From 151ca78806d3a20aa9773c7ec930ded576a37167 Mon Sep 17 00:00:00 2001
From: arnaucube <arnaucube@gmail.com>
Date: Sun, 7 Oct 2018 00:16:23 +0200
Subject: [PATCH] bn128 finite fields operations

---
 README.md                                  |  13 ++
 bn128/fq.go                                |  74 ++++++++
 bn128/fq12.go                              | 118 +++++++++++++
 bn128/fq2.go                               | 110 ++++++++++++
 bn128/fq6.go                               | 150 ++++++++++++++++
 bn128/fqn_test.go                          | 190 +++++++++++++++++++++
 ecc/ecc.go                                 |  32 ++--
 ecc/ecc_test.go                            |  20 +--
 ecc/point.go                               |  12 +-
 ecdsa/ecdsa_test.go                        |   4 +-
 elgamal/elgamal_test.go                    |   6 +-
 schnorr/schnorr_test.go                    |  18 +-
 shamirsecretsharing/shamirsecretsharing.go |   2 +-
 utils/utils.go                             |  17 ++
 14 files changed, 712 insertions(+), 54 deletions(-)
 create mode 100644 bn128/fq.go
 create mode 100644 bn128/fq12.go
 create mode 100644 bn128/fq2.go
 create mode 100644 bn128/fq6.go
 create mode 100644 bn128/fqn_test.go
 create mode 100644 utils/utils.go

diff --git a/README.md b/README.md
index 1cfc130..425cd8e 100644
--- a/README.md
+++ b/README.md
@@ -58,6 +58,19 @@ https://en.wikipedia.org/wiki/Schnorr_signature
 - [x] Sign
 - [x] Verify signature
 
+
+## 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
+- `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
+
+- [x] Fq, Fq2, Fq6, Fq12 operations
+
 ---
 
 To run all tests:
diff --git a/bn128/fq.go b/bn128/fq.go
new file mode 100644
index 0000000..630b56c
--- /dev/null
+++ b/bn128/fq.go
@@ -0,0 +1,74 @@
+package bn128
+
+import (
+	"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 {
+	sum := new(big.Int).Add(a, b)
+	return new(big.Int).Mod(sum, fq.Q)
+}
+
+// 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)
+}
+
+// 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)
+}
+
+// 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)
+}
+
+// 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)
+}
+
+// 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
+}
+
+// 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)
+}
diff --git a/bn128/fq12.go b/bn128/fq12.go
new file mode 100644
index 0000000..80d6131
--- /dev/null
+++ b/bn128/fq12.go
@@ -0,0 +1,118 @@
+package bn128
+
+import (
+	"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.One()}
+}
+
+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 substraction 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 [2]*big.Ints.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)),
+	}
+}
+
+// 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),
+	}
+}
diff --git a/bn128/fq2.go b/bn128/fq2.go
new file mode 100644
index 0000000..66d76a1
--- /dev/null
+++ b/bn128/fq2.go
@@ -0,0 +1,110 @@
+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.One()}
+}
+
+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 substraction 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)
+	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)),
+	}
+}
+
+// Inverse returns the inverse on the Fq2
+func (fq2 Fq2) Inverse(a [2]*big.Int) [2]*big.Int {
+	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 {
+	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),
+	}
+}
diff --git a/bn128/fq6.go b/bn128/fq6.go
new file mode 100644
index 0000000..90e7b4a
--- /dev/null
+++ b/bn128/fq6.go
@@ -0,0 +1,150 @@
+package bn128
+
+import (
+	"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.One(), fq6.F.One()}
+}
+
+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]),
+	}
+}
+
+// Sub performs a substraction 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),
+	}
+}
+
+// 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)),
+	}
+}
diff --git a/bn128/fqn_test.go b/bn128/fqn_test.go
new file mode 100644
index 0000000..3f17673
--- /dev/null
+++ b/bn128/fqn_test.go
@@ -0,0 +1,190 @@
+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), res)
+
+	res = fq1.Double(iToBig(5))
+	assert.Equal(t, iToBig(3), res)
+
+	res = fq1.Sub(iToBig(5), iToBig(7))
+	assert.Equal(t, iToBig(5), res)
+
+	res = fq1.Neg(iToBig(5))
+	assert.Equal(t, iToBig(2), res)
+
+	res = fq1.Mul(iToBig(5), iToBig(11))
+	assert.Equal(t, iToBig(6), 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 / Beta
+	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), res)
+
+	res = fq2.Double(iiToBig(5, 3))
+	assert.Equal(t, iiToBig(3, 6), res)
+
+	res = fq2.Sub(iiToBig(5, 3), iiToBig(7, 2))
+	assert.Equal(t, iiToBig(5, 1), res)
+
+	res = fq2.Neg(iiToBig(4, 4))
+	assert.Equal(t, iiToBig(3, 3), res)
+
+	res = fq2.Mul(iiToBig(4, 4), iiToBig(3, 4))
+	assert.Equal(t, iiToBig(3, 0), 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)
+
+	res = fq2.Square(iiToBig(4, 4))
+	assert.Equal(t, iiToBig(0, 4), res)
+	res2 := fq2.Mul(iiToBig(4, 4), iiToBig(4, 4))
+	assert.Equal(t, res, res2)
+
+	res = fq2.Square(iiToBig(3, 5))
+	assert.Equal(t, iiToBig(5, 2), res)
+	res2 = fq2.Mul(iiToBig(3, 5), iiToBig(3, 5))
+	assert.Equal(t, res, 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
+
+	fq2 := Fq2{fq1, nonResidueFq2}
+	fq6 := Fq6{fq2, nonResidueFq6}
+	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)}
+
+	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)
+}
+
+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, a, divRes)
+}
diff --git a/ecc/ecc.go b/ecc/ecc.go
index 4195326..f196248 100644
--- a/ecc/ecc.go
+++ b/ecc/ecc.go
@@ -14,10 +14,10 @@ type EC struct {
 }
 
