diff --git a/README.md b/README.md
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--- /dev/null
+++ b/README.md
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+# go-snark [![Go Report Card](https://goreportcard.com/badge/github.com/arnaucube/go-snark)](https://goreportcard.com/report/github.com/arnaucube/go-snark)
+
+Not finished, work in progress (doing this in my free time, so I don't have much time).
+
+
+
+#### Test
+```
+go test ./... -v
+```
+
+## R1CS to Quadratic Arithmetic Program
+- `Succinct Non-Interactive Zero Knowledge for a von Neumann Architecture`, Eli Ben-Sasson, Alessandro Chiesa, Eran Tromer, Madars Virza https://eprint.iacr.org/2013/879.pdf
+- Vitalik Buterin blog post about QAP https://medium.com/@VitalikButerin/quadratic-arithmetic-programs-from-zero-to-hero-f6d558cea649
+- Ariel Gabizon in Zcash blog https://z.cash/blog/snark-explain5
+- Lagrange polynomial Wikipedia article https://en.wikipedia.org/wiki/Lagrange_polynomial
+
+#### Usage
+- R1CS to QAP
+```go
+b0 := big.NewFloat(float64(0))
+b1 := big.NewFloat(float64(1))
+b5 := big.NewFloat(float64(5))
+a := [][]*big.Float{
+ []*big.Float{b0, b1, b0, b0, b0, b0},
+ []*big.Float{b0, b0, b0, b1, b0, b0},
+ []*big.Float{b0, b1, b0, b0, b1, b0},
+ []*big.Float{b5, b0, b0, b0, b0, b1},
+}
+b := [][]*big.Float{
+ []*big.Float{b0, b1, b0, b0, b0, b0},
+ []*big.Float{b0, b1, b0, b0, b0, b0},
+ []*big.Float{b1, b0, b0, b0, b0, b0},
+ []*big.Float{b1, b0, b0, b0, b0, b0},
+}
+c := [][]*big.Float{
+ []*big.Float{b0, b0, b0, b1, b0, b0},
+ []*big.Float{b0, b0, b0, b0, b1, b0},
+ []*big.Float{b0, b0, b0, b0, b0, b1},
+ []*big.Float{b0, b0, b1, b0, b0, b0},
+}
+alpha, beta, gamma, z := R1CSToQAP(a, b, c)
+fmt.Println(alpha)
+fmt.Println(beta)
+fmt.Println(gamma)
+fmt.Println(z)
+/*
+out:
+alpha: [[-5 9.166666666666666 -5 0.8333333333333334] [8 -11.333333333333332 5 -0.6666666666666666] [0 0 0 0] [-6 9.5 -4 0.5] [4 -7 3.5 -0.5] [-1 1.8333333333333333 -1 0.16666666666666666]]
+beta: [[3 -5.166666666666667 2.5 -0.33333333333333337] [-2 5.166666666666667 -2.5 0.33333333333333337] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0]]
+gamma: [[0 0 0 0] [0 0 0 0] [-1 1.8333333333333333 -1 0.16666666666666666] [4 -4.333333333333333 1.5 -0.16666666666666666] [-6 9.5 -4 0.5] [4 -7 3.5 -0.5]]
+z: [24 -50 35 -10 1]
+*/
+```
+
+## Bn128
+Implementation of the bn128 pairing.
+
+
+Implementation followng the information 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 , 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
+- `Implementing Cryptographic Pairings over Barreto-Naehrig Curves`, Augusto Jun Devegili, Michael Scott, Ricardo Dahab https://eprint.iacr.org/2007/390.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
+
+
+#### Usage
+
+- 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))
+```
diff --git a/bn128/LICENSE b/bn128/LICENSE
new file mode 100644
index 0000000..f288702
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+++ b/bn128/LICENSE
@@ -0,0 +1,674 @@
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+.
diff --git a/bn128/README.md b/bn128/README.md
new file mode 100644
index 0000000..1d4ed9f
--- /dev/null
+++ b/bn128/README.md
@@ -0,0 +1,186 @@
+## Bn128
+Implementation of the bn128 pairing.
+
+
+Implementation followng the information 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 , 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
+- `Implementing Cryptographic Pairings over Barreto-Naehrig Curves`, Augusto Jun Devegili, Michael Scott, Ricardo Dahab https://eprint.iacr.org/2007/390.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
+
+### Installation
+```
+go get github.com/arnaucube/bn128
+```
+
+#### Usage
+
+- 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))
+```
+
+#### Test
+```
+go test -v
+```
+
+##### Internal operations more deeply
+
+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))
+}
+
+func iiToBig(a, b int) [2]*big.Int {
+ return [2]*big.Int{iToBig(a), iToBig(b)}
+}
+
+func iiiToBig(a, b int) [2]*big.Int {
+ return [2]*big.Int{iToBig(a), iToBig(b)}
+}
+```
+- Finite Fields (1, 2, 6, 12) operations
+```go
+// new finite field of order 1
+fq1 := NewFq(iToBig(7))
+
+// basic operations of finite field 1
+res := fq1.Add(iToBig(4), iToBig(4))
+res = fq1.Double(iToBig(5))
+res = fq1.Sub(iToBig(5), iToBig(7))
+res = fq1.Neg(iToBig(5))
+res = fq1.Mul(iToBig(5), iToBig(11))
+res = fq1.Inverse(iToBig(4))
+res = fq1.Square(iToBig(5))
+
+// new finite field of order 2
+nonResidueFq2str := "-1" // i/j
+nonResidueFq2, ok := new(big.Int).SetString(nonResidueFq2str, 10)
+fq2 := Fq2{fq1, nonResidueFq2}
+nonResidueFq6 := iiToBig(9, 1)
+
+// basic operations of finite field of order 2
+res := fq2.Add(iiToBig(4, 4), iiToBig(3, 4))
+res = fq2.Double(iiToBig(5, 3))
+res = fq2.Sub(iiToBig(5, 3), iiToBig(7, 2))
+res = fq2.Neg(iiToBig(4, 4))
