Add initial implementation, simple flow works

.golangci.yml

@@ -0,0 +1,18 @@
+issues:
+        max-same-issues: 0
+        exclude-use-default: false
+linters:
+        enable:
+        - whitespace
+        - gosec
+        - gci
+        - misspell
+        - gomnd
+        - gofmt
+        - goimports
+        - lll
+        - golint
+        - gocyclo
+linters-settings:
+        lll:
+                line-length: 100

README.md

@@ -0,0 +1,5 @@
+# kzg-commitments-study
+
+Doing this to study and learn [KZG commitments](http://cacr.uwaterloo.ca/techreports/2010/cacr2010-10.pdf), do not use in production.
+
+Thanks to [Dankrad Feist](https://dankradfeist.de/ethereum/2020/06/16/kate-polynomial-commitments.html), [Alin Tomescu](https://alinush.github.io/2020/05/06/kzg-polynomial-commitments.html), [Tom Walton-Pocock](https://hackmd.io/@tompocock/Hk2A7BD6U) for their articles, which helped me understand a bit the KZG Commitments.

arithmetic.go

@@ -0,0 +1,238 @@
+package kzg
+
+import (
+	"bytes"
+	"crypto/rand"
+	"fmt"
+	"math/big"
+	"strconv"
+
+	bn256 "github.com/ethereum/go-ethereum/crypto/bn256/cloudflare"
+)
+
+// Q is the order of the integer field (Zq) that fits inside the snark
+var Q, _ = new(big.Int).SetString(
+	"21888242871839275222246405745257275088696311157297823662689037894645226208583", 10)
+
+// R is the mod of the finite field
+var R, _ = new(big.Int).SetString(
+	"21888242871839275222246405745257275088548364400416034343698204186575808495617", 10)
+
+func randBigInt() (*big.Int, error) {
+	maxbits := R.BitLen()
+	b := make([]byte, (maxbits/8)-1)
+	_, err := rand.Read(b)
+	if err != nil {
+		return nil, err
+	}
+	r := new(big.Int).SetBytes(b)
+	rq := new(big.Int).Mod(r, R)
+
+	return rq, nil
+}
+
+func arrayOfZeroes(n int) []*big.Int {
+	r := make([]*big.Int, n)
+	for i := 0; i < n; i++ {
+		r[i] = new(big.Int).SetInt64(0)
+	}
+	return r[:]
+}
+
+//nolint:deadcode,unused
+func arrayOfZeroesG1(n int) []*bn256.G1 {
+	r := make([]*bn256.G1, n)
+	for i := 0; i < n; i++ {
+		r[i] = new(bn256.G1).ScalarBaseMult(big.NewInt(0))
+	}
+	return r[:]
+}
+
+//nolint:deadcode,unused
+func arrayOfZeroesG2(n int) []*bn256.G2 {
+	r := make([]*bn256.G2, n)
+	for i := 0; i < n; i++ {
+		r[i] = new(bn256.G2).ScalarBaseMult(big.NewInt(0))
+	}
+	return r[:]
+}
+
+func compareBigIntArray(a, b []*big.Int) bool {
+	if len(a) != len(b) {
+		return false
+	}
+	for i := 0; i < len(a); i++ {
+		if a[i] != b[i] {
+			return false
+		}
+	}
+	return true
+}
+
+func fAdd(a, b *big.Int) *big.Int {
+	ab := new(big.Int).Add(a, b)
+	return ab.Mod(ab, R)
+}
+
+func fSub(a, b *big.Int) *big.Int {
+	ab := new(big.Int).Sub(a, b)
+	return new(big.Int).Mod(ab, R)
+}
+
+func fMul(a, b *big.Int) *big.Int {
+	ab := new(big.Int).Mul(a, b)
+	return ab.Mod(ab, R)
+}
+
+func fDiv(a, b *big.Int) *big.Int {
+	ab := new(big.Int).Mul(a, new(big.Int).ModInverse(b, R))
+	return new(big.Int).Mod(ab, R)
+}
+
+//nolint:unused,deadcode // TODO check
+func fNeg(a *big.Int) *big.Int {
+	return new(big.Int).Mod(new(big.Int).Neg(a), R)
+}
+
+//nolint:deadcode,unused
+func fInv(a *big.Int) *big.Int {
