto, Francisco Rodríguez-Henríquez, and Tadanori Teruya https://eprint.iacr.org/2010/354.pdf
- `New software speed records for cryptographic pairings`, Michael Naehrig, Ruben Niederhagen, Peter Schwabe https://cryptojedi.org/papers/dclxvi-20100714.pdf
- https://github.com/zcash/zcash/tree/master/src/snark
- https://github.com/iden3/snarkjs
- https://github.com/ethereum/py_ecc/tree/master/py_ecc/bn128
- [x] Fq, Fq2, Fq6, Fq12 operations
- [x] G1, G2 operations
- [x] preparePairing
- [x] PreComupteG1, PreComupteG2
- [x] DoubleStep, AddStep
- [x] MillerLoop
- [x] Pairing
#### Usage
First let's define three basic functions to convert integer compositions to big integer compositions:
First let's assume that we have these three basic functions to convert integer compositions to big integer compositions:
```go
func iToBig(a int) *big.Int {
return big.NewInt(int64(a))
@@ -33,6 +39,28 @@ func iiiToBig(a, b int) [2]*big.Int {
}
```
- Pairing
```go
bn128, err := NewBn128()
assert.Nil(t, err)
big25 := big.NewInt(int64(25))
big30 := big.NewInt(int64(30))
g1a := bn128.G1.MulScalar(bn128.G1.G, big25)
g2a := bn128.G2.MulScalar(bn128.G2.G, big30)
g1b := bn128.G1.MulScalar(bn128.G1.G, big30)
g2b := bn128.G2.MulScalar(bn128.G2.G, big25)
pA, err := bn128.Pairing(g1a, g2a)
assert.Nil(t, err)
pB, err := bn128.Pairing(g1b, g2b)
assert.Nil(t, err)
assert.True(t, bn128.Fq12.Equal(pA, pB))
```
- Finite Fields (1, 2, 6, 12) operations
```go
// new finite field of order 1
@@ -48,7 +76,7 @@ res = fq1.Inverse(iToBig(4))
res = fq1.Square(iToBig(5))
// new finite field of order 2
nonResidueFq2str := "-1" // i / Beta
nonResidueFq2str := "-1" // i/j
nonResidueFq2, ok := new(big.Int).SetString(nonResidueFq2str, 10)
fq2 := Fq2{fq1, nonResidueFq2}
nonResidueFq6 := iiToBig(9, 1)

357
bn128/bn128.go
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@@ -1,13 +1,13 @@
package bn128
import (
"bytes"
"errors"
"math/big"
)
type Bn128 struct {
Q *big.Int
R *big.Int
Gg1 [2]*big.Int
Gg2 [2][2]*big.Int
NonResidueFq2 *big.Int
@@ -18,6 +18,17 @@ type Bn128 struct {
Fq12 Fq12
G1 G1
G2 G2
LoopCount *big.Int
LoopCountNeg bool
TwoInv *big.Int
CoefB *big.Int
TwistCoefB [2]*big.Int
Twist [2]*big.Int
FrobeniusCoeffsC11 *big.Int
TwistMulByQX [2]*big.Int
TwistMulByQY [2]*big.Int
FinalExp *big.Int
}
func NewBn128() (Bn128, error) {
@@ -27,11 +38,6 @@ func NewBn128() (Bn128, error) {
return b, errors.New("err with q")
}
b.Q = q
r, ok := new(big.Int).SetString("21888242871839275222246405745257275088548364400416034343698204186575808495617", 10) // i
if !ok {
return b, errors.New("err with r")
}
b.R = r
b.Gg1 = [2]*big.Int{
big.NewInt(int64(1)),
@@ -54,17 +60,16 @@ func NewBn128() (Bn128, error) {
if !ok {
return b, errors.New("err with g2_00")
}
g2_0 := [2]*big.Int{
g2_00,
g2_01,
}
g2_1 := [2]*big.Int{
g2_10,
g2_11,
}
b.Gg2 = [2][2]*big.Int{
g2_0,
g2_1,
[2]*big.Int{
g2_00,
g2_01,
},
[2]*big.Int{
g2_10,
g2_11,
},
}
b.Fq1 = NewFq(q)
@@ -77,12 +82,326 @@ func NewBn128() (Bn128, error) {
big.NewInt(int64(1)),
}
b.Fq2 = Fq2{b.Fq1, b.NonResidueFq2}
b.Fq6 = Fq6{b.Fq2, b.NonResidueFq6}
b.Fq12 = Fq12{b.Fq6, b.Fq2, b.NonResidueFq6}
b.Fq2 = NewFq2(b.Fq1, b.NonResidueFq2)
b.Fq6 = NewFq6(b.Fq2, b.NonResidueFq6)
b.Fq12 = NewFq12(b.Fq6, b.Fq2, b.NonResidueFq6)
b.G1 = NewG1(b.Fq1, b.Gg1)
b.G2 = NewG2(b.Fq2, b.Gg2)
err := b.preparePairing()
if err != nil {
return b, err
}
return b, nil
}
func BigIsOdd(n *big.Int) bool {
one := big.NewInt(int64(1))
and := new(big.Int).And(n, one)
return bytes.Equal(and.Bytes(), big.NewInt(int64(1)).Bytes())
}
func (bn128 *Bn128) preparePairing() error {
var ok bool
bn128.LoopCount, ok = new(big.Int).SetString("29793968203157093288", 10)
if !ok {
return errors.New("err with LoopCount from string")
}
bn128.LoopCountNeg = false
bn128.TwoInv = bn128.Fq1.Inverse(big.NewInt(int64(2)))
bn128.CoefB = big.NewInt(int64(3))
bn128.Twist = [2]*big.Int{
big.NewInt(int64(9)),