+res = fq2.Mul(iiToBig(4, 4), iiToBig(3, 4))
+res = fq2.Inverse(iiToBig(4, 4))
+res = fq2.Div(iiToBig(4, 4), iiToBig(3, 4))
+res = fq2.Square(iiToBig(4, 4))
+
+
+// new finite field of order 6
+nonResidueFq6 := iiToBig(9, 1) // TODO
+fq6 := Fq6{fq2, nonResidueFq6}
+
+// define two new values of Finite Field 6, in order to be able to perform the operations
+a := [3][2]*big.Int{
+ iiToBig(1, 2),
+ iiToBig(3, 4),
+ iiToBig(5, 6)}
+b := [3][2]*big.Int{
+ iiToBig(12, 11),
+ iiToBig(10, 9),
+ iiToBig(8, 7)}
+
+// basic operations of finite field order 6
+res := fq6.Add(a, b)
+res = fq6.Sub(a, b)
+res = fq6.Mul(a, b)
+divRes := fq6.Div(mulRes, b)
+
+
+// new finite field of order 12
+q, ok := new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208583", 10) // i
+if !ok {
+ fmt.Println("error parsing string to big integer")
+}
+
+fq1 := NewFq(q)
+nonResidueFq2, ok := new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208582", 10) // i
+assert.True(t, ok)
+nonResidueFq6 := iiToBig(9, 1)
+
+fq2 := Fq2{fq1, nonResidueFq2}
+fq6 := Fq6{fq2, nonResidueFq6}
+fq12 := Fq12{fq6, fq2, nonResidueFq6}
+
+```
+
+- G1 operations
+```go
+bn128, err := NewBn128()
+assert.Nil(t, err)
+
+r1 := big.NewInt(int64(33))
+r2 := big.NewInt(int64(44))
+
+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)
+
+a := bn128.G1.Affine(grsum1)
+b := bn128.G1.Affine(grsum2)
+assert.Equal(t, a, b)
+assert.Equal(t, "0x2f978c0ab89ebaa576866706b14787f360c4d6c3869efe5a72f7c3651a72ff00", utils.BytesToHex(a[0].Bytes()))
+assert.Equal(t, "0x12e4ba7f0edca8b4fa668fe153aebd908d322dc26ad964d4cd314795844b62b2", utils.BytesToHex(a[1].Bytes()))
+```
+
+- G2 operations
+```go
+bn128, err := NewBn128()
+assert.Nil(t, err)
+
+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))
+
+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)
+```
diff --git a/bn128/bn128.go b/bn128/bn128.go
new file mode 100644
index 0000000..95a4ac8
--- /dev/null
+++ b/bn128/bn128.go
@@ -0,0 +1,407 @@
+package bn128
+
+import (
+ "bytes"
+ "errors"
+ "math/big"
+)
+
+type Bn128 struct {
+ Q *big.Int
+ Gg1 [2]*big.Int
+ Gg2 [2][2]*big.Int
+ NonResidueFq2 *big.Int
+ NonResidueFq6 [2]*big.Int
+ Fq1 Fq
+ Fq2 Fq2
+ Fq6 Fq6
+ 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) {
+ var b Bn128
+ q, ok := new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208583", 10) // i
+ if !ok {
+ return b, errors.New("err with q")
+ }
+ b.Q = q
+
+ b.Gg1 = [2]*big.Int{
+ big.NewInt(int64(1)),
+ big.NewInt(int64(2)),
+ }
+
+ g2_00, ok := new(big.Int).SetString("10857046999023057135944570762232829481370756359578518086990519993285655852781", 10)
+ if !ok {
+ return b, errors.New("err with g2_00")
+ }
+ g2_01, ok := new(big.Int).SetString("11559732032986387107991004021392285783925812861821192530917403151452391805634", 10)
+ if !ok {
+ return b, errors.New("err with g2_00")
+ }
+ g2_10, ok := new(big.Int).SetString("8495653923123431417604973247489272438418190587263600148770280649306958101930", 10)
+ if !ok {
+ return b, errors.New("err with g2_00")
+ }
+ g2_11, ok := new(big.Int).SetString("4082367875863433681332203403145435568316851327593401208105741076214120093531", 10)
+ if !ok {
+ return b, errors.New("err with g2_00")
+ }
+
+ b.Gg2 = [2][2]*big.Int{
+ [2]*big.Int{
+ g2_00,
+ g2_01,
+ },
+ [2]*big.Int{
+ g2_10,
+ g2_11,
+ },
+ }
+
+ b.Fq1 = NewFq(q)
+ b.NonResidueFq2, ok = new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208582", 10) // i
+ if !ok {
+ return b, errors.New("err with nonResidueFq2")
+ }
+ b.NonResidueFq6 = [2]*big.Int{
+ big.NewInt(int64(9)),
+ big.NewInt(int64(1)),
+ }
+
+ 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
+}
diff --git a/bn128/bn128_test.go b/bn128/bn128_test.go
new file mode 100644
index 0000000..f565258
--- /dev/null
+++ b/bn128/bn128_test.go
@@ -0,0 +1,66 @@
+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)
+ assert.Nil(t, err)
+
+ 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 TestBN128Pairing(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")
+}
diff --git a/bn128/fq.go b/bn128/fq.go
new file mode 100644
index 0000000..a736f83
--- /dev/null
+++ b/bn128/fq.go
@@ -0,0 +1,129 @@
+package bn128
+
+import (
+ "bytes"
+ "math/big"
+)
+
+// Fq is the Z field over modulus Q
+type Fq struct {
+ Q *big.Int // Q
+}
+
+// NewFq generates a new Fq
+func NewFq(q *big.Int) Fq {
+ return Fq{
+ q,
+ }
+}
+
+// Zero returns a Zero value on the Fq
+func (fq Fq) Zero() *big.Int {
+ return big.NewInt(int64(0))
+}
+
+// One returns a One value on the Fq
+func (fq Fq) One() *big.Int {
+ return big.NewInt(int64(1))
+}
+
+// Add performs an addition on the Fq
+func (fq Fq) Add(a, b *big.Int) *big.Int {
+ 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 {
+ r := new(big.Int).Add(a, a)
+ // return new(big.Int).Mod(r, fq.Q)
+ return r
+}
+
+// Sub performs a subtraction on the Fq
+func (fq Fq) Sub(a, b *big.Int) *big.Int {