+	return new(big.Int).ModInverse(a, R)
+}
+
+func fExp(base *big.Int, e *big.Int) *big.Int {
+	res := big.NewInt(1)
+	rem := new(big.Int).Set(e)
+	exp := base
+
+	for !bytes.Equal(rem.Bytes(), big.NewInt(int64(0)).Bytes()) {
+		// if BigIsOdd(rem) {
+		if rem.Bit(0) == 1 { // .Bit(0) returns 1 when is odd
+			res = fMul(res, exp)
+		}
+		exp = fMul(exp, exp)
+		rem.Rsh(rem, 1)
+	}
+	return res
+}
+
+func max(a, b int) int {
+	if a > b {
+		return a
+	}
+	return b
+}
+
+func polynomialAdd(a, b []*big.Int) []*big.Int {
+	r := arrayOfZeroes(max(len(a), len(b)))
+	for i := 0; i < len(a); i++ {
+		r[i] = fAdd(r[i], a[i])
+	}
+	for i := 0; i < len(b); i++ {
+		r[i] = fAdd(r[i], b[i])
+	}
+	return r
+}
+
+func polynomialSub(a, b []*big.Int) []*big.Int {
+	r := arrayOfZeroes(max(len(a), len(b)))
+	for i := 0; i < len(a); i++ {
+		r[i] = fAdd(r[i], a[i])
+	}
+	for i := 0; i < len(b); i++ {
+		r[i] = fSub(r[i], b[i])
+	}
+	return r
+}
+
+func polynomialMul(a, b []*big.Int) []*big.Int {
+	r := arrayOfZeroes(len(a) + len(b) - 1)
+	for i := 0; i < len(a); i++ {
+		for j := 0; j < len(b); j++ {
+			r[i+j] = fAdd(r[i+j], fMul(a[i], b[j]))
+		}
+	}
+	return r
+}
+
+func polynomialDiv(a, b []*big.Int) ([]*big.Int, []*big.Int) {
+	// https://en.wikipedia.org/wiki/Division_algorithm
+	r := arrayOfZeroes(len(a) - len(b) + 1)
+	rem := a
+	for len(rem) >= len(b) {
+		l := fDiv(rem[len(rem)-1], b[len(b)-1])
+		pos := len(rem) - len(b)
+		r[pos] = l
+		aux := arrayOfZeroes(pos)
+		aux1 := append(aux, l)
+		aux2 := polynomialSub(rem, polynomialMul(b, aux1))
+		rem = aux2[:len(aux2)-1]
+	}
+	return r, rem
+}
+
+// polynomialEval evaluates the polinomial over the Finite Field at the given value x
+func polynomialEval(p []*big.Int, x *big.Int) *big.Int {
+	r := big.NewInt(int64(0))
+	for i := 0; i < len(p); i++ {
+		xi := fExp(x, big.NewInt(int64(i)))
+		elem := fMul(p[i], xi)
+		r = fAdd(r, elem)
+	}
+	return r
+}
+
+// newPolZeroAt generates a new polynomial that has value zero at the given value
+func newPolZeroAt(pointPos, totalPoints int, height *big.Int) []*big.Int {
+	fac := 1
+	for i := 1; i < totalPoints+1; i++ {
+		if i != pointPos {
+			fac = fac * (pointPos - i)
+		}
+	}
+	facBig := big.NewInt(int64(fac))
+	hf := fDiv(height, facBig)
+	r := []*big.Int{hf}
+	for i := 1; i < totalPoints+1; i++ {
+		if i != pointPos {
+			ineg := big.NewInt(int64(-i))
+			b1 := big.NewInt(int64(1))
+			r = polynomialMul(r, []*big.Int{ineg, b1})
+		}
+	}
+	return r
+}
+
+var sNums = map[string]string{
+	"0": "⁰",
+	"1": "¹",
+	"2": "²",
+	"3": "³",
+	"4": "⁴",
+	"5": "⁵",
+	"6": "⁶",
+	"7": "⁷",
+	"8": "⁸",
+	"9": "⁹",
+}
+
+func intToSNum(n int) string {
+	s := strconv.Itoa(n)
+	sN := ""
+	for i := 0; i < len(s); i++ {
+		sN += sNums[string(s[i])]
+	}
+	return sN
+}
+
+func polynomialToString(p []*big.Int) string {
+	s := ""
+	for i := len(p) - 1; i >= 1; i-- {
+		if !bytes.Equal(p[i].Bytes(), big.NewInt(0).Bytes()) {
+			s += fmt.Sprintf("%sx%s + ", p[i], intToSNum(i))
+		}
+	}
+	s += p[0].String()
+	return s
+}
+
+// TODO add method to 'clean' the polynomial, to remove right-zeroes