big.NewInt(int64(1)),
}
bn128.TwistCoefB = bn128.Fq2.MulScalar(bn128.Fq2.Inverse(bn128.Twist), bn128.CoefB)
bn128.FrobeniusCoeffsC11, ok = new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208582", 10)
if !ok {
return errors.New("error parsing frobeniusCoeffsC11")
}
a, ok := new(big.Int).SetString("21575463638280843010398324269430826099269044274347216827212613867836435027261", 10)
if !ok {
return errors.New("error parsing a")
}
b, ok := new(big.Int).SetString("10307601595873709700152284273816112264069230130616436755625194854815875713954", 10)
if !ok {
return errors.New("error parsing b")
}
bn128.TwistMulByQX = [2]*big.Int{
a,
b,
}
a, ok = new(big.Int).SetString("2821565182194536844548159561693502659359617185244120367078079554186484126554", 10)
if !ok {
return errors.New("error parsing a")
}
b, ok = new(big.Int).SetString("3505843767911556378687030309984248845540243509899259641013678093033130930403", 10)
if !ok {
return errors.New("error parsing b")
}
bn128.TwistMulByQY = [2]*big.Int{
a,
b,
}
bn128.FinalExp, ok = new(big.Int).SetString("552484233613224096312617126783173147097382103762957654188882734314196910839907541213974502761540629817009608548654680343627701153829446747810907373256841551006201639677726139946029199968412598804882391702273019083653272047566316584365559776493027495458238373902875937659943504873220554161550525926302303331747463515644711876653177129578303191095900909191624817826566688241804408081892785725967931714097716709526092261278071952560171111444072049229123565057483750161460024353346284167282452756217662335528813519139808291170539072125381230815729071544861602750936964829313608137325426383735122175229541155376346436093930287402089517426973178917569713384748081827255472576937471496195752727188261435633271238710131736096299798168852925540549342330775279877006784354801422249722573783561685179618816480037695005515426162362431072245638324744480", 10)
if !ok {
return errors.New("error parsing finalExp")
}
return nil
}
func (bn128 Bn128) Pairing(p1 [3]*big.Int, p2 [3][2]*big.Int) ([2][3][2]*big.Int, error) {
pre1 := bn128.PreComputeG1(p1)
pre2, err := bn128.PreComputeG2(p2)
if err != nil {
return [2][3][2]*big.Int{}, err
}
r1 := bn128.MillerLoop(pre1, pre2)
res := bn128.FinalExponentiation(r1)
return res, nil
}
type AteG1Precomp struct {
Px *big.Int
Py *big.Int
}
func (bn128 Bn128) PreComputeG1(p [3]*big.Int) AteG1Precomp {
pCopy := bn128.G1.Affine(p)
res := AteG1Precomp{
Px: pCopy[0],
Py: pCopy[1],
}
return res
}
type EllCoeffs struct {
Ell0 [2]*big.Int
EllVW [2]*big.Int
EllVV [2]*big.Int
}
type AteG2Precomp struct {
Qx [2]*big.Int
Qy [2]*big.Int
Coeffs []EllCoeffs
}
func (bn128 Bn128) PreComputeG2(p [3][2]*big.Int) (AteG2Precomp, error) {
qCopy := bn128.G2.Affine(p)
res := AteG2Precomp{
qCopy[0],
qCopy[1],
[]EllCoeffs{},
}
r := [3][2]*big.Int{
bn128.Fq2.Copy(qCopy[0]),
bn128.Fq2.Copy(qCopy[1]),
bn128.Fq2.One(),
}
var c EllCoeffs
for i := bn128.LoopCount.BitLen() - 2; i >= 0; i-- {
bit := bn128.LoopCount.Bit(i)
c, r = bn128.DoublingStep(r)
res.Coeffs = append(res.Coeffs, c)
if bit == 1 {
c, r = bn128.MixedAdditionStep(qCopy, r)
res.Coeffs = append(res.Coeffs, c)
}
}
q1 := bn128.G2.Affine(bn128.G2MulByQ(qCopy))
if !bn128.Fq2.Equal(q1[2], bn128.Fq2.One()) {
return res, errors.New("q1[2] != Fq2.One")
}
q2 := bn128.G2.Affine(bn128.G2MulByQ(q1))
if !bn128.Fq2.Equal(q2[2], bn128.Fq2.One()) {
return res, errors.New("q2[2] != Fq2.One")
}
if bn128.LoopCountNeg {
r[1] = bn128.Fq2.Neg(r[1])
}
q2[1] = bn128.Fq2.Neg(q2[1])
c, r = bn128.MixedAdditionStep(q1, r)
res.Coeffs = append(res.Coeffs, c)
c, r = bn128.MixedAdditionStep(q2, r)
res.Coeffs = append(res.Coeffs, c)
return res, nil
}
func (bn128 Bn128) DoublingStep(current [3][2]*big.Int) (EllCoeffs, [3][2]*big.Int) {
x := current[0]
y := current[1]
z := current[2]
a := bn128.Fq2.MulScalar(bn128.Fq2.Mul(x, y), bn128.TwoInv)
b := bn128.Fq2.Square(y)