 // NewEC (y^2 = x^3 + ax + b) mod q, where q is a prime number
-func NewEC(a, b, q int) (ec EC) {
-	ec.A = big.NewInt(int64(a))
-	ec.B = big.NewInt(int64(b))
-	ec.Q = big.NewInt(int64(q))
+func NewEC(a, b, q *big.Int) (ec EC) {
+	ec.A = a
+	ec.B = b
+	ec.Q = q
 	return ec
 }
 
@@ -51,16 +51,16 @@ func (ec *EC) Neg(p Point) Point {
 
 // Add adds two points p1 and p2 and gets q, returns the negate of q
 func (ec *EC) Add(p1, p2 Point) (Point, error) {
-	if p1.Equal(zeroPoint) {
+	if p1.Equal(ZeroPoint) {
 		return p2, nil
 	}
-	if p2.Equal(zeroPoint) {
+	if p2.Equal(ZeroPoint) {
 		return p1, nil
 	}
 
 	var numerator, denominator, sRaw, s *big.Int
-	if bytes.Equal(p1.X.Bytes(), p2.X.Bytes()) && (!bytes.Equal(p1.Y.Bytes(), p2.Y.Bytes()) || bytes.Equal(p1.Y.Bytes(), bigZero.Bytes())) {
-		return zeroPoint, nil
+	if bytes.Equal(p1.X.Bytes(), p2.X.Bytes()) && (!bytes.Equal(p1.Y.Bytes(), p2.Y.Bytes()) || bytes.Equal(p1.Y.Bytes(), BigZero.Bytes())) {
+		return ZeroPoint, nil
 	} else if bytes.Equal(p1.X.Bytes(), p2.X.Bytes()) {
 		// use tangent as slope
 		// x^2
@@ -115,10 +115,10 @@ func (ec *EC) Add(p1, p2 Point) (Point, error) {
 func (ec *EC) Mul(p Point, n *big.Int) (Point, error) {
 	var err error
 	p2 := p
-	r := zeroPoint
-	for bigZero.Cmp(n) == -1 { // 0<n
-		z := new(big.Int).And(n, bigOne)            // n&1
-		if bytes.Equal(z.Bytes(), bigOne.Bytes()) { // n&1==1
+	r := ZeroPoint
+	for BigZero.Cmp(n) == -1 { // 0<n
+		z := new(big.Int).And(n, BigOne)            // n&1
+		if bytes.Equal(z.Bytes(), BigOne.Bytes()) { // n&1==1
 			r, err = ec.Add(r, p2)
 			if err != nil {
 				return p, err
@@ -137,16 +137,16 @@ func (ec *EC) Mul(p Point, n *big.Int) (Point, error) {
 func (ec *EC) Order(g Point) (*big.Int, error) {
 	// loop from i:=1 to i<ec.Q+1
 	start := big.NewInt(1)
-	end := new(big.Int).Add(ec.Q, bigOne)
-	for i := new(big.Int).Set(start); i.Cmp(end) <= 0; i.Add(i, bigOne) {
+	end := new(big.Int).Add(ec.Q, BigOne)
+	for i := new(big.Int).Set(start); i.Cmp(end) <= 0; i.Add(i, BigOne) {
 		iCopy := new(big.Int).SetBytes(i.Bytes())
 		mPoint, err := ec.Mul(g, iCopy)
 		if err != nil {
 			return i, err
 		}
-		if mPoint.Equal(zeroPoint) {
+		if mPoint.Equal(ZeroPoint) {
 			return i, nil
 		}
 	}
-	return bigZero, errors.New("invalid order")
+	return BigZero, errors.New("invalid order")
 }
diff --git a/ecc/ecc_test.go b/ecc/ecc_test.go
index 91a7022..329ac23 100644
--- a/ecc/ecc_test.go
+++ b/ecc/ecc_test.go
@@ -8,7 +8,7 @@ import (
 )
 