+ 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 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 {
+ return fq.Mul(base, e)
+}
+
+// Inverse returns the inverse on the Fq
+func (fq Fq) Inverse(a *big.Int) *big.Int {
+ return new(big.Int).ModInverse(a, fq.Q)
+ // 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
+func (fq Fq) Square(a *big.Int) *big.Int {
+ m := new(big.Int).Mul(a, a)
+ return new(big.Int).Mod(m, fq.Q)
+}
+
+func (fq Fq) IsZero(a *big.Int) bool {
+ return bytes.Equal(a.Bytes(), fq.Zero().Bytes())
+}
+
+func (fq Fq) Copy(a *big.Int) *big.Int {
+ return new(big.Int).SetBytes(a.Bytes())
+}
+
+func (fq Fq) Affine(a *big.Int) *big.Int {
+ 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 {
+ aAff := fq.Affine(a)
+ bAff := fq.Affine(b)
+ return bytes.Equal(aAff.Bytes(), bAff.Bytes())
+}
diff --git a/bn128/fq12.go b/bn128/fq12.go
new file mode 100644
index 0000000..490ad97
--- /dev/null
+++ b/bn128/fq12.go
@@ -0,0 +1,161 @@
+package bn128
+
+import (
+ "bytes"
+ "math/big"
+)
+
+// Fq12 uses the same algorithms than Fq2, but with [2][3][2]*big.Int data structure
+
+// Fq12 is Field 12
+type Fq12 struct {
+ F Fq6
+ Fq2 Fq2
+ NonResidue [2]*big.Int
+}
+
+// NewFq12 generates a new Fq12
+func NewFq12(f Fq6, fq2 Fq2, nonResidue [2]*big.Int) Fq12 {
+ fq12 := Fq12{
+ f,
+ fq2,
+ nonResidue,
+ }
+ return fq12
+}
+
+// Zero returns a Zero value on the Fq12
+func (fq12 Fq12) Zero() [2][3][2]*big.Int {
+ return [2][3][2]*big.Int{fq12.F.Zero(), fq12.F.Zero()}
+}
+
+// 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.Zero()}
+}
+
+func (fq12 Fq12) mulByNonResidue(a [3][2]*big.Int) [3][2]*big.Int {
+ return [3][2]*big.Int{
+ fq12.Fq2.Mul(fq12.NonResidue, a[2]),
+ a[0],
+ a[1],
+ }
+}
+
+// Add performs an addition on the Fq12
+func (fq12 Fq12) Add(a, b [2][3][2]*big.Int) [2][3][2]*big.Int {
+ return [2][3][2]*big.Int{
+ fq12.F.Add(a[0], b[0]),
+ fq12.F.Add(a[1], b[1]),
+ }
+}
+
+// Double performs a doubling on the Fq12
+func (fq12 Fq12) Double(a [2][3][2]*big.Int) [2][3][2]*big.Int {
+ return fq12.Add(a, a)
+}
+
+// Sub performs a subtraction on the Fq12
+func (fq12 Fq12) Sub(a, b [2][3][2]*big.Int) [2][3][2]*big.Int {
+ return [2][3][2]*big.Int{
+ fq12.F.Sub(a[0], b[0]),
+ fq12.F.Sub(a[1], b[1]),
+ }
+}
+
+// Neg performs a negation on the Fq12
+func (fq12 Fq12) Neg(a [2][3][2]*big.Int) [2][3][2]*big.Int {
+ return fq12.Sub(fq12.Zero(), a)
+}
+
+// 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 .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{
+ fq12.F.Add(v0, fq12.mulByNonResidue(v1)),
+ fq12.F.Sub(
+ fq12.F.Mul(
+ fq12.F.Add(a[0], a[1]),
+ fq12.F.Add(b[0], b[1])),
+ fq12.F.Add(v0, v1)),
+ }
+}
+
+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
+
+ 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 = fq12.Add(res, exp)
+ }
+ exp = fq12.Double(exp)
+ rem = rem.Rsh(rem, 1) // rem = rem >> 1
+ }
+ return res
+}
+
+// Inverse returns the inverse on the Fq12
+func (fq12 Fq12) Inverse(a [2][3][2]*big.Int) [2][3][2]*big.Int {
+ t0 := fq12.F.Square(a[0])
+ t1 := fq12.F.Square(a[1])
+ t2 := fq12.F.Sub(t0, fq12.mulByNonResidue(t1))
+ t3 := fq12.F.Inverse(t2)
+ return [2][3][2]*big.Int{
+ fq12.F.Mul(a[0], t3),
+ fq12.F.Neg(fq12.F.Mul(a[1], t3)),
+ }
+}
+
+// Div performs a division on the Fq12
+func (fq12 Fq12) Div(a, b [2][3][2]*big.Int) [2][3][2]*big.Int {
+ return fq12.Mul(a, fq12.Inverse(b))
+}
+
+// Square performs a square operation on the Fq12
+func (fq12 Fq12) Square(a [2][3][2]*big.Int) [2][3][2]*big.Int {
+ ab := fq12.F.Mul(a[0], a[1])
+
+ return [2][3][2]*big.Int{
+ fq12.F.Sub(
+ fq12.F.Mul(
+ fq12.F.Add(a[0], a[1]),
+ fq12.F.Add(
+ a[0],
+ fq12.mulByNonResidue(a[1]))),
+ fq12.F.Add(
+ ab,
+ fq12.mulByNonResidue(ab))),
+ 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])
+}
diff --git a/bn128/fq2.go b/bn128/fq2.go
new file mode 100644
index 0000000..a36497e
--- /dev/null
+++ b/bn128/fq2.go
@@ -0,0 +1,154 @@
+package bn128
+
+import (
+ "math/big"
+)
+
+// Fq2 is Field 2
+type Fq2 struct {
+ F Fq
+ NonResidue *big.Int
+}
+
+// NewFq2 generates a new Fq2
+func NewFq2(f Fq, nonResidue *big.Int) Fq2 {
+ fq2 := Fq2{
+ f,
+ nonResidue,
+ }
+ return fq2
+}
+
+// Zero returns a Zero value on the Fq2
+func (fq2 Fq2) Zero() [2]*big.Int {
+ return [2]*big.Int{fq2.F.Zero(), fq2.F.Zero()}
+}
+
+// One returns a One value on the Fq2
+func (fq2 Fq2) One() [2]*big.Int {
+ return [2]*big.Int{fq2.F.One(), fq2.F.Zero()}
+}
+
+func (fq2 Fq2) mulByNonResidue(a *big.Int) *big.Int {
+ return fq2.F.Mul(fq2.NonResidue, a)
+}
+
+// Add performs an addition on the Fq2
+func (fq2 Fq2) Add(a, b [2]*big.Int) [2]*big.Int {
+ return [2]*big.Int{
+ fq2.F.Add(a[0], b[0]),
+ fq2.F.Add(a[1], b[1]),
+ }
+}
+
+// Double performs a doubling on the Fq2
+func (fq2 Fq2) Double(a [2]*big.Int) [2]*big.Int {
+ return fq2.Add(a, a)
+}
+
+// Sub performs a subtraction on the Fq2