arithmetic_test.go

@@ -0,0 +1,111 @@
+package kzg
+
+import (
+	"bytes"
+	"crypto/rand"
+	"math/big"
+	"testing"
+
+	cryptoConstants "github.com/iden3/go-iden3-crypto/constants"
+	"github.com/stretchr/testify/assert"
+)
+
+func randBI() *big.Int {
+	maxbits := 256
+	b := make([]byte, (maxbits/8)-1)
+	_, err := rand.Read(b)
+	if err != nil {
+		panic(err)
+	}
+	r := new(big.Int).SetBytes(b)
+	return new(big.Int).Mod(r, cryptoConstants.Q)
+}
+
+func neg(a *big.Int) *big.Int {
+	return new(big.Int).Neg(a)
+}
+
+func TestPolynomial(t *testing.T) {
+	b0 := big.NewInt(int64(0))
+	b1 := big.NewInt(int64(1))
+	b2 := big.NewInt(int64(2))
+	b3 := big.NewInt(int64(3))
+	b4 := big.NewInt(int64(4))
+	b5 := big.NewInt(int64(5))
+	b6 := big.NewInt(int64(6))
+	b16 := big.NewInt(int64(16))
+
+	a := []*big.Int{b1, b0, b5}
+	b := []*big.Int{b3, b0, b1}
+
+	// new Finite Field
+	r, ok := new(big.Int).SetString("21888242871839275222246405745257275088548364400416034343698204186575808495617", 10) //nolint:lll
+	assert.True(nil, ok)
+
+	// polynomial multiplication
+	o := polynomialMul(a, b)
+	assert.Equal(t, o, []*big.Int{b3, b0, b16, b0, b5})
+
+	// polynomial division
+	quo, rem := polynomialDiv(a, b)
+	assert.Equal(t, quo[0].Int64(), int64(5))
+	// check the rem result without modulo
+	assert.Equal(t, new(big.Int).Sub(rem[0], r).Int64(), int64(-14))
+
+	c := []*big.Int{neg(b4), b0, neg(b2), b1}
+	d := []*big.Int{neg(b3), b1}
+	quo2, rem2 := polynomialDiv(c, d)
+	assert.Equal(t, quo2, []*big.Int{b3, b1, b1})
+	assert.Equal(t, rem2[0].Int64(), int64(5))
+
+	// polynomial addition
+	o = polynomialAdd(a, b)
+	assert.Equal(t, o, []*big.Int{b4, b0, b6})
+
+	// polynomial subtraction
+	o1 := polynomialSub(a, b)
+	o2 := polynomialSub(b, a)
+	o = polynomialAdd(o1, o2)
+	assert.True(t, bytes.Equal(b0.Bytes(), o[0].Bytes()))
+	assert.True(t, bytes.Equal(b0.Bytes(), o[1].Bytes()))
+	assert.True(t, bytes.Equal(b0.Bytes(), o[2].Bytes()))
+
+	c = []*big.Int{b5, b6, b1}
+	d = []*big.Int{b1, b3}
+	o = polynomialSub(c, d)
+	assert.Equal(t, o, []*big.Int{b4, b3, b1})
+
+	// NewPolZeroAt
+	o = newPolZeroAt(3, 4, b4)
+	assert.Equal(t, polynomialEval(o, big.NewInt(3)), b4)
+	o = newPolZeroAt(2, 4, b3)
+	assert.Equal(t, polynomialEval(o, big.NewInt(2)), b3)
+}
+
+func BenchmarkArithmetic(b *testing.B) {
+	// generate arrays with bigint
+	var p, q []*big.Int
+	for i := 0; i < 1000; i++ {
+		pi := randBI()
+		p = append(p, pi)
+	}
+	for i := 1000 - 1; i >= 0; i-- {
+		q = append(q, p[i])
+	}
+
+	b.Run("polynomialSub", func(b *testing.B) {
+		for i := 0; i < b.N; i++ {
+			polynomialSub(p, q)
+		}
+	})
+	b.Run("polynomialMul", func(b *testing.B) {
+		for i := 0; i < b.N; i++ {
+			polynomialMul(p, q)
+		}
+	})
+	b.Run("polynomialDiv", func(b *testing.B) {
+		for i := 0; i < b.N; i++ {
+			polynomialDiv(p, q)
+		}
+	})
+}