 func TestECC(t *testing.T) {
-	ec := NewEC(0, 7, 11)
+	ec := NewEC(big.NewInt(int64(0)), big.NewInt(int64(7)), big.NewInt(int64(11)))
 	p1, p1i, err := ec.At(big.NewInt(int64(7)))
 	assert.Nil(t, err)
 
@@ -20,7 +20,7 @@ func TestECC(t *testing.T) {
 	}
 }
 func TestNeg(t *testing.T) {
-	ec := NewEC(0, 7, 11)
+	ec := NewEC(big.NewInt(int64(0)), big.NewInt(int64(7)), big.NewInt(int64(11)))
 	p1, p1i, err := ec.At(big.NewInt(int64(7)))
 	assert.Nil(t, err)
 
@@ -32,7 +32,7 @@ func TestNeg(t *testing.T) {
 }
 
 func TestAdd(t *testing.T) {
-	ec := NewEC(0, 7, 11)
+	ec := NewEC(big.NewInt(int64(0)), big.NewInt(int64(7)), big.NewInt(int64(11)))
 	p1 := Point{big.NewInt(int64(4)), big.NewInt(int64(7))}
 	p2 := Point{big.NewInt(int64(2)), big.NewInt(int64(2))}
 	q, err := ec.Add(p1, p2)
@@ -53,7 +53,7 @@ func TestAdd(t *testing.T) {
 }
 
 func TestAddSamePoint(t *testing.T) {
-	ec := NewEC(0, 7, 11)
+	ec := NewEC(big.NewInt(int64(0)), big.NewInt(int64(7)), big.NewInt(int64(11)))
 	p1 := Point{big.NewInt(int64(4)), big.NewInt(int64(7))}
 	p1i := Point{big.NewInt(int64(4)), big.NewInt(int64(4))}
 
@@ -74,7 +74,7 @@ func TestAddSamePoint(t *testing.T) {
 }
 
 func TestMulPoint1(t *testing.T) {
-	ec := NewEC(0, 7, 29)
+	ec := NewEC(big.NewInt(int64(0)), big.NewInt(int64(7)), big.NewInt(int64(29)))
 	p := Point{big.NewInt(int64(11)), big.NewInt(int64(27))}
 
 	q, err := ec.Mul(p, big.NewInt(int64(1)))
@@ -107,7 +107,7 @@ func TestMulPoint1(t *testing.T) {
 }
 
 func TestMulPoint2(t *testing.T) {
-	ec := NewEC(0, 7, 29)
+	ec := NewEC(big.NewInt(int64(0)), big.NewInt(int64(7)), big.NewInt(int64(29)))
 	p1 := Point{big.NewInt(int64(4)), big.NewInt(int64(19))}
 	q3, err := ec.Mul(p1, big.NewInt(int64(3)))
 	assert.Nil(t, err)
@@ -132,7 +132,7 @@ func TestMulPoint2(t *testing.T) {
 
 func TestMulPoint3(t *testing.T) {
 	// in this test we will multiply by a high number
-	ec := NewEC(0, 7, 11)
+	ec := NewEC(big.NewInt(int64(0)), big.NewInt(int64(7)), big.NewInt(int64(11)))
 	p := Point{big.NewInt(int64(7)), big.NewInt(int64(3))}
 
 	q, err := ec.Mul(p, big.NewInt(int64(100)))
@@ -149,7 +149,7 @@ func TestMulPoint3(t *testing.T) {
 }
 
 func TestMulEqualSelfAdd(t *testing.T) {
-	ec := NewEC(0, 7, 29)
+	ec := NewEC(big.NewInt(int64(0)), big.NewInt(int64(7)), big.NewInt(int64(29)))
 	p1 := Point{big.NewInt(int64(11)), big.NewInt(int64(27))}
 