+func (fq2 Fq2) Sub(a, b [2]*big.Int) [2]*big.Int {
+ return [2]*big.Int{
+ fq2.F.Sub(a[0], b[0]),
+ fq2.F.Sub(a[1], b[1]),
+ }
+}
+
+// Neg performs a negation on the Fq2
+func (fq2 Fq2) Neg(a [2]*big.Int) [2]*big.Int {
+ return fq2.Sub(fq2.Zero(), a)
+}
+
+// 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{
+ fq2.F.Add(v0, fq2.mulByNonResidue(v1)),
+ fq2.F.Sub(
+ fq2.F.Mul(
+ fq2.F.Add(a[0], a[1]),
+ fq2.F.Add(b[0], b[1])),
+ fq2.F.Add(v0, v1)),
+ }
+}
+
+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)
+ }
+ }
+ 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))
+ t3 := fq2.F.Inverse(t2)
+ return [2]*big.Int{
+ fq2.F.Mul(a[0], t3),
+ fq2.F.Neg(fq2.F.Mul(a[1], t3)),
+ }
+}
+
+// Div performs a division on the Fq2
+func (fq2 Fq2) Div(a, b [2]*big.Int) [2]*big.Int {
+ return fq2.Mul(a, fq2.Inverse(b))
+}
+
+// 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(
+ fq2.F.Add(a[0], a[1]),
+ fq2.F.Add(
+ a[0],
+ fq2.mulByNonResidue(a[1]))),
+ fq2.F.Add(
+ ab,
+ fq2.mulByNonResidue(ab))),
+ fq2.F.Add(ab, ab),
+ }
+}
+
+func (fq2 Fq2) IsZero(a [2]*big.Int) bool {
+ return fq2.F.IsZero(a[0]) && fq2.F.IsZero(a[1])
+}
+
+func (fq2 Fq2) Affine(a [2]*big.Int) [2]*big.Int {
+ return [2]*big.Int{
+ fq2.F.Affine(a[0]),
+ fq2.F.Affine(a[1]),
+ }
+}
+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]),
+ }
+}
diff --git a/bn128/fq6.go b/bn128/fq6.go
new file mode 100644
index 0000000..2d401f5
--- /dev/null
+++ b/bn128/fq6.go
@@ -0,0 +1,192 @@
+package bn128
+
+import (
+ "bytes"
+ "math/big"
+)
+
+// Fq6 is Field 6
+type Fq6 struct {
+ F Fq2
+ NonResidue [2]*big.Int
+}
+
+// NewFq6 generates a new Fq6
+func NewFq6(f Fq2, nonResidue [2]*big.Int) Fq6 {
+ fq6 := Fq6{
+ f,
+ nonResidue,
+ }
+ return fq6
+}
+
+// Zero returns a Zero value on the Fq6
+func (fq6 Fq6) Zero() [3][2]*big.Int {
+ return [3][2]*big.Int{fq6.F.Zero(), fq6.F.Zero(), fq6.F.Zero()}
+}
+
+// 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.Zero(), fq6.F.Zero()}
+}
+
+func (fq6 Fq6) mulByNonResidue(a [2]*big.Int) [2]*big.Int {
+ return fq6.F.Mul(fq6.NonResidue, a)
+}
+
+// Add performs an addition on the Fq6
+func (fq6 Fq6) Add(a, b [3][2]*big.Int) [3][2]*big.Int {
+ return [3][2]*big.Int{
+ fq6.F.Add(a[0], b[0]),
+ fq6.F.Add(a[1], b[1]),
+ fq6.F.Add(a[2], b[2]),
+ }
+}
+
+func (fq6 Fq6) Double(a [3][2]*big.Int) [3][2]*big.Int {
+ return fq6.Add(a, a)
+}
+
+// Sub performs a subtraction on the Fq6
+func (fq6 Fq6) Sub(a, b [3][2]*big.Int) [3][2]*big.Int {
+ return [3][2]*big.Int{
+ fq6.F.Sub(a[0], b[0]),
+ fq6.F.Sub(a[1], b[1]),
+ fq6.F.Sub(a[2], b[2]),
+ }
+}
+
+// Neg performs a negation on the Fq6
+func (fq6 Fq6) Neg(a [3][2]*big.Int) [3][2]*big.Int {
+ return fq6.Sub(fq6.Zero(), a)
+}
+
+// Mul performs a multiplication on the Fq6
+func (fq6 Fq6) Mul(a, b [3][2]*big.Int) [3][2]*big.Int {
+ v0 := fq6.F.Mul(a[0], b[0])
+ v1 := fq6.F.Mul(a[1], b[1])
+ v2 := fq6.F.Mul(a[2], b[2])
+ return [3][2]*big.Int{
+ fq6.F.Add(
+ v0,
+ fq6.mulByNonResidue(
+ fq6.F.Sub(
+ fq6.F.Mul(
+ fq6.F.Add(a[1], a[2]),
+ fq6.F.Add(b[1], b[2])),
+ fq6.F.Add(v1, v2)))),
+
+ fq6.F.Add(
+ fq6.F.Sub(
+ fq6.F.Mul(
+ fq6.F.Add(a[0], a[1]),
+ fq6.F.Add(b[0], b[1])),
+ fq6.F.Add(v0, v1)),
+ fq6.mulByNonResidue(v2)),
+
+ fq6.F.Add(
+ fq6.F.Sub(
+ fq6.F.Mul(
+ fq6.F.Add(a[0], a[2]),
+ fq6.F.Add(b[0], b[2])),
+ fq6.F.Add(v0, v2)),
+ v1),
+ }
+}
+
+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
+
+ 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 = fq6.Add(res, exp)
+ }
+ exp = fq6.Double(exp)
+ rem = rem.Rsh(rem, 1) // rem = rem >> 1
+ }
+ return res
+}
+
+// Inverse returns the inverse on the Fq6
+func (fq6 Fq6) Inverse(a [3][2]*big.Int) [3][2]*big.Int {
+ t0 := fq6.F.Square(a[0])
+ t1 := fq6.F.Square(a[1])
+ t2 := fq6.F.Square(a[2])
+ t3 := fq6.F.Mul(a[0], a[1])
+ t4 := fq6.F.Mul(a[0], a[2])
+ t5 := fq6.F.Mul(a[1], a[2])
+
+ c0 := fq6.F.Sub(t0, fq6.mulByNonResidue(t5))
+ c1 := fq6.F.Sub(fq6.mulByNonResidue(t2), t3)
+ c2 := fq6.F.Sub(t1, t4)
+
+ t6 := fq6.F.Inverse(
+ fq6.F.Add(
+ fq6.F.Mul(a[0], c0),
+ fq6.mulByNonResidue(
+ fq6.F.Add(
+ fq6.F.Mul(a[2], c1),
+ fq6.F.Mul(a[1], c2)))))
+ return [3][2]*big.Int{
+ fq6.F.Mul(t6, c0),
+ fq6.F.Mul(t6, c1),
+ fq6.F.Mul(t6, c2),
+ }
+}
+
+// Div performs a division on the Fq6
+func (fq6 Fq6) Div(a, b [3][2]*big.Int) [3][2]*big.Int {
+ return fq6.Mul(a, fq6.Inverse(b))
+}
+
+// Square performs a square operation on the Fq6
+func (fq6 Fq6) Square(a [3][2]*big.Int) [3][2]*big.Int {
+ s0 := fq6.F.Square(a[0])
+ ab := fq6.F.Mul(a[0], a[1])
+ s1 := fq6.F.Add(ab, ab)
+ s2 := fq6.F.Square(
+ fq6.F.Add(
+ fq6.F.Sub(a[0], a[1]),
+ a[2]))
+ bc := fq6.F.Mul(a[1], a[2])
+ s3 := fq6.F.Add(bc, bc)
+ s4 := fq6.F.Square(a[2])
+
+ return [3][2]*big.Int{
+ fq6.F.Add(
+ s0,
+ fq6.mulByNonResidue(s3)),
+ fq6.F.Add(
+ s1,
+ fq6.mulByNonResidue(s4)),
+ fq6.F.Sub(
+ fq6.F.Add(
