├── .github └── workflows │ ├── lint.yml │ └── test.yml ├── .golangci.yml ├── LICENSE ├── README.md ├── arithmetic.go ├── arithmetic_test.go ├── go.mod ├── go.sum ├── kzg.go └── kzg_test.go /.github/workflows/lint.yml: -------------------------------------------------------------------------------- 1 | name: Lint 2 | on: [ push, pull_request ] 3 | jobs: 4 | lint: 5 | runs-on: ubuntu-latest 6 | steps: 7 | - name: Install Go 8 | uses: actions/setup-go@v1 9 | with: 10 | go-version: 1.16.x 11 | - name: Checkout code 12 | uses: actions/checkout@v2 13 | - name: Lint 14 | run: | 15 | curl -sSfL https://raw.githubusercontent.com/golangci/golangci-lint/master/install.sh | sh -s -- -b $(go env GOPATH)/bin v1.30.0 16 | $(go env GOPATH)/bin/golangci-lint run --timeout=5m -c .golangci.yml 17 | -------------------------------------------------------------------------------- /.github/workflows/test.yml: -------------------------------------------------------------------------------- 1 | name: Test 2 | on: [push, pull_request] 3 | jobs: 4 | test: 5 | # matrix strategy from: https://github.com/mvdan/github-actions-golang/blob/master/.github/workflows/test.yml 6 | strategy: 7 | matrix: 8 | go-version: [1.16.x] 9 | platform: [ubuntu-latest] 10 | runs-on: ${{ matrix.platform }} 11 | steps: 12 | - name: Install Go 13 | uses: actions/setup-go@v1 14 | with: 15 | go-version: ${{ matrix.go-version }} 16 | - name: Checkout code 17 | uses: actions/checkout@v2 18 | - name: Go test 19 | run: go test ./... 20 | -------------------------------------------------------------------------------- /.golangci.yml: -------------------------------------------------------------------------------- 1 | issues: 2 | max-same-issues: 0 3 | exclude-use-default: false 4 | linters: 5 | enable: 6 | - whitespace 7 | - gosec 8 | - gci 9 | - misspell 10 | - gomnd 11 | - gofmt 12 | - goimports 13 | - lll 14 | - golint 15 | - gocyclo 16 | linters-settings: 17 | lll: 18 | line-length: 100 19 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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Interpretation of Sections 15 and 16. 613 | 614 | If the disclaimer of warranty and limitation of liability provided 615 | above cannot be given local legal effect according to their terms, 616 | reviewing courts shall apply local law that most closely approximates 617 | an absolute waiver of all civil liability in connection with the 618 | Program, unless a warranty or assumption of liability accompanies a 619 | copy of the Program in return for a fee. 620 | 621 | END OF TERMS AND CONDITIONS 622 | 623 | How to Apply These Terms to Your New Programs 624 | 625 | If you develop a new program, and you want it to be of the greatest 626 | possible use to the public, the best way to achieve this is to make it 627 | free software which everyone can redistribute and change under these terms. 628 | 629 | To do so, attach the following notices to the program. It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # kzg-commitments-study [![GoDoc](https://godoc.org/github.com/arnaucube/kzg-commitments-study?status.svg)](https://godoc.org/github.com/arnaucube/kzg-commitments-study) [![Go Report Card](https://goreportcard.com/badge/github.com/arnaucube/kzg-commitments-study)](https://goreportcard.com/report/github.com/arnaucube/kzg-commitments-study) [![Test](https://github.com/arnaucube/kzg-commitments-study/workflows/Test/badge.svg)](https://github.com/arnaucube/kzg-commitments-study/actions?query=workflow%3ATest) 2 | 3 | Doing this to study and learn [KZG commitments](http://cacr.uwaterloo.ca/techreports/2010/cacr2010-10.pdf), do not use in production. More details at https://arnaucube.com/blog/kzg-commitments.html . 4 | 5 | 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. 6 | 7 | It uses the [ethereum bn256](https://github.com/ethereum/go-ethereum/tree/master/crypto/bn256/cloudflare). 