c := bn128.Fq2.Square(z)
d := bn128.Fq2.Add(c, bn128.Fq2.Add(c, c))
e := bn128.Fq2.Mul(bn128.TwistCoefB, d)
f := bn128.Fq2.Add(e, bn128.Fq2.Add(e, e))
g := bn128.Fq2.MulScalar(bn128.Fq2.Add(b, f), bn128.TwoInv)
h := bn128.Fq2.Sub(
bn128.Fq2.Square(bn128.Fq2.Add(y, z)),
bn128.Fq2.Add(b, c))
i := bn128.Fq2.Sub(e, b)
j := bn128.Fq2.Square(x)
eSqr := bn128.Fq2.Square(e)
current[0] = bn128.Fq2.Mul(a, bn128.Fq2.Sub(b, f))
current[1] = bn128.Fq2.Sub(bn128.Fq2.Sub(bn128.Fq2.Square(g), eSqr),
bn128.Fq2.Add(eSqr, eSqr))
current[2] = bn128.Fq2.Mul(b, h)
res := EllCoeffs{
Ell0: bn128.Fq2.Mul(i, bn128.Twist),
EllVW: bn128.Fq2.Neg(h),
EllVV: bn128.Fq2.Add(j, bn128.Fq2.Add(j, j)),
}
return res, current
}
func (bn128 Bn128) MixedAdditionStep(base, current [3][2]*big.Int) (EllCoeffs, [3][2]*big.Int) {
x1 := current[0]
y1 := current[1]
z1 := current[2]
x2 := base[0]
y2 := base[1]
d := bn128.Fq2.Sub(x1, bn128.Fq2.Mul(x2, z1))
e := bn128.Fq2.Sub(y1, bn128.Fq2.Mul(y2, z1))
f := bn128.Fq2.Square(d)
g := bn128.Fq2.Square(e)
h := bn128.Fq2.Mul(d, f)
i := bn128.Fq2.Mul(x1, f)
j := bn128.Fq2.Sub(
bn128.Fq2.Add(h, bn128.Fq2.Mul(z1, g)),
bn128.Fq2.Add(i, i))
current[0] = bn128.Fq2.Mul(d, j)
current[1] = bn128.Fq2.Sub(
bn128.Fq2.Mul(e, bn128.Fq2.Sub(i, j)),
bn128.Fq2.Mul(h, y1))
current[2] = bn128.Fq2.Mul(z1, h)
coef := EllCoeffs{
Ell0: bn128.Fq2.Mul(
bn128.Twist,
bn128.Fq2.Sub(
bn128.Fq2.Mul(e, x2),
bn128.Fq2.Mul(d, y2))),
EllVW: d,
EllVV: bn128.Fq2.Neg(e),
}
return coef, current
}
func (bn128 Bn128) G2MulByQ(p [3][2]*big.Int) [3][2]*big.Int {
fmx := [2]*big.Int{
p[0][0],
bn128.Fq1.Mul(p[0][1], bn128.Fq1.Copy(bn128.FrobeniusCoeffsC11)),
}
fmy := [2]*big.Int{
p[1][0],
bn128.Fq1.Mul(p[1][1], bn128.Fq1.Copy(bn128.FrobeniusCoeffsC11)),
}
fmz := [2]*big.Int{
p[2][0],
bn128.Fq1.Mul(p[2][1], bn128.Fq1.Copy(bn128.FrobeniusCoeffsC11)),
}
return [3][2]*big.Int{
bn128.Fq2.Mul(bn128.TwistMulByQX, fmx),
bn128.Fq2.Mul(bn128.TwistMulByQY, fmy),
fmz,
}
}
func (bn128 Bn128) MillerLoop(pre1 AteG1Precomp, pre2 AteG2Precomp) [2][3][2]*big.Int {
// https://cryptojedi.org/papers/dclxvi-20100714.pdf
// https://eprint.iacr.org/2008/096.pdf
idx := 0
var c EllCoeffs
f := bn128.Fq12.One()
for i := bn128.LoopCount.BitLen() - 2; i >= 0; i-- {
bit := bn128.LoopCount.Bit(i)
c = pre2.Coeffs[idx]
idx++
f = bn128.Fq12.Square(f)
f = bn128.MulBy024(f,
c.Ell0,
bn128.Fq2.MulScalar(c.EllVW, pre1.Py),
bn128.Fq2.MulScalar(c.EllVV, pre1.Px))
if bit == 1 {
c = pre2.Coeffs[idx]
idx++
f = bn128.MulBy024(
f,
c.Ell0,
bn128.Fq2.MulScalar(c.EllVW, pre1.Py),
bn128.Fq2.MulScalar(c.EllVV, pre1.Px))
}
}
if bn128.LoopCountNeg {
f = bn128.Fq12.Inverse(f)
}
c = pre2.Coeffs[idx]
idx++
f = bn128.MulBy024(
f,
c.Ell0,
bn128.Fq2.MulScalar(c.EllVW, pre1.Py),
bn128.Fq2.MulScalar(c.EllVV, pre1.Px))
c = pre2.Coeffs[idx]
idx++
f = bn128.MulBy024(
f,
c.Ell0,
bn128.Fq2.MulScalar(c.EllVW, pre1.Py),
bn128.Fq2.MulScalar(c.EllVV, pre1.Px))
return f
}
func (bn128 Bn128) MulBy024(a [2][3][2]*big.Int, ell0, ellVW, ellVV [2]*big.Int) [2][3][2]*big.Int {
b := [2][3][2]*big.Int{
[3][2]*big.Int{
ell0,
bn128.Fq2.Zero(),
ellVV,
},
[3][2]*big.Int{
bn128.Fq2.Zero(),
ellVW,
bn128.Fq2.Zero(),
},
}
return bn128.Fq12.Mul(a, b)
}
func (bn128 Bn128) FinalExponentiation(r [2][3][2]*big.Int) [2][3][2]*big.Int {
res := bn128.Fq12.Exp(r, bn128.FinalExp)
return res
}

65
bn128/bn128_test.go Normal file
View File

@@ -0,0 +1,65 @@
package bn128
import (
"math/big"
"testing"
"github.com/stretchr/testify/assert"
)
func TestBN128(t *testing.T) {
bn128, err := NewBn128()
assert.Nil(t, err)
big40 := big.NewInt(int64(40))
big75 := big.NewInt(int64(75))
g1a := bn128.G1.MulScalar(bn128.G1.G, bn128.Fq1.Copy(big40))
g2a := bn128.G2.MulScalar(bn128.G2.G, bn128.Fq1.Copy(big75))
g1b := bn128.G1.MulScalar(bn128.G1.G, bn128.Fq1.Copy(big75))
g2b := bn128.G2.MulScalar(bn128.G2.G, bn128.Fq1.Copy(big40))
pre1a := bn128.PreComputeG1(g1a)
pre2a, err := bn128.PreComputeG2(g2a)