+ fq6.F.Add(s1, s2),
+ s3),
+ 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]),
+ }
+}
diff --git a/bn128/fqn_test.go b/bn128/fqn_test.go
new file mode 100644
index 0000000..a22e5eb
--- /dev/null
+++ b/bn128/fqn_test.go
@@ -0,0 +1,160 @@
+package bn128
+
+import (
+ "math/big"
+ "testing"
+
+ "github.com/stretchr/testify/assert"
+)
+
+func iToBig(a int) *big.Int {
+ return big.NewInt(int64(a))
+}
+
+func iiToBig(a, b int) [2]*big.Int {
+ return [2]*big.Int{iToBig(a), iToBig(b)}
+}
+
+func iiiToBig(a, b int) [2]*big.Int {
+ return [2]*big.Int{iToBig(a), iToBig(b)}
+}
+
+func TestFq1(t *testing.T) {
+ fq1 := NewFq(iToBig(7))
+
+ res := fq1.Add(iToBig(4), iToBig(4))
+ assert.Equal(t, iToBig(1), fq1.Affine(res))
+
+ res = fq1.Double(iToBig(5))
+ assert.Equal(t, iToBig(3), fq1.Affine(res))
+
+ res = fq1.Sub(iToBig(5), iToBig(7))
+ assert.Equal(t, iToBig(5), fq1.Affine(res))
+
+ res = fq1.Neg(iToBig(5))
+ assert.Equal(t, iToBig(2), fq1.Affine(res))
+
+ res = fq1.Mul(iToBig(5), iToBig(11))
+ assert.Equal(t, iToBig(6), fq1.Affine(res))
+
+ res = fq1.Inverse(iToBig(4))
+ assert.Equal(t, iToBig(2), res)
+
+ res = fq1.Square(iToBig(5))
+ assert.Equal(t, iToBig(4), res)
+}
+
+func TestFq2(t *testing.T) {
+ fq1 := NewFq(iToBig(7))
+ nonResidueFq2str := "-1" // i/j
+ nonResidueFq2, ok := new(big.Int).SetString(nonResidueFq2str, 10)
+ assert.True(t, ok)
+ assert.Equal(t, nonResidueFq2.String(), nonResidueFq2str)
+
+ fq2 := Fq2{fq1, nonResidueFq2}
+
+ res := fq2.Add(iiToBig(4, 4), iiToBig(3, 4))
+ assert.Equal(t, iiToBig(0, 1), fq2.Affine(res))
+
+ res = fq2.Double(iiToBig(5, 3))
+ assert.Equal(t, iiToBig(3, 6), fq2.Affine(res))
+
+ res = fq2.Sub(iiToBig(5, 3), iiToBig(7, 2))
+ assert.Equal(t, iiToBig(5, 1), fq2.Affine(res))
+
+ res = fq2.Neg(iiToBig(4, 4))
+ assert.Equal(t, iiToBig(3, 3), fq2.Affine(res))
+
+ res = fq2.Mul(iiToBig(4, 4), iiToBig(3, 4))
+ assert.Equal(t, iiToBig(3, 0), fq2.Affine(res))
+
+ res = fq2.Inverse(iiToBig(4, 4))
+ assert.Equal(t, iiToBig(1, 6), fq2.Affine(res))
+
+ res = fq2.Square(iiToBig(4, 4))
+ assert.Equal(t, iiToBig(0, 4), fq2.Affine(res))
+ res2 := fq2.Mul(iiToBig(4, 4), iiToBig(4, 4))
+ 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), fq2.Affine(res))
+ res2 = fq2.Mul(iiToBig(3, 5), iiToBig(3, 5))
+ assert.Equal(t, fq2.Affine(res), fq2.Affine(res2))
+}
+
+func TestFq6(t *testing.T) {
+ bn128, err := NewBn128()
+ assert.Nil(t, err)
+
+ a := [3][2]*big.Int{
+ iiToBig(1, 2),
+ iiToBig(3, 4),
+ iiToBig(5, 6)}
+ b := [3][2]*big.Int{
+ iiToBig(12, 11),
+ iiToBig(10, 9),
+ iiToBig(8, 7)}
+
+ 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) {
+ q, ok := new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208583", 10) // i
+ assert.True(t, ok)
+ fq1 := NewFq(q)
+ nonResidueFq2, ok := new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208582", 10) // i
+ assert.True(t, ok)
+ nonResidueFq6 := iiToBig(9, 1)
+
+ fq2 := Fq2{fq1, nonResidueFq2}
+ fq6 := Fq6{fq2, nonResidueFq6}
+ fq12 := Fq12{fq6, fq2, nonResidueFq6}
+
+ a := [2][3][2]*big.Int{
+ {
+ iiToBig(1, 2),
+ iiToBig(3, 4),
+ iiToBig(5, 6),
+ },
+ {
+ iiToBig(7, 8),
+ iiToBig(9, 10),
+ iiToBig(11, 12),
+ },
+ }
+ b := [2][3][2]*big.Int{
+ {
+ iiToBig(12, 11),
+ iiToBig(10, 9),
+ iiToBig(8, 7),
+ },
+ {
+ iiToBig(6, 5),
+ iiToBig(4, 3),
+ iiToBig(2, 1),
+ },
+ }
+
+ res := fq12.Add(a, b)
+ assert.Equal(t,
+ [2][3][2]*big.Int{
+ {
+ iiToBig(13, 13),
+ iiToBig(13, 13),
+ iiToBig(13, 13),
+ },
+ {
+ iiToBig(13, 13),
+ iiToBig(13, 13),
+ iiToBig(13, 13),
+ },
+ },
+ res)
+
+ mulRes := fq12.Mul(a, b)
+ divRes := fq12.Div(mulRes, b)
+ assert.Equal(t, fq12.Affine(a), fq12.Affine(divRes))
+}
diff --git a/bn128/g1.go b/bn128/g1.go
new file mode 100644
index 0000000..4edd043
--- /dev/null
+++ b/bn128/g1.go
@@ -0,0 +1,191 @@
+package bn128
+
+import (
+ "math/big"
+)
+
+type G1 struct {
+ F Fq
+ G [3]*big.Int
+}
+
+func NewG1(f Fq, g [2]*big.Int) G1 {
+ var g1 G1
+ g1.F = f
+ g1.G = [3]*big.Int{
+ g[0],
+ g[1],
+ g1.F.One(),
+ }
+ return g1
+}
+
+func (g1 G1) Zero() [2]*big.Int {
+ return [2]*big.Int{g1.F.Zero(), g1.F.Zero()}
+}
+func (g1 G1) IsZero(p [3]*big.Int) bool {
+ return g1.F.IsZero(p[2])
+}
+
+func (g1 G1) Add(p1, p2 [3]*big.Int) [3]*big.Int {
+
+ // https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates
+ // https://github.com/zcash/zcash/blob/master/src/snark/libsnark/algebra/curves/alt_bn128/alt_bn128_g1.cpp#L208
+ // http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/addition/add-2007-bl.op3
+
+ if g1.IsZero(p1) {
+ return p2
+ }
+ if g1.IsZero(p2) {
+ return p1
+ }
+
+ x1 := p1[0]
+ y1 := p1[1]
+ z1 := p1[2]
+ x2 := p2[0]
+ y2 := p2[1]
+ z2 := p2[2]
+
+ z1z1 := g1.F.Square(z1)
+ z2z2 := g1.F.Square(z2)
+
+ u1 := g1.F.Mul(x1, z2z2)
+ u2 := g1.F.Mul(x2, z1z1)
+
+ t0 := g1.F.Mul(z2, z2z2)
+ s1 := g1.F.Mul(y1, t0)
+
+ t1 := g1.F.Mul(z1, z1z1)
+ s2 := g1.F.Mul(y2, t1)
+
