bn128 pairing implemented

README.md

@@ -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:

bn128/README.md

@@ -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)

bn128/bn128.go

@@ -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
+}

bn128/bn128_test.go

@@ -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")
+}

bn128/fq.go

@@ -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())
 }

bn128/fq12.go

@@ -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])
+}

bn128/fq2.go

@@ -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]),
+	}
+}

bn128/fq6.go

@@ -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]),
+	}
+}

bn128/fqn_test.go

@@ -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))
 }

bn128/g1.go

@@ -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
-
-	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)
+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)
 		}
-		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)
+}

bn128/g1_test.go

@@ -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)

bn128/g2.go

@@ -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)
 }

bn128/g2_test.go

@@ -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))
 }

paillier/paillier.go

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

shamirsecretsharing/shamirsecretsharing_test.go

@@ -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")
 	}