assert.Nil(t, err)
pre1b := bn128.PreComputeG1(g1b)
pre2b, err := bn128.PreComputeG2(g2b)
r1 := bn128.MillerLoop(pre1a, pre2a)
r2 := bn128.MillerLoop(pre1b, pre2b)
rbe := bn128.Fq12.Mul(r1, bn128.Fq12.Inverse(r2))
res := bn128.FinalExponentiation(rbe)
a := bn128.Fq12.Affine(res)
b := bn128.Fq12.Affine(bn128.Fq12.One())
assert.True(t, bn128.Fq12.Equal(a, b))
assert.True(t, bn128.Fq12.Equal(res, bn128.Fq12.One()))
}
func TestBN128_PairingFunction(t *testing.T) {
bn128, err := NewBn128()
assert.Nil(t, err)
big25 := big.NewInt(int64(25))
big30 := big.NewInt(int64(30))
g1a := bn128.G1.MulScalar(bn128.G1.G, big25)
g2a := bn128.G2.MulScalar(bn128.G2.G, big30)
g1b := bn128.G1.MulScalar(bn128.G1.G, big30)
g2b := bn128.G2.MulScalar(bn128.G2.G, big25)
pA, err := bn128.Pairing(g1a, g2a)
assert.Nil(t, err)
pB, err := bn128.Pairing(g1b, g2b)
assert.Nil(t, err)
assert.True(t, bn128.Fq12.Equal(pA, pB))
assert.Equal(t, pA[0][0][0].String(), "73680848340331011700282047627232219336104151861349893575958589557226556635706")
assert.Equal(t, bn128.Fq12.Affine(pA)[0][0][0].String(), "8016119724813186033542830391460394070015218389456422587891475873290878009957")
}

65
bn128/fq.go
View File

@@ -29,32 +29,37 @@ func (fq Fq) One() *big.Int {
// Add performs an addition on the Fq
func (fq Fq) Add(a, b *big.Int) *big.Int {
sum := new(big.Int).Add(a, b)
return new(big.Int).Mod(sum, fq.Q)
r := new(big.Int).Add(a, b)
// return new(big.Int).Mod(r, fq.Q)
return r
}
// Double performs a doubling on the Fq
func (fq Fq) Double(a *big.Int) *big.Int {
sum := new(big.Int).Add(a, a)
return new(big.Int).Mod(sum, fq.Q)
r := new(big.Int).Add(a, a)
// return new(big.Int).Mod(r, fq.Q)
return r
}
// Sub performs a substraction on the Fq
func (fq Fq) Sub(a, b *big.Int) *big.Int {
sum := new(big.Int).Sub(a, b)
return new(big.Int).Mod(sum, fq.Q)
r := new(big.Int).Sub(a, b)
// return new(big.Int).Mod(r, fq.Q)
return r
}
// Neg performs a negation on the Fq
func (fq Fq) Neg(a *big.Int) *big.Int {
m := new(big.Int).Neg(a)
return new(big.Int).Mod(m, fq.Q)
// return new(big.Int).Mod(m, fq.Q)
return m
}
// Mul performs a multiplication on the Fq
func (fq Fq) Mul(a, b *big.Int) *big.Int {
m := new(big.Int).Mul(a, b)
return new(big.Int).Mod(m, fq.Q)
// return m
}
func (fq Fq) MulScalar(base, e *big.Int) *big.Int {
@@ -64,12 +69,24 @@ func (fq Fq) MulScalar(base, e *big.Int) *big.Int {
// Inverse returns the inverse on the Fq
func (fq Fq) Inverse(a *big.Int) *big.Int {
return new(big.Int).ModInverse(a, fq.Q)
}
// Div performs a division on the Fq
func (fq Fq) Div(a, b *big.Int) *big.Int {
// not used in fq1, method added to fit the interface
return a
// q := bigCopy(fq.Q)
// t := big.NewInt(int64(0))
// r := fq.Q
// newt := big.NewInt(int64(0))
// newr := fq.Affine(a)
// for !bytes.Equal(newr.Bytes(), big.NewInt(int64(0)).Bytes()) {
// q := new(big.Int).Div(bigCopy(r), bigCopy(newr))
//
// t = bigCopy(newt)
// newt = fq.Sub(t, fq.Mul(q, newt))
//
// r = bigCopy(newr)
// newr = fq.Sub(r, fq.Mul(q, newr))
// }
// if t.Cmp(big.NewInt(0)) == -1 { // t< 0
// t = fq.Add(t, q)
// }
// return t
}
// Square performs a square operation on the Fq
@@ -87,8 +104,26 @@ func (fq Fq) Copy(a *big.Int) *big.Int {
}
func (fq Fq) Affine(a *big.Int) *big.Int {
return a
nq := fq.Neg(fq.Q)
aux := a
if aux.Cmp(big.NewInt(int64(0))) == -1 { // negative value
if aux.Cmp(nq) != 1 { // aux less or equal nq
aux = new(big.Int).Mod(aux, fq.Q)
}
if aux.Cmp(big.NewInt(int64(0))) == -1 { // negative value
aux = new(big.Int).Add(aux, fq.Q)
}
} else {
if aux.Cmp(fq.Q) != -1 { // aux greater or equal nq
aux = new(big.Int).Mod(aux, fq.Q)
}
}
return aux
}
func (fq Fq) Equal(a, b *big.Int) bool {
return bytes.Equal(a.Bytes(), b.Bytes())
aAff := fq.Affine(a)
bAff := fq.Affine(b)
return bytes.Equal(aAff.Bytes(), bAff.Bytes())
}

30
bn128/fq12.go
View File

@@ -31,7 +31,7 @@ func (fq12 Fq12) Zero() [2][3][2]*big.Int {