+ h := g1.F.Sub(u2, u1)
+ t2 := g1.F.Add(h, h)
+ i := g1.F.Square(t2)
+ j := g1.F.Mul(h, i)
+ t3 := g1.F.Sub(s2, s1)
+ r := g1.F.Add(t3, t3)
+ v := g1.F.Mul(u1, i)
+ t4 := g1.F.Square(r)
+ t5 := g1.F.Add(v, v)
+ t6 := g1.F.Sub(t4, j)
+ x3 := g1.F.Sub(t6, t5)
+ t7 := g1.F.Sub(v, x3)
+ t8 := g1.F.Mul(s1, j)
+ t9 := g1.F.Add(t8, t8)
+ t10 := g1.F.Mul(r, t7)
+
+ y3 := g1.F.Sub(t10, t9)
+
+ t11 := g1.F.Add(z1, z2)
+ t12 := g1.F.Square(t11)
+ t13 := g1.F.Sub(t12, z1z1)
+ t14 := g1.F.Sub(t13, z2z2)
+ z3 := g1.F.Mul(t14, h)
+
+ return [3]*big.Int{x3, y3, z3}
+}
+
+func (g1 G1) Neg(p [3]*big.Int) [3]*big.Int {
+ return [3]*big.Int{
+ p[0],
+ g1.F.Neg(p[1]),
+ p[2],
+ }
+}
+func (g1 G1) Sub(a, b [3]*big.Int) [3]*big.Int {
+ return g1.Add(a, g1.Neg(b))
+}
+func (g1 G1) Double(p [3]*big.Int) [3]*big.Int {
+
+ // https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates
+ // http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/doubling/dbl-2009-l.op3
+ // https://github.com/zcash/zcash/blob/master/src/snark/libsnark/algebra/curves/alt_bn128/alt_bn128_g1.cpp#L325
+
+ if g1.IsZero(p) {
+ return p
+ }
+
+ a := g1.F.Square(p[0])
+ b := g1.F.Square(p[1])
+ c := g1.F.Square(b)
+
+ t0 := g1.F.Add(p[0], b)
+ t1 := g1.F.Square(t0)
+ t2 := g1.F.Sub(t1, a)
+ t3 := g1.F.Sub(t2, c)
+
+ d := g1.F.Double(t3)
+ e := g1.F.Add(g1.F.Add(a, a), a)
+ f := g1.F.Square(e)
+
+ t4 := g1.F.Double(d)
+ x3 := g1.F.Sub(f, t4)
+
+ t5 := g1.F.Sub(d, x3)
+ twoC := g1.F.Add(c, c)
+ fourC := g1.F.Add(twoC, twoC)
+ t6 := g1.F.Add(fourC, fourC)
+ t7 := g1.F.Mul(e, t5)
+ y3 := g1.F.Sub(t7, t6)
+
+ t8 := g1.F.Mul(p[1], p[2])
+ z3 := g1.F.Double(t8)
+
+ return [3]*big.Int{x3, y3, z3}
+}
+
+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()
+
+ 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)
+ }
+ }
+
+ return q
+}
+
+func (g1 G1) Affine(p [3]*big.Int) [2]*big.Int {
+ if g1.IsZero(p) {
+ return g1.Zero()
+ }
+
+ zinv := g1.F.Inverse(p[2])
+ zinv2 := g1.F.Square(zinv)
+ x := g1.F.Mul(p[0], zinv2)
+
+ zinv3 := g1.F.Mul(zinv2, zinv)
+ y := g1.F.Mul(p[1], zinv3)
+
+ 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)
+}
diff --git a/bn128/g1_test.go b/bn128/g1_test.go
new file mode 100644
index 0000000..e2bf533
--- /dev/null
+++ b/bn128/g1_test.go
@@ -0,0 +1,31 @@
+package bn128
+
+import (
+ "math/big"
+ "testing"
+
+ "github.com/arnaucube/cryptofun/utils"
+ "github.com/stretchr/testify/assert"
+)
+
+func TestG1(t *testing.T) {
+ bn128, err := NewBn128()
+ assert.Nil(t, err)
+
+ r1 := big.NewInt(int64(33))
+ r2 := big.NewInt(int64(44))
+
+ 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) // 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)
+ assert.Equal(t, "0x2f978c0ab89ebaa576866706b14787f360c4d6c3869efe5a72f7c3651a72ff00", utils.BytesToHex(a[0].Bytes()))
+ assert.Equal(t, "0x12e4ba7f0edca8b4fa668fe153aebd908d322dc26ad964d4cd314795844b62b2", utils.BytesToHex(a[1].Bytes()))
+}
diff --git a/bn128/g2.go b/bn128/g2.go
new file mode 100644
index 0000000..f6b7cad
--- /dev/null
+++ b/bn128/g2.go
@@ -0,0 +1,221 @@
+package bn128
+
+import (
+ "math/big"
+)
+
+type G2 struct {
+ F Fq2
+ G [3][2]*big.Int
+}
+
+func NewG2(f Fq2, g [2][2]*big.Int) G2 {
+ var g2 G2
+ g2.F = f
+ g2.G = [3][2]*big.Int{
+ g[0],
+ g[1],
+ g2.F.One(),
+ }
+ return g2
+}
+
+func (g2 G2) Zero() [3][2]*big.Int {
+ return [3][2]*big.Int{g2.F.Zero(), g2.F.One(), g2.F.Zero()}
+}
+func (g2 G2) IsZero(p [3][2]*big.Int) bool {
+ return g2.F.IsZero(p[2])
+}
+
+func (g2 G2) Add(p1, p2 [3][2]*big.Int) [3][2]*big.Int {
+
+ // https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates
+ // https://github.com/zcash/zcash/blob/master/src/snark/libsnark/algebra/curves/alt_bn128/alt_bn128_g2.cpp#L208
+ // http://hyperelliptic.org/EFD/g2p/auto-code/shortw/jacobian-0/addition/add-2007-bl.op3
+
+ if g2.IsZero(p1) {
+ return p2
+ }
+ if g2.IsZero(p2) {
+ return p1
+ }
+
+ x1 := p1[0]
+ y1 := p1[1]
+ z1 := p1[2]
+ x2 := p2[0]
+ y2 := p2[1]
+ z2 := p2[2]
+
+ z1z1 := g2.F.Square(z1)
+ z2z2 := g2.F.Square(z2)
+
+ u1 := g2.F.Mul(x1, z2z2)
+ u2 := g2.F.Mul(x2, z1z1)
+
+ t0 := g2.F.Mul(z2, z2z2)
+ s1 := g2.F.Mul(y1, t0)
+
+ t1 := g2.F.Mul(z1, z1z1)
+ s2 := g2.F.Mul(y2, t1)
+
+ h := g2.F.Sub(u2, u1)
+ t2 := g2.F.Add(h, h)
+ i := g2.F.Square(t2)
+ j := g2.F.Mul(h, i)
+ t3 := g2.F.Sub(s2, s1)
+ r := g2.F.Add(t3, t3)
+ v := g2.F.Mul(u1, i)
+ t4 := g2.F.Square(r)
+ t5 := g2.F.Add(v, v)
+ t6 := g2.F.Sub(t4, j)
+ x3 := g2.F.Sub(t6, t5)
+ t7 := g2.F.Sub(v, x3)
+ t8 := g2.F.Mul(s1, j)
+ t9 := g2.F.Add(t8, t8)
+ t10 := g2.F.Mul(r, t7)
+
+ y3 := g2.F.Sub(t10, t9)
+
+ t11 := g2.F.Add(z1, z2)
+ t12 := g2.F.Square(t11)
+ t13 := g2.F.Sub(t12, z1z1)
+ t14 := g2.F.Sub(t13, z2z2)
+ z3 := g2.F.Mul(t14, h)
+
+ return [3][2]*big.Int{x3, y3, z3}
+}
+
+func (g2 G2) Neg(p [3][2]*big.Int) [3][2]*big.Int {
+ return [3][2]*big.Int{
+ p[0],
+ g2.F.Neg(p[1]),
+ p[2],
+ }
+}
+