 	p1_2, err := ec.Add(p1, p1)
@@ -185,7 +185,7 @@ func TestMulEqualSelfAdd(t *testing.T) {
 }
 
 func TestOrder(t *testing.T) {
-	ec := NewEC(0, 7, 11)
+	ec := NewEC(big.NewInt(int64(0)), big.NewInt(int64(7)), big.NewInt(int64(11)))
 	g := Point{big.NewInt(int64(7)), big.NewInt(int64(8))}
 	order, err := ec.Order(g)
 	assert.Nil(t, err)
@@ -198,7 +198,7 @@ func TestOrder(t *testing.T) {
 	assert.Equal(t, order.Int64(), int64(4))
 
 	// another test with another curve
-	ec = NewEC(0, 7, 29)
+	ec = NewEC(big.NewInt(int64(0)), big.NewInt(int64(7)), big.NewInt(int64(29)))
 	g = Point{big.NewInt(int64(6)), big.NewInt(int64(22))}
 	order, err = ec.Order(g)
 	assert.Nil(t, err)
diff --git a/ecc/point.go b/ecc/point.go
index 271d117..882f7bf 100644
--- a/ecc/point.go
+++ b/ecc/point.go
@@ -6,9 +6,9 @@ import (
 )
 
 var (
-	bigZero   = big.NewInt(int64(0))
-	bigOne    = big.NewInt(int64(1))
-	zeroPoint = Point{bigZero, bigZero}
+	BigZero   = big.NewInt(int64(0))
+	BigOne    = big.NewInt(int64(1))
+	ZeroPoint = Point{BigZero, BigZero}
 )
 
 // Point is the data structure for a point, containing the X and Y coordinates
@@ -18,11 +18,11 @@ type Point struct {
 }
 
 // Equal compares the X and Y coord of a Point and returns true if are the same
-func (c1 *Point) Equal(c2 Point) bool {
-	if !bytes.Equal(c1.X.Bytes(), c2.X.Bytes()) {
+func (p1 *Point) Equal(p2 Point) bool {
+	if !bytes.Equal(p1.X.Bytes(), p2.X.Bytes()) {
 		return false
 	}
-	if !bytes.Equal(c1.Y.Bytes(), c2.Y.Bytes()) {
+	if !bytes.Equal(p1.Y.Bytes(), p2.Y.Bytes()) {
 		return false
 	}
 	return true
diff --git a/ecdsa/ecdsa_test.go b/ecdsa/ecdsa_test.go
index b7f5a76..f006587 100644
--- a/ecdsa/ecdsa_test.go
+++ b/ecdsa/ecdsa_test.go
@@ -9,7 +9,7 @@ import (
 )
 
 func TestNewECDSA(t *testing.T) {
-	ec := ecc.NewEC(1, 18, 19)
+	ec := ecc.NewEC(big.NewInt(int64(1)), big.NewInt(int64(18)), big.NewInt(int64(19)))
 	g := ecc.Point{big.NewInt(int64(7)), big.NewInt(int64(11))}
 	dsa, err := NewDSA(ec, g)
 	assert.Nil(t, err)
@@ -24,7 +24,7 @@ func TestNewECDSA(t *testing.T) {
 }
 
 func TestECDSASignAndVerify(t *testing.T) {
-	ec := ecc.NewEC(1, 18, 19)
+	ec := ecc.NewEC(big.NewInt(int64(1)), big.NewInt(int64(18)), big.NewInt(int64(19)))
 	g := ecc.Point{big.NewInt(int64(7)), big.NewInt(int64(11))}
 	dsa, err := NewDSA(ec, g)
 	assert.Nil(t, err)
diff --git a/elgamal/elgamal_test.go b/elgamal/elgamal_test.go
index 70bc8e2..7bbaa23 100644
--- a/elgamal/elgamal_test.go
+++ b/elgamal/elgamal_test.go
@@ -9,7 +9,7 @@ import (
 )
 
 func TestNewEG(t *testing.T) {
-	ec := ecc.NewEC(1, 18, 19)
+	ec := ecc.NewEC(big.NewInt(int64(1)), big.NewInt(int64(18)), big.NewInt(int64(19)))
 	g := ecc.Point{big.NewInt(int64(7)), big.NewInt(int64(11))}
 	eg, err := NewEG(ec, g)
 	assert.Nil(t, err)
@@ -23,7 +23,7 @@ func TestNewEG(t *testing.T) {
 	}
 }
 func TestEGEncrypt(t *testing.T) {
-	ec := ecc.NewEC(1, 18, 19)
+	ec := ecc.NewEC(big.NewInt(int64(1)), big.NewInt(int64(18)), big.NewInt(int64(19)))
 	g := ecc.Point{big.NewInt(int64(7)), big.NewInt(int64(11))}
 	eg, err := NewEG(ec, g)
 	assert.Nil(t, err)
@@ -46,7 +46,7 @@ func TestEGEncrypt(t *testing.T) {
 }
 