// One returns a One value on the Fq12
func (fq12 Fq12) One() [2][3][2]*big.Int {
return [2][3][2]*big.Int{fq12.F.One(), fq12.F.One()}
return [2][3][2]*big.Int{fq12.F.One(), fq12.F.Zero()}
}
func (fq12 Fq12) mulByNonResidue(a [3][2]*big.Int) [3][2]*big.Int {
@@ -70,7 +70,7 @@ func (fq12 Fq12) Neg(a [2][3][2]*big.Int) [2][3][2]*big.Int {
// Mul performs a multiplication on the Fq12
func (fq12 Fq12) Mul(a, b [2][3][2]*big.Int) [2][3][2]*big.Int {
// Multiplication and Squaring on Pairing-Friendly [2]*big.Ints.pdf; Section 3 (Karatsuba)
// Multiplication and Squaring on Pairing-Friendly .pdf; Section 3 (Karatsuba)
v0 := fq12.F.Mul(a[0], b[0])
v1 := fq12.F.Mul(a[1], b[1])
return [2][3][2]*big.Int{
@@ -84,6 +84,8 @@ func (fq12 Fq12) Mul(a, b [2][3][2]*big.Int) [2][3][2]*big.Int {
}
func (fq12 Fq12) MulScalar(base [2][3][2]*big.Int, e *big.Int) [2][3][2]*big.Int {
// for more possible implementations see g2.go file, at the function g2.MulScalar()
res := fq12.Zero()
rem := e
exp := base
@@ -133,3 +135,27 @@ func (fq12 Fq12) Square(a [2][3][2]*big.Int) [2][3][2]*big.Int {
fq12.F.Add(ab, ab),
}
}
func (fq12 Fq12) Exp(base [2][3][2]*big.Int, e *big.Int) [2][3][2]*big.Int {
res := fq12.One()
rem := fq12.Fq2.F.Copy(e)
exp := base
for !bytes.Equal(rem.Bytes(), big.NewInt(int64(0)).Bytes()) {
if BigIsOdd(rem) {
res = fq12.Mul(res, exp)
}
exp = fq12.Square(exp)
rem = new(big.Int).Rsh(rem, 1)
}
return res
}
func (fq12 Fq12) Affine(a [2][3][2]*big.Int) [2][3][2]*big.Int {
return [2][3][2]*big.Int{
fq12.F.Affine(a[0]),
fq12.F.Affine(a[1]),
}
}
func (fq12 Fq12) Equal(a, b [2][3][2]*big.Int) bool {
return fq12.F.Equal(a[0], b[0]) && fq12.F.Equal(a[1], b[1])
}

42
bn128/fq2.go
View File

@@ -1,7 +1,6 @@
package bn128
import (
"bytes"
"math/big"
)
@@ -27,7 +26,7 @@ func (fq2 Fq2) Zero() [2]*big.Int {
// One returns a One value on the Fq2
func (fq2 Fq2) One() [2]*big.Int {
return [2]*big.Int{fq2.F.One(), fq2.F.One()}
return [2]*big.Int{fq2.F.One(), fq2.F.Zero()}
}
func (fq2 Fq2) mulByNonResidue(a *big.Int) *big.Int {
@@ -63,6 +62,7 @@ func (fq2 Fq2) Neg(a [2]*big.Int) [2]*big.Int {
// Mul performs a multiplication on the Fq2
func (fq2 Fq2) Mul(a, b [2]*big.Int) [2]*big.Int {
// Multiplication and Squaring on Pairing-Friendly.pdf; Section 3 (Karatsuba)
// https://pdfs.semanticscholar.org/3e01/de88d7428076b2547b60072088507d881bf1.pdf
v0 := fq2.F.Mul(a[0], b[0])
v1 := fq2.F.Mul(a[1], b[1])
return [2]*big.Int{
@@ -74,24 +74,31 @@ func (fq2 Fq2) Mul(a, b [2]*big.Int) [2]*big.Int {
fq2.F.Add(v0, v1)),
}
}
func (fq2 Fq2) MulScalar(base [2]*big.Int, e *big.Int) [2]*big.Int {
res := fq2.Zero()
rem := e
exp := base
for !bytes.Equal(rem.Bytes(), big.NewInt(int64(0)).Bytes()) {
// if rem % 2 == 1
if bytes.Equal(new(big.Int).Rem(rem, big.NewInt(int64(2))).Bytes(), big.NewInt(int64(1)).Bytes()) {
res = fq2.Add(res, exp)
func (fq2 Fq2) MulScalar(p [2]*big.Int, e *big.Int) [2]*big.Int {
// for more possible implementations see g2.go file, at the function g2.MulScalar()
q := fq2.Zero()
d := fq2.F.Copy(e)
r := p
foundone := false
for i := d.BitLen(); i >= 0; i-- {
if foundone {
q = fq2.Double(q)
}
if d.Bit(i) == 1 {
foundone = true
q = fq2.Add(q, r)
}
exp = fq2.Double(exp)
rem = rem.Rsh(rem, 1) // rem = rem >> 1
}
return res
return q
}
// Inverse returns the inverse on the Fq2
func (fq2 Fq2) Inverse(a [2]*big.Int) [2]*big.Int {
// High-Speed Software Implementation of the Optimal Ate Pairing over Barreto–Naehrig Curves .pdf
// https://eprint.iacr.org/2010/354.pdf , algorithm 8
t0 := fq2.F.Square(a[0])
t1 := fq2.F.Square(a[1])
t2 := fq2.F.Sub(t0, fq2.mulByNonResidue(t1))
@@ -109,8 +116,8 @@ func (fq2 Fq2) Div(a, b [2]*big.Int) [2]*big.Int {
// Square performs a square operation on the Fq2
func (fq2 Fq2) Square(a [2]*big.Int) [2]*big.Int {
// https://pdfs.semanticscholar.org/3e01/de88d7428076b2547b60072088507d881bf1.pdf , complex squaring
ab := fq2.F.Mul(a[0], a[1])