+func (g2 G2) Sub(a, b [3][2]*big.Int) [3][2]*big.Int {
+ return g2.Add(a, g2.Neg(b))
+}
+
+func (g2 G2) Double(p [3][2]*big.Int) [3][2]*big.Int {
+
+ // https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates
+ // http://hyperelliptic.org/EFD/g2p/auto-code/shortw/jacobian-0/doubling/dbl-2009-l.op3
+ // https://github.com/zcash/zcash/blob/master/src/snark/libsnark/algebra/curves/alt_bn128/alt_bn128_g2.cpp#L325
+
+ if g2.IsZero(p) {
+ return p
+ }
+
+ a := g2.F.Square(p[0])
+ b := g2.F.Square(p[1])
+ c := g2.F.Square(b)
+
+ t0 := g2.F.Add(p[0], b)
+ t1 := g2.F.Square(t0)
+ t2 := g2.F.Sub(t1, a)
+ t3 := g2.F.Sub(t2, c)
+
+ d := g2.F.Double(t3)
+ e := g2.F.Add(g2.F.Add(a, a), a)
+ f := g2.F.Square(e)
+
+ t4 := g2.F.Double(d)
+ x3 := g2.F.Sub(f, t4)
+
+ t5 := g2.F.Sub(d, x3)
+ twoC := g2.F.Add(c, c)
+ fourC := g2.F.Add(twoC, twoC)
+ t6 := g2.F.Add(fourC, fourC)
+ t7 := g2.F.Mul(e, t5)
+ y3 := g2.F.Sub(t7, t6)
+
+ t8 := g2.F.Mul(p[1], p[2])
+ z3 := g2.F.Double(t8)
+
+ return [3][2]*big.Int{x3, y3, z3}
+}
+
+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
+
+ 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)
+ }
+ }
+
+ /* 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 {
+ if g2.IsZero(p) {
+ return g2.Zero()
+ }
+
+ zinv := g2.F.Inverse(p[2])
+ zinv2 := g2.F.Square(zinv)
+ x := g2.F.Mul(p[0], zinv2)
+
+ zinv3 := g2.F.Mul(zinv2, zinv)
+ y := g2.F.Mul(p[1], zinv3)
+
+ return [3][2]*big.Int{
+ 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)
+}
diff --git a/bn128/g2_test.go b/bn128/g2_test.go
new file mode 100644
index 0000000..0782ae0
--- /dev/null
+++ b/bn128/g2_test.go
@@ -0,0 +1,24 @@
+package bn128
+
+import (
+ "math/big"
+ "testing"
+
+ "github.com/stretchr/testify/assert"
+)
+
+func TestG2(t *testing.T) {
+ bn128, err := NewBn128()
+ assert.Nil(t, err)
+
+ r1 := big.NewInt(int64(33))
+ r2 := big.NewInt(int64(44))
+
+ 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.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))
+}
diff --git a/go.mod b/go.mod
new file mode 100644
index 0000000..001be98
--- /dev/null
+++ b/go.mod
@@ -0,0 +1,6 @@
+module github.com/arnaucube/go-snark
+
+require (
+ github.com/arnaucube/cryptofun v0.0.0-20181124004321-9b11ae8280bd
+ github.com/stretchr/testify v1.2.2
+)
diff --git a/go.sum b/go.sum
new file mode 100644
index 0000000..9fa27b1
--- /dev/null
+++ b/go.sum
@@ -0,0 +1,8 @@
+github.com/arnaucube/cryptofun v0.0.0-20181124004321-9b11ae8280bd h1:NDpNBTFeHNE2IHya+msmKlCzIPGzn8qN3Z2jtegFYT0=
+github.com/arnaucube/cryptofun v0.0.0-20181124004321-9b11ae8280bd/go.mod h1:PZE8kKpHPD1UMrS3mTfAMmEEinGtijSwjxLRqRcD64A=
+github.com/davecgh/go-spew v1.1.1 h1:vj9j/u1bqnvCEfJOwUhtlOARqs3+rkHYY13jYWTU97c=
+github.com/davecgh/go-spew v1.1.1/go.mod h1:J7Y8YcW2NihsgmVo/mv3lAwl/skON4iLHjSsI+c5H38=
+github.com/pmezard/go-difflib v1.0.0 h1:4DBwDE0NGyQoBHbLQYPwSUPoCMWR5BEzIk/f1lZbAQM=
+github.com/pmezard/go-difflib v1.0.0/go.mod h1:iKH77koFhYxTK1pcRnkKkqfTogsbg7gZNVY4sRDYZ/4=
+github.com/stretchr/testify v1.2.2 h1:bSDNvY7ZPG5RlJ8otE/7V6gMiyenm9RtJ7IUVIAoJ1w=
+github.com/stretchr/testify v1.2.2/go.mod h1:a8OnRcib4nhh0OaRAV+Yts87kKdq0PP7pXfy6kDkUVs=
diff --git a/r1csqap.go b/r1csqap.go
new file mode 100644
index 0000000..02684e4
--- /dev/null
+++ b/r1csqap.go
@@ -0,0 +1,145 @@
+package sn
+
+import (
+ "math/big"
+)
+
+func Transpose(matrix [][]*big.Float) [][]*big.Float {
+ var r [][]*big.Float
+ for i := 0; i < len(matrix[0]); i++ {
+ var row []*big.Float
+ for j := 0; j < len(matrix); j++ {
+ row = append(row, matrix[j][i])
+ }
+ r = append(r, row)
+ }
+ return r
+}
+
+func ArrayOfBigZeros(num int) []*big.Float {
+ bigZero := big.NewFloat(float64(0))
+ var r []*big.Float
+ for i := 0; i < num; i++ {
+ r = append(r, bigZero)
+ }
+ return r
+}
+
+func PolMul(a, b []*big.Float) []*big.Float {
+ r := ArrayOfBigZeros(len(a) + len(b) - 1)
+ for i := 0; i < len(a); i++ {
+ for j := 0; j < len(b); j++ {
+ r[i+j] = new(big.Float).Add(
+ r[i+j],
+ new(big.Float).Mul(a[i], b[j]))
+ }
+ }
+ return r
+}
+
+func max(a, b int) int {
+ if a > b {
+ return a
+ }
+ return b
+}
+
+func PolAdd(a, b []*big.Float) []*big.Float {
+ r := ArrayOfBigZeros(max(len(a), len(b)))
+ for i := 0; i < len(a); i++ {
+ r[i] = new(big.Float).Add(r[i], a[i])
+ }
+ for i := 0; i < len(b); i++ {
+ r[i] = new(big.Float).Add(r[i], b[i])
+ }
+ return r
+}
+
+func PolSub(a, b []*big.Float) []*big.Float {
+ r := ArrayOfBigZeros(max(len(a), len(b)))
+ for i := 0; i < len(a); i++ {
+ r[i] = new(big.Float).Add(r[i], a[i])
+ }
+ for i := 0; i < len(b); i++ {
+ bneg := new(big.Float).Mul(b[i], big.NewFloat(float64(-1)))
+ r[i] = new(big.Float).Add(r[i], bneg)
+ }
+ return r
+
+}
+
+func FloatPow(a *big.Float, e int) *big.Float {
+ if e == 0 {
+ return big.NewFloat(float64(1))
+ }
+ result := new(big.Float).Copy(a)
+ for i := 0; i < e-1; i++ {
+ result = new(big.Float).Mul(result, a)
+ }