8 | 9 | ### Usage 10 | 11 | ```go 12 | // p(x) = x^3 + x + 5 13 | p := []*big.Int{ 14 | big.NewInt(5), 15 | big.NewInt(1), // x^1 16 | big.NewInt(0), // x^2 17 | big.NewInt(1), // x^3 18 | } 19 | assert.Equal(t, "1x³ + 1x¹ + 5", PolynomialToString(p)) 20 | 21 | // TrustedSetup 22 | ts, err := NewTrustedSetup(p) 23 | assert.Nil(t, err) 24 | 25 | // Commit 26 | c := Commit(ts, p) 27 | 28 | // p(z)=y --> p(3)=35 29 | z := big.NewInt(3) 30 | y := big.NewInt(35) 31 | 32 | // z & y: to prove an evaluation p(z)=y 33 | proof, err := EvaluationProof(ts, p, z, y) 34 | assert.Nil(t, err) 35 | 36 | // verification 37 | v := Verify(ts, c, proof, z, y) 38 | assert.True(t, v) 39 | ``` 40 | 41 | Batch Proofs: 42 | ```go 43 | // zs & ys contain the f(z_i)=y_i values that will be proved inside a batch proof 44 | zs := []*big.Int{z0, z1, z2} 45 | ys := []*big.Int{y0, y1, y2} 46 | 47 | // prove an evaluation of the multiple z_i & y_i 48 | proof, err := EvaluationBatchProof(ts, p, zs, ys) 49 | assert.Nil(t, err) 50 | 51 | // batch proof verification 52 | v := VerifyBatchProof(ts, c, proof, zs, ys) 53 | assert.True(t, v) 54 | ``` 55 | -------------------------------------------------------------------------------- /arithmetic.go: -------------------------------------------------------------------------------- 1 | package kzg 2 | 3 | import ( 4 | "bytes" 5 | "crypto/rand" 6 | "fmt" 7 | "math/big" 8 | "strconv" 9 | 10 | bn256 "github.com/ethereum/go-ethereum/crypto/bn256/cloudflare" 11 | ) 12 | 13 | // Q is the order of the integer field (Zq) that fits inside the snark 14 | var Q, _ = new(big.Int).SetString( 15 | "21888242871839275222246405745257275088696311157297823662689037894645226208583", 10) 16 | 17 | // R is the mod of the finite field 18 | var R, _ = new(big.Int).SetString( 19 | "21888242871839275222246405745257275088548364400416034343698204186575808495617", 10) 20 | 21 | func randBigInt() (*big.Int, error) { 22 | maxbits := R.BitLen() 23 | b := make([]byte, (maxbits/8)-1) 24 | _, err := rand.Read(b) 25 | if err != nil { 26 | return nil, err 27 | } 28 | r := new(big.Int).SetBytes(b) 29 | rq := new(big.Int).Mod(r, R) 30 | 31 | return rq, nil 32 | } 33 | 34 | func arrayOfZeroes(n int) []*big.Int { 35 | r := make([]*big.Int, n) 36 | for i := 0; i < n; i++ { 37 | r[i] = new(big.Int).SetInt64(0) 38 | } 39 | return r[:] 40 | } 41 | 42 | //nolint:deadcode,unused 43 | func arrayOfZeroesG1(n int) []*bn256.G1 { 44 | r := make([]*bn256.G1, n) 45 | for i := 0; i < n; i++ { 46 | r[i] = new(bn256.G1).ScalarBaseMult(big.NewInt(0)) 47 | } 48 | return r[:] 49 | } 50 | 51 | //nolint:deadcode,unused 52 | func arrayOfZeroesG2(n int) []*bn256.G2 { 53 | r := make([]*bn256.G2, n) 54 | for i := 0; i < n; i++ { 55 | r[i] = new(bn256.G2).ScalarBaseMult(big.NewInt(0)) 56 | } 57 | return r[:] 58 | } 59 | 60 | func compareBigIntArray(a, b []*big.Int) bool { 61 | if len(a) != len(b) { 62 | return false 63 | } 64 | for i := 0; i < len(a); i++ { 65 | if a[i] != b[i] { 66 | return false 67 | } 68 | } 69 | return true 70 | } 71 | 72 | //nolint:deadcode,unused 73 | func checkArrayOfZeroes(a []*big.Int) bool { 74 | z := arrayOfZeroes(len(a)) 75 | return compareBigIntArray(a, z) 76 | } 77 | 78 | func fAdd(a, b *big.Int) *big.Int { 79 | ab := new(big.Int).Add(a, b) 80 | return ab.Mod(ab, R) 81 | } 82 | 83 | func fSub(a, b *big.Int) *big.Int { 84 | ab := new(big.Int).Sub(a, b) 85 | return new(big.Int).Mod(ab, R) 86 | } 87 | 88 | func fMul(a, b *big.Int) *big.Int { 89 | ab := new(big.Int).Mul(a, b) 90 | return ab.Mod(ab, R) 91 | } 92 | 93 | func fDiv(a, b *big.Int) *big.Int { 94 | ab := new(big.Int).Mul(a, new(big.Int).ModInverse(b, R)) 95 | return new(big.Int).Mod(ab, R) 96 | } 97 | 98 | func fNeg(a *big.Int) *big.Int { 99 | return new(big.Int).Mod(new(big.Int).Neg(a), R) 