return [2]*big.Int{
fq2.F.Sub(
fq2.F.Mul(
@@ -138,3 +145,10 @@ func (fq2 Fq2) Affine(a [2]*big.Int) [2]*big.Int {
func (fq2 Fq2) Equal(a, b [2]*big.Int) bool {
return fq2.F.Equal(a[0], b[0]) && fq2.F.Equal(a[1], b[1])
}
func (fq2 Fq2) Copy(a [2]*big.Int) [2]*big.Int {
return [2]*big.Int{
fq2.F.Copy(a[0]),
fq2.F.Copy(a[1]),
}
}

23
bn128/fq6.go
View File

@@ -27,7 +27,7 @@ func (fq6 Fq6) Zero() [3][2]*big.Int {
// One returns a One value on the Fq6
func (fq6 Fq6) One() [3][2]*big.Int {
return [3][2]*big.Int{fq6.F.One(), fq6.F.One(), fq6.F.One()}
return [3][2]*big.Int{fq6.F.One(), fq6.F.Zero(), fq6.F.Zero()}
}
func (fq6 Fq6) mulByNonResidue(a [2]*big.Int) [2]*big.Int {
@@ -95,6 +95,8 @@ func (fq6 Fq6) Mul(a, b [3][2]*big.Int) [3][2]*big.Int {
}
func (fq6 Fq6) MulScalar(base [3][2]*big.Int, e *big.Int) [3][2]*big.Int {
// for more possible implementations see g2.go file, at the function g2.MulScalar()
res := fq6.Zero()
rem := e
exp := base
@@ -169,3 +171,22 @@ func (fq6 Fq6) Square(a [3][2]*big.Int) [3][2]*big.Int {
fq6.F.Add(s0, s4)),
}
}
func (fq6 Fq6) Affine(a [3][2]*big.Int) [3][2]*big.Int {
return [3][2]*big.Int{
fq6.F.Affine(a[0]),
fq6.F.Affine(a[1]),
fq6.F.Affine(a[2]),
}
}
func (fq6 Fq6) Equal(a, b [3][2]*big.Int) bool {
return fq6.F.Equal(a[0], b[0]) && fq6.F.Equal(a[1], b[1]) && fq6.F.Equal(a[2], b[2])
}
func (fq6 Fq6) Copy(a [3][2]*big.Int) [3][2]*big.Int {
return [3][2]*big.Int{
fq6.F.Copy(a[0]),
fq6.F.Copy(a[1]),
fq6.F.Copy(a[2]),
}
}

76
bn128/fqn_test.go
View File

@@ -23,19 +23,19 @@ func TestFq1(t *testing.T) {
fq1 := NewFq(iToBig(7))
res := fq1.Add(iToBig(4), iToBig(4))
assert.Equal(t, iToBig(1), res)
assert.Equal(t, iToBig(1), fq1.Affine(res))
res = fq1.Double(iToBig(5))
assert.Equal(t, iToBig(3), res)
assert.Equal(t, iToBig(3), fq1.Affine(res))
res = fq1.Sub(iToBig(5), iToBig(7))
assert.Equal(t, iToBig(5), res)
assert.Equal(t, iToBig(5), fq1.Affine(res))
res = fq1.Neg(iToBig(5))
assert.Equal(t, iToBig(2), res)
assert.Equal(t, iToBig(2), fq1.Affine(res))
res = fq1.Mul(iToBig(5), iToBig(11))
assert.Equal(t, iToBig(6), res)
assert.Equal(t, iToBig(6), fq1.Affine(res))
res = fq1.Inverse(iToBig(4))
assert.Equal(t, iToBig(2), res)
@@ -46,7 +46,7 @@ func TestFq1(t *testing.T) {
func TestFq2(t *testing.T) {
fq1 := NewFq(iToBig(7))
nonResidueFq2str := "-1" // i / Beta
nonResidueFq2str := "-1" // i/j
nonResidueFq2, ok := new(big.Int).SetString(nonResidueFq2str, 10)
assert.True(t, ok)
assert.Equal(t, nonResidueFq2.String(), nonResidueFq2str)
@@ -54,45 +54,39 @@ func TestFq2(t *testing.T) {
fq2 := Fq2{fq1, nonResidueFq2}
res := fq2.Add(iiToBig(4, 4), iiToBig(3, 4))
assert.Equal(t, iiToBig(0, 1), res)
assert.Equal(t, iiToBig(0, 1), fq2.Affine(res))
res = fq2.Double(iiToBig(5, 3))
assert.Equal(t, iiToBig(3, 6), res)
assert.Equal(t, iiToBig(3, 6), fq2.Affine(res))
res = fq2.Sub(iiToBig(5, 3), iiToBig(7, 2))
assert.Equal(t, iiToBig(5, 1), res)
assert.Equal(t, iiToBig(5, 1), fq2.Affine(res))
res = fq2.Neg(iiToBig(4, 4))
assert.Equal(t, iiToBig(3, 3), res)
assert.Equal(t, iiToBig(3, 3), fq2.Affine(res))
res = fq2.Mul(iiToBig(4, 4), iiToBig(3, 4))
assert.Equal(t, iiToBig(3, 0), res)
assert.Equal(t, iiToBig(3, 0), fq2.Affine(res))
res = fq2.Inverse(iiToBig(4, 4))
assert.Equal(t, iiToBig(1, 6), res)
res = fq2.Div(iiToBig(4, 4), iiToBig(3, 4))
assert.Equal(t, iiToBig(0, 6), res)
assert.Equal(t, iiToBig(1, 6), fq2.Affine(res))
res = fq2.Square(iiToBig(4, 4))
assert.Equal(t, iiToBig(0, 4), res)
assert.Equal(t, iiToBig(0, 4), fq2.Affine(res))
res2 := fq2.Mul(iiToBig(4, 4), iiToBig(4, 4))
assert.Equal(t, res, res2)
assert.Equal(t, fq2.Affine(res), fq2.Affine(res2))
assert.True(t, fq2.Equal(res, res2))
res = fq2.Square(iiToBig(3, 5))
assert.Equal(t, iiToBig(5, 2), res)
assert.Equal(t, iiToBig(5, 2), fq2.Affine(res))
res2 = fq2.Mul(iiToBig(3, 5), iiToBig(3, 5))
assert.Equal(t, res, res2)
assert.Equal(t, fq2.Affine(res), fq2.Affine(res2))