go.mod

@@ -0,0 +1,9 @@
+module github.com/arnaucube/kzg-commitments-study
+
+go 1.16
+
+require (
+	github.com/ethereum/go-ethereum v1.10.4
+	github.com/iden3/go-iden3-crypto v0.0.5
+	github.com/stretchr/testify v1.7.0
+)

go.sum

@@ -0,0 +1,555 @@
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kzg.go

@@ -0,0 +1,101 @@
+package kzg
+
+import (
+	"fmt"
+	"math/big"
+
+	bn256 "github.com/ethereum/go-ethereum/crypto/bn256/cloudflare"
+)
+
+// TrustedSetup also named Reference String
+type TrustedSetup struct {
+	Tau1 []*bn256.G1
+	Tau2 []*bn256.G2
+}
+
+// NewTrustedSetup returns a new trusted setup. This step should be done in a
+// secure & distributed way
+func NewTrustedSetup(p []*big.Int) (*TrustedSetup, error) {
+	// compute random s
+	s, err := randBigInt()
+	if err != nil {
+		return nil, err
+	}
+
+	// Notation: [x]₁=xG ∈ 𝔾₁, [x]₂=xH ∈ 𝔾₂
+	// τ₁: [x₀]₁, [x₁]₁, [x₂]₁, ..., [x n₋₁]₁
+	// τ₂: [x₀]₂, [x₁]₂, [x₂]₂, ..., [x n₋₁]₂
+
+	// sPow := make([]*big.Int, len(p))
+	tauG1 := make([]*bn256.G1, len(p))
+	tauG2 := make([]*bn256.G2, len(p))
+	for i := 0; i < len(p); i++ {
+		sPow := fExp(s, big.NewInt(int64(i)))
+		tauG1[i] = new(bn256.G1).ScalarBaseMult(sPow)
+		tauG2[i] = new(bn256.G2).ScalarBaseMult(sPow)
+	}
+
+	return &TrustedSetup{tauG1, tauG2}, nil
+}
+
+// Commit generates the commitment to the polynomial p(x)
+func Commit(ts *TrustedSetup, p []*big.Int) *bn256.G1 {
+	c := evaluateG1(ts, p)
+	return c
+}
+
+func evaluateG1(ts *TrustedSetup, p []*big.Int) *bn256.G1 {
+	c := new(bn256.G1).ScalarMult(ts.Tau1[0], p[0])
+	for i := 1; i < len(p); i++ {
+		sp := new(bn256.G1).ScalarMult(ts.Tau1[i], p[i])
+		c = new(bn256.G1).Add(c, sp)
+	}
+	return c
+}
+
+//nolint:deadcode,unused
+func evaluateG2(ts *TrustedSetup, p []*big.Int) *bn256.G2 {
+	c := new(bn256.G2).ScalarMult(ts.Tau2[0], p[0])
+	for i := 1; i < len(p); i++ {
+		sp := new(bn256.G2).ScalarMult(ts.Tau2[i], p[i])
+		c = new(bn256.G2).Add(c, sp)
+	}
+	return c
+}
+