+ return result
+}
+
+func PolEval(v []*big.Float, x *big.Float) *big.Float {
+ r := big.NewFloat(float64(0))
+ for i := 0; i < len(v); i++ {
+ xi := FloatPow(x, i)
+ elem := new(big.Float).Mul(v[i], xi)
+ r = new(big.Float).Add(r, elem)
+ }
+ return r
+}
+
+func NewPolZeroAt(pointPos, totalPoints int, height *big.Float) []*big.Float {
+ fac := 1
+ for i := 1; i < totalPoints+1; i++ {
+ if i != pointPos {
+ fac = fac * (pointPos - i)
+ }
+ }
+ facBig := big.NewFloat(float64(fac))
+ hf := new(big.Float).Quo(height, facBig)
+ r := []*big.Float{hf}
+ for i := 1; i < totalPoints+1; i++ {
+ if i != pointPos {
+ ineg := big.NewFloat(float64(-i))
+ b1 := big.NewFloat(float64(1))
+ r = PolMul(r, []*big.Float{ineg, b1})
+ }
+ }
+ return r
+}
+
+func LagrangeInterpolation(v []*big.Float) []*big.Float {
+ // https://en.wikipedia.org/wiki/Lagrange_polynomial
+ var r []*big.Float
+ for i := 0; i < len(v); i++ {
+ r = PolAdd(r, NewPolZeroAt(i+1, len(v), v[i]))
+ }
+ //
+ return r
+}
+
+func R1CSToQAP(a, b, c [][]*big.Float) ([][]*big.Float, [][]*big.Float, [][]*big.Float, []*big.Float) {
+ aT := Transpose(a)
+ bT := Transpose(b)
+ cT := Transpose(c)
+ var alpha [][]*big.Float
+ for i := 0; i < len(aT); i++ {
+ alpha = append(alpha, LagrangeInterpolation(aT[i]))
+ }
+ var beta [][]*big.Float
+ for i := 0; i < len(bT); i++ {
+ beta = append(beta, LagrangeInterpolation(bT[i]))
+ }
+ var gamma [][]*big.Float
+ for i := 0; i < len(cT); i++ {
+ gamma = append(gamma, LagrangeInterpolation(cT[i]))
+ }
+ z := []*big.Float{big.NewFloat(float64(1))}
+ for i := 1; i < len(aT[0])+1; i++ {
+ ineg := big.NewFloat(float64(-i))
+ b1 := big.NewFloat(float64(1))
+ z = PolMul(z, []*big.Float{ineg, b1})
+ }
+ return alpha, beta, gamma, z
+}
diff --git a/r1csqap_test.go b/r1csqap_test.go
new file mode 100644
index 0000000..a46f09e
--- /dev/null
+++ b/r1csqap_test.go
@@ -0,0 +1,112 @@
+package sn
+
+import (
+ "fmt"
+ "math/big"
+ "testing"
+
+ "github.com/stretchr/testify/assert"
+)
+
+func TestTranspose(t *testing.T) {
+ b0 := big.NewFloat(float64(0))
+ b1 := big.NewFloat(float64(1))
+ bFive := big.NewFloat(float64(5))
+ a := [][]*big.Float{
+ []*big.Float{b0, b1, b0, b0, b0, b0},
+ []*big.Float{b0, b0, b0, b1, b0, b0},
+ []*big.Float{b0, b1, b0, b0, b1, b0},
+ []*big.Float{bFive, b0, b0, b0, b0, b1},
+ }
+ aT := Transpose(a)
+ assert.Equal(t, aT, [][]*big.Float{
+ []*big.Float{b0, b0, b0, bFive},
+ []*big.Float{b1, b0, b1, b0},
+ []*big.Float{b0, b0, b0, b0},
+ []*big.Float{b0, b1, b0, b0},
+ []*big.Float{b0, b0, b1, b0},
+ []*big.Float{b0, b0, b0, b1},
+ })
+}
+
+func TestPol(t *testing.T) {
+ b0 := big.NewFloat(float64(0))
+ b1 := big.NewFloat(float64(1))
+ // b1neg := big.NewFloat(float64(-1))
+ // b2 := big.NewFloat(float64(2))
+ b2neg := big.NewFloat(float64(-2))
+ b3 := big.NewFloat(float64(3))
+ b4 := big.NewFloat(float64(4))
+ b5 := big.NewFloat(float64(5))
+ b6 := big.NewFloat(float64(6))
+ b16 := big.NewFloat(float64(16))
+
+ a := []*big.Float{b1, b0, b5}
+ b := []*big.Float{b3, b0, b1}
+
+ // polynomial multiplication
+ c := PolMul(a, b)
+ assert.Equal(t, c, []*big.Float{b3, b0, b16, b0, b5})
+
+ // polynomial addition
+ c = PolAdd(a, b)
+ assert.Equal(t, c, []*big.Float{b4, b0, b6})
+
+ // polynomial substraction
+ c = PolSub(a, b)
+ assert.Equal(t, c, []*big.Float{b2neg, b0, b4})
+
+ // FloatPow
+ p := FloatPow(big.NewFloat(float64(5)), 3)
+ assert.Equal(t, p, big.NewFloat(float64(125)))
+ p = FloatPow(big.NewFloat(float64(5)), 0)
+ assert.Equal(t, p, big.NewFloat(float64(1)))
+
+ // NewPolZeroAt
+ r := NewPolZeroAt(3, 4, b4)
+ assert.Equal(t, PolEval(r, big.NewFloat(3)), b4)
+ r = NewPolZeroAt(2, 4, b3)
+ assert.Equal(t, PolEval(r, big.NewFloat(2)), b3)
+}
+
+func TestLagrangeInterpolation(t *testing.T) {
+ b0 := big.NewFloat(float64(0))
+ b5 := big.NewFloat(float64(5))
+ a := []*big.Float{b0, b0, b0, b5}
+ alpha := LagrangeInterpolation(a)
+
+ assert.Equal(t, PolEval(alpha, big.NewFloat(4)), b5)
+ aux, _ := PolEval(alpha, big.NewFloat(3)).Int64()
+ assert.Equal(t, aux, int64(0))
+
+}
+
+func TestR1CSToQAP(t *testing.T) {
+ b0 := big.NewFloat(float64(0))
+ b1 := big.NewFloat(float64(1))
+ b5 := big.NewFloat(float64(5))
+ a := [][]*big.Float{
+ []*big.Float{b0, b1, b0, b0, b0, b0},
+ []*big.Float{b0, b0, b0, b1, b0, b0},
+ []*big.Float{b0, b1, b0, b0, b1, b0},
+ []*big.Float{b5, b0, b0, b0, b0, b1},
+ }
+ b := [][]*big.Float{
+ []*big.Float{b0, b1, b0, b0, b0, b0},
+ []*big.Float{b0, b1, b0, b0, b0, b0},
+ []*big.Float{b1, b0, b0, b0, b0, b0},
+ []*big.Float{b1, b0, b0, b0, b0, b0},
+ }
+ c := [][]*big.Float{
+ []*big.Float{b0, b0, b0, b1, b0, b0},
+ []*big.Float{b0, b0, b0, b0, b1, b0},
+ []*big.Float{b0, b0, b0, b0, b0, b1},
+ []*big.Float{b0, b0, b1, b0, b0, b0},
+ }
+ alpha, beta, gamma, z := R1CSToQAP(a, b, c)
+ fmt.Println(alpha)
+ fmt.Println(beta)
+ fmt.Println(gamma)
+ fmt.Println(z)
+
+}