}
func TestFq6(t *testing.T) {
fq1 := NewFq(big.NewInt(int64(7)))
nonResidueFq2, ok := new(big.Int).SetString("-1", 10) // i
assert.True(t, ok)
nonResidueFq6 := iiToBig(9, 1) // TODO
bn128, err := NewBn128()
assert.Nil(t, err)
fq2 := Fq2{fq1, nonResidueFq2}
fq6 := Fq6{fq2, nonResidueFq6}
a := [3][2]*big.Int{
iiToBig(1, 2),
iiToBig(3, 4),
@@ -102,33 +96,9 @@ func TestFq6(t *testing.T) {
iiToBig(10, 9),
iiToBig(8, 7)}
res := fq6.Add(a, b)
assert.Equal(t,
[3][2]*big.Int{
iiToBig(6, 6),
iiToBig(6, 6),
iiToBig(6, 6)},
res)
res = fq6.Sub(a, b)
assert.Equal(t,
[3][2]*big.Int{
iiToBig(3, 5),
iiToBig(0, 2),
iiToBig(4, 6)},
res)
res = fq6.Mul(a, b)
assert.Equal(t,
[3][2]*big.Int{
iiToBig(5, 0),
iiToBig(2, 1),
iiToBig(3, 0)},
res)
mulRes := fq6.Mul(a, b)
divRes := fq6.Div(mulRes, b)
assert.Equal(t, a, divRes)
mulRes := bn128.Fq6.Mul(a, b)
divRes := bn128.Fq6.Div(mulRes, b)
assert.Equal(t, bn128.Fq6.Affine(a), bn128.Fq6.Affine(divRes))
}
func TestFq12(t *testing.T) {
@@ -186,5 +156,5 @@ func TestFq12(t *testing.T) {
mulRes := fq12.Mul(a, b)
divRes := fq12.Div(mulRes, b)
assert.Equal(t, a, divRes)
assert.Equal(t, fq12.Affine(a), fq12.Affine(divRes))
}

50
bn128/g1.go
View File

@@ -1,7 +1,6 @@
package bn128
import (
"bytes"
"math/big"
)
@@ -117,7 +116,7 @@ func (g1 G1) Double(p [3]*big.Int) [3]*big.Int {
t3 := g1.F.Sub(t2, c)
d := g1.F.Double(t3)
e := g1.F.Add(g1.F.Add(a, a), a) // e = 3*a
e := g1.F.Add(g1.F.Add(a, a), a)
f := g1.F.Square(e)
t4 := g1.F.Double(d)
@@ -136,21 +135,21 @@ func (g1 G1) Double(p [3]*big.Int) [3]*big.Int {
return [3]*big.Int{x3, y3, z3}
}
func (g1 G1) MulScalar(base [3]*big.Int, e *big.Int) [3]*big.Int {
// res := g1.Zero()
res := [3]*big.Int{g1.F.Zero(), g1.F.Zero(), g1.F.Zero()}
rem := e
exp := base
func (g1 G1) MulScalar(p [3]*big.Int, e *big.Int) [3]*big.Int {
// https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication#Double-and-add
// for more possible implementations see g2.go file, at the function g2.MulScalar()
for !bytes.Equal(rem.Bytes(), big.NewInt(int64(0)).Bytes()) {
// if rem % 2 == 1
if bytes.Equal(new(big.Int).Rem(rem, big.NewInt(int64(2))).Bytes(), big.NewInt(int64(1)).Bytes()) {
res = g1.Add(res, exp)
q := [3]*big.Int{g1.F.Zero(), g1.F.Zero(), g1.F.Zero()}
d := g1.F.Copy(e)
r := p
for i := d.BitLen() - 1; i >= 0; i-- {
q = g1.Double(q)
if d.Bit(i) == 1 {
q = g1.Add(q, r)
}
exp = g1.Double(exp)
rem = rem.Rsh(rem, 1) // rem = rem >> 1
}
return res
return q
}
func (g1 G1) Affine(p [3]*big.Int) [2]*big.Int {
@@ -167,3 +166,26 @@ func (g1 G1) Affine(p [3]*big.Int) [2]*big.Int {
return [2]*big.Int{x, y}
}
func (g1 G1) Equal(p1, p2 [3]*big.Int) bool {
if g1.IsZero(p1) {
return g1.IsZero(p2)
}
if g1.IsZero(p2) {
return g1.IsZero(p1)
}
z1z1 := g1.F.Square(p1[2])
z2z2 := g1.F.Square(p2[2])
u1 := g1.F.Mul(p1[0], z2z2)
u2 := g1.F.Mul(p2[0], z1z1)
z1cub := g1.F.Mul(p1[2], z1z1)
z2cub := g1.F.Mul(p2[2], z2z2)
s1 := g1.F.Mul(p1[1], z2cub)
s2 := g1.F.Mul(p2[1], z1cub)
return g1.F.Equal(u1, u2) && g1.F.Equal(s1, s2)
}

7
bn128/g1_test.go
View File

@@ -18,10 +18,11 @@ func TestG1(t *testing.T) {
gr1 := bn128.G1.MulScalar(bn128.G1.G, bn128.Fq1.Copy(r1))
gr2 := bn128.G1.MulScalar(bn128.G1.G, bn128.Fq1.Copy(r2))
grsum1 := bn128.G1.Add(gr1, gr2)
r1r2 := bn128.Fq1.Add(r1, r2)
grsum2 := bn128.G1.MulScalar(bn128.G1.G, r1r2)
grsum1 := bn128.G1.Add(gr1, gr2) // g*33 + g*44
r1r2 := bn128.Fq1.Add(r1, r2) // 33 + 44
grsum2 := bn128.G1.MulScalar(bn128.G1.G, r1r2) // g * (33+44)
assert.True(t, bn128.G1.Equal(grsum1, grsum2))
a := bn128.G1.Affine(grsum1)
b := bn128.G1.Affine(grsum2)
assert.Equal(t, a, b)

83
bn128/g2.go
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@@ -1,7 +1,6 @@
package bn128
import (
"bytes"
"math/big"
)
@@ -119,7 +118,7 @@ func (g2 G2) Double(p [3][2]*big.Int) [3][2]*big.Int {
t3 := g2.F.Sub(t2, c)
d := g2.F.Double(t3)
e := g2.F.Add(g2.F.Add(a, a), a) // e = 3*a