100 | } 101 | 102 | //nolint:deadcode,unused 103 | func fInv(a *big.Int) *big.Int { 104 | return new(big.Int).ModInverse(a, R) 105 | } 106 | 107 | func fExp(base *big.Int, e *big.Int) *big.Int { 108 | res := big.NewInt(1) 109 | rem := new(big.Int).Set(e) 110 | exp := base 111 | 112 | for !bytes.Equal(rem.Bytes(), big.NewInt(int64(0)).Bytes()) { 113 | // if BigIsOdd(rem) { 114 | if rem.Bit(0) == 1 { // .Bit(0) returns 1 when is odd 115 | res = fMul(res, exp) 116 | } 117 | exp = fMul(exp, exp) 118 | rem.Rsh(rem, 1) 119 | } 120 | return res 121 | } 122 | 123 | func max(a, b int) int { 124 | if a > b { 125 | return a 126 | } 127 | return b 128 | } 129 | 130 | func polynomialAdd(a, b []*big.Int) []*big.Int { 131 | r := arrayOfZeroes(max(len(a), len(b))) 132 | for i := 0; i < len(a); i++ { 133 | r[i] = fAdd(r[i], a[i]) 134 | } 135 | for i := 0; i < len(b); i++ { 136 | r[i] = fAdd(r[i], b[i]) 137 | } 138 | return r 139 | } 140 | 141 | func polynomialSub(a, b []*big.Int) []*big.Int { 142 | r := arrayOfZeroes(max(len(a), len(b))) 143 | for i := 0; i < len(a); i++ { 144 | r[i] = fAdd(r[i], a[i]) 145 | } 146 | for i := 0; i < len(b); i++ { 147 | r[i] = fSub(r[i], b[i]) 148 | } 149 | return r 150 | } 151 | 152 | func polynomialMul(a, b []*big.Int) []*big.Int { 153 | r := arrayOfZeroes(len(a) + len(b) - 1) 154 | for i := 0; i < len(a); i++ { 155 | for j := 0; j < len(b); j++ { 156 | r[i+j] = fAdd(r[i+j], fMul(a[i], b[j])) 157 | } 158 | } 159 | return r 160 | } 161 | 162 | func polynomialDiv(a, b []*big.Int) ([]*big.Int, []*big.Int) { 163 | // https://en.wikipedia.org/wiki/Division_algorithm 164 | r := arrayOfZeroes(len(a) - len(b) + 1) 165 | rem := a 166 | for len(rem) >= len(b) { 167 | l := fDiv(rem[len(rem)-1], b[len(b)-1]) 168 | pos := len(rem) - len(b) 169 | r[pos] = l 170 | aux := arrayOfZeroes(pos) 171 | aux1 := append(aux, l) 172 | aux2 := polynomialSub(rem, polynomialMul(b, aux1)) 173 | rem = aux2[:len(aux2)-1] 174 | } 175 | return r, rem 176 | } 177 | 178 | func polynomialMulByConstant(a []*big.Int, c *big.Int) []*big.Int { 179 | for i := 0; i < len(a); i++ { 180 | a[i] = fMul(a[i], c) 181 | } 182 | return a 183 | } 184 | func polynomialDivByConstant(a []*big.Int, c *big.Int) []*big.Int { 185 | for i := 0; i < len(a); i++ { 186 | a[i] = fDiv(a[i], c) 187 | } 188 | return a 189 | } 190 | 191 | // polynomialEval evaluates the polinomial over the Finite Field at the given value x 192 | func polynomialEval(p []*big.Int, x *big.Int) *big.Int { 193 | r := big.NewInt(int64(0)) 194 | for i := 0; i < len(p); i++ { 195 | xi := fExp(x, big.NewInt(int64(i))) 196 | elem := fMul(p[i], xi) 197 | r = fAdd(r, elem) 198 | } 199 | return r 200 | } 201 | 202 | // newPolZeroAt generates a new polynomial that has value zero at the given value 203 | func newPolZeroAt(pointPos, totalPoints int, height *big.Int) []*big.Int { 204 | fac := 1 205 | for i := 1; i < totalPoints+1; i++ { 206 | if i != pointPos { 207 | fac = fac * (pointPos - i) 208 | } 209 | } 210 | facBig := big.NewInt(int64(fac)) 211 | hf := fDiv(height, facBig) 212 | r := []*big.Int{hf} 213 | for i := 1; i < totalPoints+1; i++ { 214 | if i != pointPos { 215 | ineg := big.NewInt(int64(-i)) 216 | b1 := big.NewInt(int64(1)) 217 | r = polynomialMul(r, []*big.Int{ineg, b1}) 218 | } 219 | } 220 | return r 221 | } 222 | 223 | // zeroPolynomial returns the zero polynomial: 224 | // z(x) = (x - z_0) (x - z_1) ... (x - z_{k-1}) 225 | func zeroPolynomial(zs []*big.Int) []*big.Int { 226 | z := []*big.Int{fNeg(zs[0]), big.NewInt(1)} // (x - z0) 227 | for i := 1; i < len(zs); i++ { 228 | z = polynomialMul(z, []*big.Int{fNeg(zs[i]), big.NewInt(1)}) // (x - zi) 229 | } 230 | return z 231 | } 232 | 233 | var sNums = map[string]string{ 234 | "0": "⁰", 235 | "1": "¹", 236 | "2": "²", 237 | "3": "³", 238 | "4": "⁴", 239 | "5": "⁵", 240 | "6": "⁶", 241 | "7": "⁷", 242 | "8": "⁸", 243 | "9": "⁹", 244 | } 245 | 246 | func intToSNum(n int) string { 247 | s := strconv.Itoa(n) 248 | sN := "" 249 | for i := 0; i < len(s); i++ { 250 | sN += sNums[string(s[i])] 251 | } 252 | return sN 253 | } 254 | 255 | // PolynomialToString converts a polynomial represented by a *big.Int array, 256 | // into its string human readable representation 257 | func PolynomialToString(p []*big.Int) string { 258 | s := "" 259 | for i := len(p) - 1; i >= 1; i-- { 260 | if bytes.Equal(p[i].Bytes(), big.NewInt(1).Bytes()) { 261 | s += fmt.Sprintf("x%s + ", intToSNum(i)) 262 | } else if !bytes.Equal(p[i].Bytes(), big.NewInt(0).Bytes()) { 263 | s += fmt.Sprintf("%sx%s + ", p[i], intToSNum(i)) 264 | } 265 | } 266 | s += p[0].String() 267 | return s 268 | } 269 | 270 | // LagrangeInterpolation implements the Lagrange interpolation: 271 | // https://en.wikipedia.org/wiki/Lagrange_polynomial 272 | func LagrangeInterpolation(x, y []*big.Int) ([]*big.Int, error) { 273 | // p(x) will be the interpoled polynomial 274 | // var p []*big.Int 275 | if len(x) != len(y) { 276 | return nil, fmt.Errorf("len(x)!=len(y): %d, %d", len(x), len(y)) 277 | } 278 | p := arrayOfZeroes(len(x)) 279 | k := len(x) 280 | 281 | for j := 0; j < k; j++ { 282 | // jPol is the Lagrange basis polynomial for each point 283 | var jPol []*big.Int 284 | for m := 0; m < k; m++ { 285 | // if x[m] == x[j] { 286 | if m == j { 287 | continue 288 | } 289 | // numerator & denominator of the current iteration 290 | num := []*big.Int{fNeg(x[m]), big.NewInt(1)} // (x^1 - x_m) 291 | den := fSub(x[j], x[m]) // x_j-x_m 292 | mPol := polynomialDivByConstant(num, den) 293 | if len(jPol) == 0 { 294 | // first j iteration 295 | jPol = mPol 296 | continue 297 | } 298 | jPol = polynomialMul(jPol, mPol) 299 | } 300 | p = polynomialAdd(p, polynomialMulByConstant(jPol, y[j])) 301 | } 302 | 303 | return p, nil 304 | } 305 | 306 | // TODO add method to 'clean' the polynomial, to remove right-zeroes 307 | -------------------------------------------------------------------------------- /arithmetic_test.go: -------------------------------------------------------------------------------- 1 | package kzg 2 | 3 | import ( 4 | "bytes" 5 | "crypto/rand" 6 | "math/big" 7 | "testing" 8 | 9 | "github.com/stretchr/testify/assert" 10 | ) 11 | 12 | func randBI() *big.Int { 13 | maxbits := 256 14 | b := make([]byte, (maxbits/8)-1) 15 | _, err := rand.Read(b) 16 | if err != nil { 17 | panic(err) 18 | } 19 | r := new(big.Int).SetBytes(b) 20 | return new(big.Int).Mod(r, Q) 21 | } 22 | 23 | func neg(a *big.Int) *big.Int { 24 | return new(big.Int).Neg(a) 25 | } 26 | 27 | func TestPolynomial(t *testing.T) { 28 | b0 := big.NewInt(int64(0)) 29 | b1 := big.NewInt(int64(1)) 30 | b2 := big.NewInt(int64(2)) 31 | b3 := big.NewInt(int64(3)) 32 | b4 := big.NewInt(int64(4)) 33 | b5 := big.NewInt(int64(5)) 34 | b6 := big.NewInt(int64(6)) 35 | b16 := big.NewInt(int64(16)) 36 | 37 | a := []*big.Int{b1, b0, b5} 38 | b := []*big.Int{b3, b0, b1} 39 | 40 | // new Finite Field 41 | r, ok := new(big.Int).SetString("21888242871839275222246405745257275088548364400416034343698204186575808495617", 10) //nolint:lll 42 | assert.True(nil, ok) 43 | 44 | // polynomial multiplication 45 | o := polynomialMul(a, b) 46 | assert.Equal(t, o, []*big.Int{b3, b0, b16, b0, b5}) 47 | 48 | // polynomial division 49 | quo, rem := polynomialDiv(a, b) 50 | assert.Equal(t, quo[0].Int64(), int64(5)) 51 | // check the rem result without modulo 52 | assert.Equal(t, new(big.Int).Sub(rem[0], r).Int64(), int64(-14)) 53 | 54 | c := []*big.Int{neg(b4), b0, neg(b2), b1} 55 | d := []*big.Int{neg(b3), b1} 56 | quo2, rem2 := polynomialDiv(c, d) 57 | assert.Equal(t, quo2, []*big.Int{b3, b1, b1}) 58 | assert.Equal(t, rem2[0].Int64(), int64(5)) 59 | 60 | // polynomial addition 61 | o = polynomialAdd(a, b) 62 | assert.Equal(t, o, []*big.Int{b4, b0, b6}) 63 | 64 | // polynomial subtraction 65 | o1 := polynomialSub(a, b) 66 | o2 := polynomialSub(b, a) 67 | o = polynomialAdd(o1, o2) 68 | assert.True(t, bytes.Equal(b0.Bytes(), o[0].Bytes())) 69 | assert.True(t, bytes.Equal(b0.Bytes(), o[1].Bytes())) 70 | assert.True(t, bytes.Equal(b0.Bytes(), o[2].Bytes())) 71 | 72 | c = []*big.Int{b5, b6, b1} 73 | d = []*big.Int{b1, b3} 74 | o = polynomialSub(c, d) 75 | assert.Equal(t, o, []*big.Int{b4, b3, b1}) 76 | 77 | // NewPolZeroAt 78 | o = newPolZeroAt(3, 4, b4) 79 | assert.Equal(t, polynomialEval(o, big.NewInt(3)), b4) 80 | o = newPolZeroAt(2, 4, b3) 81 | assert.Equal(t, polynomialEval(o, big.NewInt(2)), b3) 82 | 83 | // polynomialEval 84 | // p(x) = x^3 + x + 5 85 | p := []*big.Int{ 86 | big.NewInt(5), 87 | big.NewInt(1), // x^1 88 | big.NewInt(0), // x^2 89 | big.NewInt(1), // x^3 90 | } 91 | assert.Equal(t, "x³ + x¹ + 5", PolynomialToString(p)) 92 | assert.Equal(t, "35", polynomialEval(p, big.NewInt(3)).String()) 93 | assert.Equal(t, "1015", polynomialEval(p, big.NewInt(10)).String()) 94 | assert.Equal(t, "16777477", polynomialEval(p, big.NewInt(256)).String()) 95 | assert.Equal(t, "125055", polynomialEval(p, big.NewInt(50)).String()) 96 | assert.Equal(t, "7", polynomialEval(p, big.NewInt(1)).String()) 97 | } 98 | 99 | func BenchmarkArithmetic(b *testing.B) { 100 | // generate arrays with bigint 101 | var p, q []*big.Int 102 | for i := 0; i < 1000; i++ { 103 | pi := randBI() 104 | p = append(p, pi) 105 | } 106 | for i := 1000 - 1; i >= 0; i-- { 107 | q = append(q, p[i]) 108 | } 109 | 110 | b.Run("polynomialSub", func(b *testing.B) { 111 | for i := 0; i < b.N; i++ { 112 | polynomialSub(p, q) 113 | } 114 | }) 115 | b.Run("polynomialMul", func(b *testing.B) { 116 | for i := 0; i < b.N; i++ { 117 | polynomialMul(p, q) 118 | } 119 | }) 120 | b.Run("polynomialDiv", func(b *testing.B) { 121 | for i := 0; i < b.N; i++ { 122 | polynomialDiv(p, q) 123 | } 124 | }) 125 | } 126 | 127 | func TestLagrangeInterpolation(t *testing.T) { 128 | x0 := big.NewInt(3) 129 | y0 := big.NewInt(35) 130 | x1 := big.NewInt(10) 131 | y1 := big.NewInt(1015) 132 | x2 := big.NewInt(256) 133 | y2 := big.NewInt(16777477) 134 | x3 := big.NewInt(50) 135 | y3 := big.NewInt(125055) 136 | 137 | xs := []*big.Int{x0, x1, x2, x3} 138 | ys := []*big.Int{y0, y1, y2, y3} 139 | 140 | p, err := LagrangeInterpolation(xs, ys) 141 | assert.Nil(t, err) 142 | assert.Equal(t, "x³ + x¹ + 5", PolynomialToString(p)) 143 | 144 | assert.Equal(t, y0, polynomialEval(p, x0)) 145 | assert.Equal(t, y1, polynomialEval(p, x1)) 146 | assert.Equal(t, y2, polynomialEval(p, x2)) 147 | } 148 | 149 | func TestZeroPolynomial(t *testing.T) { 150 | x0 := big.NewInt(1) 151 | x1 := big.NewInt(40) 152 | x2 := big.NewInt(512) 153 | xs := []*big.Int{x0, x1, x2} 154 | 155 | z := zeroPolynomial(xs) 156 | assert.Equal(t, "x³ "+ 157 | "+ 21888242871839275222246405745257275088548364400416034343698204186575808495064x² "+ 158 | "+ 21032x¹ + 21888242871839275222246405745257275088548364400416034343698204186575808475137", 159 | PolynomialToString(z)) 160 | 161 | assert.Equal(t, "0", polynomialEval(z, x0).String()) 162 | assert.Equal(t, "0", polynomialEval(z, x1).String()) 163 | assert.Equal(t, "0", polynomialEval(z, x2).String()) 164 | } 165 | -------------------------------------------------------------------------------- /go.mod: -------------------------------------------------------------------------------- 1 | module github.com/arnaucube/kzg-commitments-study 2 | 3 | go 1.16 4 | 5 | require ( 6 | github.com/ethereum/go-ethereum v1.10.4 7 | github.com/stretchr/testify v1.7.0 8 | ) 9 | -------------------------------------------------------------------------------- /go.sum: -------------------------------------------------------------------------------- 