 func TestEGDecrypt(t *testing.T) {
-	ec := ecc.NewEC(1, 18, 19)
+	ec := ecc.NewEC(big.NewInt(int64(1)), big.NewInt(int64(18)), big.NewInt(int64(19)))
 	g := ecc.Point{big.NewInt(int64(7)), big.NewInt(int64(11))}
 	eg, err := NewEG(ec, g)
 	assert.Nil(t, err)
diff --git a/schnorr/schnorr_test.go b/schnorr/schnorr_test.go
index 185a5d5..2bb9de9 100644
--- a/schnorr/schnorr_test.go
+++ b/schnorr/schnorr_test.go
@@ -8,20 +8,6 @@ import (
 	"github.com/stretchr/testify/assert"
 )
 
-// func TestNewSystem(t *testing.T) {
-//
-// 	ec := ecc.NewEC(0, 7, 11)
-// 	g := ecc.Point{big.NewInt(int64(7)), big.NewInt(int64(8))} // Generator
-// 	r := big.NewInt(int64(7))                                  // random r
-// 	schnorr, sk, err := Gen(ec, g, r)
-// 	assert.Nil(t, err)
-//
-// 	fmt.Print("schnorr")
-// 	fmt.Println(schnorr)
-// 	fmt.Print("sk")
-// 	fmt.Println(sk)
-// }
-
 func TestHash(t *testing.T) {
 	c := ecc.Point{big.NewInt(int64(7)), big.NewInt(int64(8))} // Generator
 	h := Hash([]byte("hola"), c)
@@ -29,7 +15,7 @@ func TestHash(t *testing.T) {
 }
 
 func TestSign(t *testing.T) {
-	ec := ecc.NewEC(0, 7, 11)
+	ec := ecc.NewEC(big.NewInt(int64(0)), big.NewInt(int64(7)), big.NewInt(int64(11)))
 	g := ecc.Point{big.NewInt(int64(7)), big.NewInt(int64(8))} // Generator
 	r := big.NewInt(int64(7))                                  // random r
 	schnorr, sk, err := Gen(ec, g, r)
@@ -47,7 +33,7 @@ func TestSign(t *testing.T) {
 }
 
 func TestSign2(t *testing.T) {
-	ec := ecc.NewEC(0, 7, 29)
+	ec := ecc.NewEC(big.NewInt(int64(0)), big.NewInt(int64(7)), big.NewInt(int64(29)))
 	g := ecc.Point{big.NewInt(int64(11)), big.NewInt(int64(27))} // Generator
 	r := big.NewInt(int64(23))                                   // random r
 	schnorr, sk, err := Gen(ec, g, r)
diff --git a/shamirsecretsharing/shamirsecretsharing.go b/shamirsecretsharing/shamirsecretsharing.go
index cd4bb58..dddf968 100644
--- a/shamirsecretsharing/shamirsecretsharing.go
+++ b/shamirsecretsharing/shamirsecretsharing.go
@@ -10,11 +10,11 @@ const (
 	bits = 1024
 )
 
+// Create calculates the secrets to share from given parameters
 // t: number of secrets needed
 // n: number of shares
 // p: random point
 // k: secret to share
-// Create calculates the secrets to share from given parameters
 func Create(t, n, p, k *big.Int) (result [][]*big.Int, err error) {
 	if k.Cmp(p) > 0 {
 		return nil, errors.New("Error: need k<p. k: " + k.String() + ", p: " + p.String())
diff --git a/utils/utils.go b/utils/utils.go
new file mode 100644
index 0000000..3155754
--- /dev/null
+++ b/utils/utils.go
@@ -0,0 +1,17 @@
+package utils
+
+import "encoding/hex"
+
+// BytesToHex converts from an array of bytes to a hex encoded string
+func BytesToHex(bytesArray []byte) string {
+	r := "0x"
+	h := hex.EncodeToString(bytesArray)
+	r = r + h
+	return r
+}
+
+// HexToBytes converts from a hex string into an array of bytes
+func HexToBytes(h string) ([]byte, error) {
+	b, err := hex.DecodeString(h[2:])
+	return b, err
+}