e := g2.F.Add(g2.F.Add(a, a), a)
f := g2.F.Square(e)
t4 := g2.F.Double(d)
@@ -138,21 +137,45 @@ func (g2 G2) Double(p [3][2]*big.Int) [3][2]*big.Int {
return [3][2]*big.Int{x3, y3, z3}
}
func (g2 G2) MulScalar(base [3][2]*big.Int, e *big.Int) [3][2]*big.Int {
// res := g2.Zero()
res := [3][2]*big.Int{g2.F.Zero(), g2.F.Zero(), g2.F.Zero()}
rem := e
exp := base
func (g2 G2) MulScalar(p [3][2]*big.Int, e *big.Int) [3][2]*big.Int {
// https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication#Double-and-add
for !bytes.Equal(rem.Bytes(), big.NewInt(int64(0)).Bytes()) {
// if rem % 2 == 1
if bytes.Equal(new(big.Int).Rem(rem, big.NewInt(int64(2))).Bytes(), big.NewInt(int64(1)).Bytes()) {
res = g2.Add(res, exp)
q := [3][2]*big.Int{g2.F.Zero(), g2.F.Zero(), g2.F.Zero()}
d := g2.F.F.Copy(e) // d := e
r := p
/*
here are three possible implementations:
*/
/* index decreasing: */
for i := d.BitLen() - 1; i >= 0; i-- {
q = g2.Double(q)
if d.Bit(i) == 1 {
q = g2.Add(q, r)
}
exp = g2.Double(exp)
rem = rem.Rsh(rem, 1) // rem = rem >> 1
}
return res
/* index increasing: */
// for i := 0; i <= d.BitLen(); i++ {
// if d.Bit(i) == 1 {
// q = g2.Add(q, r)
// }
// r = g2.Double(r)
// }
// foundone := false
// for i := d.BitLen(); i >= 0; i-- {
// if foundone {
// q = g2.Double(q)
// }
// if d.Bit(i) == 1 {
// foundone = true
// q = g2.Add(q, r)
// }
// }
return q
}
func (g2 G2) Affine(p [3][2]*big.Int) [3][2]*big.Int {
@@ -168,11 +191,31 @@ func (g2 G2) Affine(p [3][2]*big.Int) [3][2]*big.Int {
y := g2.F.Mul(p[1], zinv3)
return [3][2]*big.Int{
x,
y,
[2]*big.Int{
big.NewInt(int64(0)),
big.NewInt(int64(0)),
},
g2.F.Affine(x),
g2.F.Affine(y),
g2.F.One(),
}
}
func (g2 G2) Equal(p1, p2 [3][2]*big.Int) bool {
if g2.IsZero(p1) {
return g2.IsZero(p2)
}
if g2.IsZero(p2) {
return g2.IsZero(p1)
}
z1z1 := g2.F.Square(p1[2])
z2z2 := g2.F.Square(p2[2])
u1 := g2.F.Mul(p1[0], z2z2)
u2 := g2.F.Mul(p2[0], z1z1)
z1cub := g2.F.Mul(p1[2], z1z1)
z2cub := g2.F.Mul(p2[2], z2z2)
s1 := g2.F.Mul(p1[1], z2cub)
s2 := g2.F.Mul(p2[1], z1cub)
return g2.F.Equal(u1, u2) && g2.F.Equal(s1, s2)
}

15
bn128/g2_test.go
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@@ -14,14 +14,11 @@ func TestG2(t *testing.T) {
r1 := big.NewInt(int64(33))
r2 := big.NewInt(int64(44))
gr1 := bn128.G2.MulScalar(bn128.G2.G, bn128.Fq1.Copy(r1))
gr2 := bn128.G2.MulScalar(bn128.G2.G, bn128.Fq1.Copy(r2))
gr1 := bn128.G2.Affine(bn128.G2.MulScalar(bn128.G2.G, r1))
gr2 := bn128.G2.Affine(bn128.G2.MulScalar(bn128.G2.G, r2))
grsum1 := bn128.G2.Add(gr1, gr2)
r1r2 := bn128.Fq1.Add(r1, r2)
grsum2 := bn128.G2.MulScalar(bn128.G2.G, r1r2)
a := bn128.G2.Affine(grsum1)
b := bn128.G2.Affine(grsum2)
assert.Equal(t, a, b)
grsum1 := bn128.G2.Affine(bn128.G2.Add(gr1, gr2))
r1r2 := bn128.Fq1.Affine(bn128.Fq1.Add(r1, r2))
grsum2 := bn128.G2.Affine(bn128.G2.MulScalar(bn128.G2.G, r1r2))
assert.True(t, bn128.G2.Equal(grsum1, grsum2))
}

2
paillier/paillier.go
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@@ -10,7 +10,7 @@ import (
)
const (
bits = 16
bits = 18
)
// PublicKey stores the public key data

17
shamirsecretsharing/shamirsecretsharing_test.go
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@@ -3,7 +3,6 @@ package shamirsecretsharing
import (
"bytes"
"crypto/rand"
"fmt"
"math/big"
"testing"
@@ -31,14 +30,14 @@ func TestCreate(t *testing.T) {
sharesToUse = append(sharesToUse, shares[0])
secr := LagrangeInterpolation(sharesToUse, p)
fmt.Print("original secret: ")
fmt.Println(k)
fmt.Print("p: ")
fmt.Println(p)
fmt.Print("shares: ")
fmt.Println(shares)
fmt.Print("secret result: ")
fmt.Println(secr)
// fmt.Print("original secret: ")
// fmt.Println(k)
// fmt.Print("p: ")
// fmt.Println(p)
// fmt.Print("shares: ")
// fmt.Println(shares)
// fmt.Print("secret result: ")
// fmt.Println(secr)
if !bytes.Equal(k.Bytes(), secr.Bytes()) {
t.Errorf("reconstructed secret not correspond to original secret")
}