1 | cloud.google.com/go v0.26.0/go.mod h1:aQUYkXzVsufM+DwF1aE+0xfcU+56JwCaLick0ClmMTw= 2 | cloud.google.com/go v0.34.0/go.mod h1:aQUYkXzVsufM+DwF1aE+0xfcU+56JwCaLick0ClmMTw= 3 | cloud.google.com/go v0.38.0/go.mod h1:990N+gfupTy94rShfmMCWGDn0LpTmnzTp2qbd1dvSRU= 4 | cloud.google.com/go v0.43.0/go.mod h1:BOSR3VbTLkk6FDC/TcffxP4NF/FFBGA5ku+jvKOP7pg= 5 | cloud.google.com/go v0.44.1/go.mod h1:iSa0KzasP4Uvy3f1mN/7PiObzGgflwredwwASm/v6AU= 6 | cloud.google.com/go v0.44.2/go.mod h1:60680Gw3Yr4ikxnPRS/oxxkBccT6SA1yMk63TGekxKY= 7 | cloud.google.com/go v0.45.1/go.mod h1:RpBamKRgapWJb87xiFSdk4g1CME7QZg3uwTez+TSTjc= 8 | cloud.google.com/go v0.46.3/go.mod h1:a6bKKbmY7er1mI7TEI4lsAkts/mkhTSZK8w33B4RAg0= 9 | cloud.google.com/go v0.50.0/go.mod h1:r9sluTvynVuxRIOHXQEHMFffphuXHOMZMycpNR5e6To= 10 | cloud.google.com/go v0.51.0/go.mod h1:hWtGJ6gnXH+KgDv+V0zFGDvpi07n3z8ZNj3T1RW0Gcw= 11 | cloud.google.com/go/bigquery v1.0.1/go.mod h1:i/xbL2UlR5RvWAURpBYZTtm/cXjCha9lbfbpx4poX+o= 12 | cloud.google.com/go/bigquery v1.3.0/go.mod h1:PjpwJnslEMmckchkHFfq+HTD2DmtT67aNFKH1/VBDHE= 13 | cloud.google.com/go/bigtable v1.2.0/go.mod h1:JcVAOl45lrTmQfLj7T6TxyMzIN/3FGGcFm+2xVAli2o= 14 | cloud.google.com/go/datastore v1.0.0/go.mod h1:LXYbyblFSglQ5pkeyhO+Qmw7ukd3C+pD7TKLgZqpHYE= 15 | cloud.google.com/go/pubsub v1.0.1/go.mod h1:R0Gpsv3s54REJCy4fxDixWD93lHJMoZTyQ2kNxGRt3I= 16 | cloud.google.com/go/pubsub v1.1.0/go.mod h1:EwwdRX2sKPjnvnqCa270oGRyludottCI76h+R3AArQw= 17 | cloud.google.com/go/storage v1.0.0/go.mod h1:IhtSnM/ZTZV8YYJWCY8RULGVqBDmpoyjwiyrjsg+URw= 18 | cloud.google.com/go/storage v1.5.0/go.mod h1:tpKbwo567HUNpVclU5sGELwQWBDZ8gh0ZeosJ0Rtdos= 19 | collectd.org v0.3.0/go.mod h1:A/8DzQBkF6abtvrT2j/AU/4tiBgJWYyh0y/oB/4MlWE= 20 | dmitri.shuralyov.com/gpu/mtl v0.0.0-20190408044501-666a987793e9/go.mod h1:H6x//7gZCb22OMCxBHrMx7a5I7Hp++hsVxbQ4BYO7hU= 21 | github.com/Azure/azure-pipeline-go v0.2.1/go.mod h1:UGSo8XybXnIGZ3epmeBw7Jdz+HiUVpqIlpz/HKHylF4= 22 | github.com/Azure/azure-pipeline-go v0.2.2/go.mod h1:4rQ/NZncSvGqNkkOsNpOU1tgoNuIlp9AfUH5G1tvCHc= 23 | github.com/Azure/azure-storage-blob-go v0.7.0/go.mod h1:f9YQKtsG1nMisotuTPpO0tjNuEjKRYAcJU8/ydDI++4= 24 | github.com/Azure/go-autorest/autorest v0.9.0/go.mod h1:xyHB1BMZT0cuDHU7I0+g046+BFDTQ8rEZB0s4Yfa6bI= 25 | github.com/Azure/go-autorest/autorest/adal v0.5.0/go.mod h1:8Z9fGy2MpX0PvDjB1pEgQTmVqjGhiHBW7RJJEciWzS0= 26 | github.com/Azure/go-autorest/autorest/adal v0.8.0/go.mod h1:Z6vX6WXXuyieHAXwMj0S6HY6e6wcHn37qQMBQlvY3lc= 27 | github.com/Azure/go-autorest/autorest/date v0.1.0/go.mod h1:plvfp3oPSKwf2DNjlBjWF/7vwR+cUD/ELuzDCXwHUVA= 28 | github.com/Azure/go-autorest/autorest/date v0.2.0/go.mod h1:vcORJHLJEh643/Ioh9+vPmf1Ij9AEBM5FuBIXLmIy0g= 29 | github.com/Azure/go-autorest/autorest/mocks v0.1.0/go.mod h1:OTyCOPRA2IgIlWxVYxBee2F5Gr4kF2zd2J5cFRaIDN0= 30 | github.com/Azure/go-autorest/autorest/mocks v0.2.0/go.mod h1:OTyCOPRA2IgIlWxVYxBee2F5Gr4kF2zd2J5cFRaIDN0= 31 | github.com/Azure/go-autorest/autorest/mocks v0.3.0/go.mod h1:a8FDP3DYzQ4RYfVAxAN3SVSiiO77gL2j2ronKKP0syM= 32 | github.com/Azure/go-autorest/logger v0.1.0/go.mod h1:oExouG+K6PryycPJfVSxi/koC6LSNgds39diKLz7Vrc= 33 | github.com/Azure/go-autorest/tracing v0.5.0/go.mod h1:r/s2XiOKccPW3HrqB+W0TQzfbtp2fGCgRFtBroKn4Dk= 34 | github.com/BurntSushi/toml v0.3.1/go.mod h1:xHWCNGjB5oqiDr8zfno3MHue2Ht5sIBksp03qcyfWMU= 35 | github.com/BurntSushi/xgb v0.0.0-20160522181843-27f122750802/go.mod h1:IVnqGOEym/WlBOVXweHU+Q+/VP0lqqI8lqeDx9IjBqo= 36 | github.com/DATA-DOG/go-sqlmock v1.3.3/go.mod h1:f/Ixk793poVmq4qj/V1dPUg2JEAKC73Q5eFN3EC/SaM= 37 | github.com/OneOfOne/xxhash v1.2.2/go.mod h1:HSdplMjZKSmBqAxg5vPj2TmRDmfkzw+cTzAElWljhcU= 38 | github.com/StackExchange/wmi v0.0.0-20180116203802-5d049714c4a6/go.mod 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v0.0.0-20180321164747-3a771d992973/go.mod