+// EvaluationProof generates the evaluation proof
+func EvaluationProof(ts *TrustedSetup, p []*big.Int, z, y *big.Int) (*bn256.G1, error) {
+	n := polynomialSub(p, []*big.Int{y})    // p-y
+	d := []*big.Int{fNeg(z), big.NewInt(1)} // x-z
+	q, rem := polynomialDiv(n, d)
+	if compareBigIntArray(rem, arrayOfZeroes(len(rem))) {
+		return nil,
+			fmt.Errorf("remainder should be 0, instead is %d", rem)
+	}
+	fmt.Println("q(x):", polynomialToString(q)) // TMP DBG
+
+	// proof: e = [q(s)]₁
+	e := evaluateG1(ts, q)
+	return e, nil
+}
+
+// Verify computes the KZG commitment verification
+func Verify(ts *TrustedSetup, c, proof *bn256.G1, z, y *big.Int) bool {
+	s2 := ts.Tau2[1] // [s]₂ = sG ∈ 𝔾₂ = Tau2[1]
+	zG2Neg := new(bn256.G2).Neg(
+		new(bn256.G2).ScalarBaseMult(z)) // [z]₂ = zG ∈ 𝔾₂
+	// [s]₂ - [z]₂
+	sz := new(bn256.G2).Add(s2, zG2Neg)
+
+	yG1Neg := new(bn256.G1).Neg(
+		new(bn256.G1).ScalarBaseMult(y)) // [y]₁ = yG ∈ 𝔾₁
+	// c - [y]₁
+	cy := new(bn256.G1).Add(c, yG1Neg)
+
+	h := new(bn256.G2).ScalarBaseMult(big.NewInt(1)) // H ∈ 𝔾₂
+
+	// e(proof, [s]₂ - [z]₂) == e(c - [y]₁, H)
+	e1 := bn256.Pair(proof, sz)
+	e2 := bn256.Pair(cy, h)
+	return e1.String() == e2.String()
+}

kzg_test.go

@@ -0,0 +1,41 @@
+package kzg
+
+import (
+	"math/big"
+	"testing"
+
+	"github.com/stretchr/testify/assert"
+)
+
+func TestSimpleFlow(t *testing.T) {
+	// p(x) = x^3 + x + 5
+	p := []*big.Int{
+		big.NewInt(5),
+		big.NewInt(1), // x^1
+		big.NewInt(0), // x^2
+		big.NewInt(1), // x^3
+	}
+	// fmt.Println("p(x):", polynomialToString(p))
+	assert.Equal(t, "1x³ + 1x¹ + 5", polynomialToString(p))
+
+	// TrustedSetup
+	ts, err := NewTrustedSetup(p)
+	assert.Nil(t, err)
+
+	// Commit
+	c := Commit(ts, p)
+
+	// p(z)=y --> p(3)=35
+	z := big.NewInt(3)
+	y := big.NewInt(35)
+
+	// z & y: to prove an evaluation p(z)=y
+	proof, err := EvaluationProof(ts, p, z, y)
+	assert.Nil(t, err)
+
+	v := Verify(ts, c, proof, z, y)
+	assert.True(t, v)
+
+	v = Verify(ts, c, proof, big.NewInt(4), y)
+	assert.False(t, v)
+}