h1:Dwedo/Wpr24TaqPxmxbtue+5NUziq4I4S80YR8gNf3Q= 57 | github.com/beorn7/perks v1.0.0/go.mod h1:KWe93zE9D1o94FZ5RNwFwVgaQK1VOXiVxmqh+CedLV8= 58 | github.com/bmizerany/pat v0.0.0-20170815010413-6226ea591a40/go.mod h1:8rLXio+WjiTceGBHIoTvn60HIbs7Hm7bcHjyrSqYB9c= 59 | github.com/boltdb/bolt v1.3.1/go.mod h1:clJnj/oiGkjum5o1McbSZDSLxVThjynRyGBgiAx27Ps= 60 | github.com/btcsuite/btcd v0.20.1-beta/go.mod h1:wVuoA8VJLEcwgqHBwHmzLRazpKxTv13Px/pDuV7OomQ= 61 | github.com/btcsuite/btclog v0.0.0-20170628155309-84c8d2346e9f/go.mod h1:TdznJufoqS23FtqVCzL0ZqgP5MqXbb4fg/WgDys70nA= 62 | github.com/btcsuite/btcutil v0.0.0-20190425235716-9e5f4b9a998d/go.mod h1:+5NJ2+qvTyV9exUAL/rxXi3DcLg2Ts+ymUAY5y4NvMg= 63 | github.com/btcsuite/go-socks v0.0.0-20170105172521-4720035b7bfd/go.mod h1:HHNXQzUsZCxOoE+CPiyCTO6x34Zs86zZUiwtpXoGdtg= 64 | github.com/btcsuite/goleveldb v0.0.0-20160330041536-7834afc9e8cd/go.mod h1:F+uVaaLLH7j4eDXPRvw78tMflu7Ie2bzYOH4Y8rRKBY= 65 | 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v1.1.0/go.mod h1:J7Y8YcW2NihsgmVo/mv3lAwl/skON4iLHjSsI+c5H38= 84 | github.com/davecgh/go-spew v1.1.1 h1:vj9j/u1bqnvCEfJOwUhtlOARqs3+rkHYY13jYWTU97c= 85 | github.com/davecgh/go-spew v1.1.1/go.mod h1:J7Y8YcW2NihsgmVo/mv3lAwl/skON4iLHjSsI+c5H38= 86 | github.com/deckarep/golang-set v0.0.0-20180603214616-504e848d77ea/go.mod h1:93vsz/8Wt4joVM7c2AVqh+YRMiUSc14yDtF28KmMOgQ= 87 | github.com/dgrijalva/jwt-go v3.2.0+incompatible/go.mod h1:E3ru+11k8xSBh+hMPgOLZmtrrCbhqsmaPHjLKYnJCaQ= 88 | github.com/dgryski/go-bitstream v0.0.0-20180413035011-3522498ce2c8/go.mod h1:VMaSuZ+SZcx/wljOQKvp5srsbCiKDEb6K2wC4+PiBmQ= 89 | github.com/dgryski/go-sip13 v0.0.0-20181026042036-e10d5fee7954/go.mod h1:vAd38F8PWV+bWy6jNmig1y/TA+kYO4g3RSRF0IAv0no= 90 | github.com/dlclark/regexp2 v1.2.0/go.mod h1:2pZnwuY/m+8K6iRw6wQdMtk+rH5tNGR1i55kozfMjCc= 91 | github.com/docker/docker v1.4.2-0.20180625184442-8e610b2b55bf/go.mod h1:eEKB0N0r5NX/I1kEveEz05bcu8tLC/8azJZsviup8Sk= 92 | github.com/dop251/goja 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golang.org/x/sys v0.0.0-20190412213103-97732733099d/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs= 384 | golang.org/x/sys v0.0.0-20190502145724-3ef323f4f1fd/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs= 385 | golang.org/x/sys v0.0.0-20190507160741-ecd444e8653b/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs= 386 | golang.org/x/sys v0.0.0-20190606165138-5da285871e9c/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs= 387 | golang.org/x/sys v0.0.0-20190624142023-c5567b49c5d0/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs= 388 | golang.org/x/sys v0.0.0-20190726091711-fc99dfbffb4e/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs= 389 | golang.org/x/sys v0.0.0-20190904154756-749cb33beabd/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs= 390 | golang.org/x/sys v0.0.0-20191005200804-aed5e4c7ecf9/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs= 391 | golang.org/x/sys v0.0.0-20191120155948-bd437916bb0e/go.mod 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golang.org/x/xerrors v0.0.0-20200804184101-5ec99f83aff1/go.mod h1:I/5z698sn9Ka8TeJc9MKroUUfqBBauWjQqLJ2OPfmY0= 445 | gonum.org/v1/gonum v0.0.0-20180816165407-929014505bf4/go.mod h1:Y+Yx5eoAFn32cQvJDxZx5Dpnq+c3wtXuadVZAcxbbBo= 446 | gonum.org/v1/gonum v0.0.0-20181121035319-3f7ecaa7e8ca/go.mod h1:Y+Yx5eoAFn32cQvJDxZx5Dpnq+c3wtXuadVZAcxbbBo= 447 | gonum.org/v1/gonum v0.6.0/go.mod h1:9mxDZsDKxgMAuccQkewq682L+0eCu4dCN2yonUJTCLU= 448 | gonum.org/v1/netlib v0.0.0-20181029234149-ec6d1f5cefe6/go.mod h1:wa6Ws7BG/ESfp6dHfk7C6KdzKA7wR7u/rKwOGE66zvw= 449 | gonum.org/v1/netlib v0.0.0-20190313105609-8cb42192e0e0/go.mod h1:wa6Ws7BG/ESfp6dHfk7C6KdzKA7wR7u/rKwOGE66zvw= 450 | gonum.org/v1/plot v0.0.0-20190515093506-e2840ee46a6b/go.mod h1:Wt8AAjI+ypCyYX3nZBvf6cAIx93T+c/OS2HFAYskSZc= 451 | google.golang.org/api v0.4.0/go.mod h1:8k5glujaEP+g9n7WNsDg8QP6cUVNI86fCNMcbazEtwE= 452 | google.golang.org/api v0.7.0/go.mod h1:WtwebWUNSVBH/HAw79HIFXZNqEvBhG+Ra+ax0hx3E3M= 453 | google.golang.org/api v0.8.0/go.mod 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h1:Co6ibVJAznAaIkqp8huTwlJQCZ016jof/cbN4VW5Yz0= 490 | gopkg.in/check.v1 v1.0.0-20180628173108-788fd7840127 h1:qIbj1fsPNlZgppZ+VLlY7N33q108Sa+fhmuc+sWQYwY= 491 | gopkg.in/check.v1 v1.0.0-20180628173108-788fd7840127/go.mod h1:Co6ibVJAznAaIkqp8huTwlJQCZ016jof/cbN4VW5Yz0= 492 | gopkg.in/errgo.v2 v2.1.0/go.mod h1:hNsd1EY+bozCKY1Ytp96fpM3vjJbqLJn88ws8XvfDNI= 493 | gopkg.in/fsnotify.v1 v1.4.7/go.mod h1:Tz8NjZHkW78fSQdbUxIjBTcgA1z1m8ZHf0WmKUhAMys= 494 | gopkg.in/natefinch/npipe.v2 v2.0.0-20160621034901-c1b8fa8bdcce/go.mod h1:5AcXVHNjg+BDxry382+8OKon8SEWiKktQR07RKPsv1c= 495 | gopkg.in/olebedev/go-duktape.v3 v3.0.0-20200619000410-60c24ae608a6/go.mod h1:uAJfkITjFhyEEuUfm7bsmCZRbW5WRq8s9EY8HZ6hCns= 496 | gopkg.in/tomb.v1 v1.0.0-20141024135613-dd632973f1e7/go.mod h1:dt/ZhP58zS4L8KSrWDmTeBkI65Dw0HsyUHuEVlX15mw= 497 | gopkg.in/urfave/cli.v1 v1.20.0/go.mod h1:vuBzUtMdQeixQj8LVd+/98pzhxNGQoyuPBlsXHOQNO0= 498 | gopkg.in/yaml.v2 v2.2.1/go.mod h1:hI93XBmqTisBFMUTm0b8Fm+jr3Dg1NNxqwp+5A1VGuI= 499 | gopkg.in/yaml.v2 v2.2.2/go.mod h1:hI93XBmqTisBFMUTm0b8Fm+jr3Dg1NNxqwp+5A1VGuI= 500 | gopkg.in/yaml.v2 v2.2.3/go.mod h1:hI93XBmqTisBFMUTm0b8Fm+jr3Dg1NNxqwp+5A1VGuI= 501 | gopkg.in/yaml.v2 v2.2.4/go.mod h1:hI93XBmqTisBFMUTm0b8Fm+jr3Dg1NNxqwp+5A1VGuI= 502 | gopkg.in/yaml.v2 v2.2.8/go.mod h1:hI93XBmqTisBFMUTm0b8Fm+jr3Dg1NNxqwp+5A1VGuI= 503 | gopkg.in/yaml.v2 v2.3.0/go.mod h1:hI93XBmqTisBFMUTm0b8Fm+jr3Dg1NNxqwp+5A1VGuI= 504 | gopkg.in/yaml.v3 v3.0.0-20200313102051-9f266ea9e77c h1:dUUwHk2QECo/6vqA44rthZ8ie2QXMNeKRTHCNY2nXvo= 505 | gopkg.in/yaml.v3 v3.0.0-20200313102051-9f266ea9e77c/go.mod h1:K4uyk7z7BCEPqu6E+C64Yfv1cQ7kz7rIZviUmN+EgEM= 506 | gotest.tools v2.2.0+incompatible/go.mod h1:DsYFclhRJ6vuDpmuTbkuFWG+y2sxOXAzmJt81HFBacw= 507 | honnef.co/go/tools v0.0.0-20190102054323-c2f93a96b099/go.mod h1:rf3lG4BRIbNafJWhAfAdb/ePZxsR/4RtNHQocxwk9r4= 508 | honnef.co/go/tools v0.0.0-20190106161140-3f1c8253044a/go.mod h1:rf3lG4BRIbNafJWhAfAdb/ePZxsR/4RtNHQocxwk9r4= 509 | honnef.co/go/tools v0.0.0-20190418001031-e561f6794a2a/go.mod h1:rf3lG4BRIbNafJWhAfAdb/ePZxsR/4RtNHQocxwk9r4= 510 | honnef.co/go/tools v0.0.0-20190523083050-ea95bdfd59fc/go.mod h1:rf3lG4BRIbNafJWhAfAdb/ePZxsR/4RtNHQocxwk9r4= 511 | honnef.co/go/tools v0.0.1-2019.2.3/go.mod h1:a3bituU0lyd329TUQxRnasdCoJDkEUEAqEt0JzvZhAg= 512 | honnef.co/go/tools v0.1.3/go.mod h1:NgwopIslSNH47DimFoV78dnkksY2EFtX0ajyb3K/las= 513 | rsc.io/binaryregexp v0.2.0/go.mod h1:qTv7/COck+e2FymRvadv62gMdZztPaShugOCi3I+8D8= 514 | rsc.io/pdf v0.1.1/go.mod h1:n8OzWcQ6Sp37PL01nO98y4iUCRdTGarVfzxY20ICaU4= 515 | -------------------------------------------------------------------------------- /kzg.go: -------------------------------------------------------------------------------- 1 | package kzg 2 | 3 | import ( 4 | "fmt" 5 | "math/big" 6 | 7 | bn256 "github.com/ethereum/go-ethereum/crypto/bn256/cloudflare" 8 | ) 9 | 10 | // TrustedSetup also named Reference String 11 | type TrustedSetup struct { 12 | Tau1 []*bn256.G1 13 | Tau2 []*bn256.G2 14 | } 15 | 16 | // NewTrustedSetup returns a new trusted setup. This step should be done in a 17 | // secure & distributed way 18 | func NewTrustedSetup(l int) (*TrustedSetup, error) { 19 | // compute random s 20 | s, err := randBigInt() 21 | if err != nil { 22 | return nil, err 23 | } 24 | 25 | // Notation: [x]₁=xG ∈ 𝔾₁, [x]₂=xH ∈ 𝔾₂ 26 | // τ₁: [x₀]₁, [x₁]₁, [x₂]₁, ..., [x n₋₁]₁ 27 | // τ₂: [x₀]₂, [x₁]₂, [x₂]₂, ..., [x n₋₁]₂ 28 | 29 | // sPow := make([]*big.Int, l) 30 | tauG1 := make([]*bn256.G1, l) 31 | tauG2 := make([]*bn256.G2, l) 32 | for i := 0; i < l; i++ { 33 | sPow := fExp(s, big.NewInt(int64(i))) 34 | tauG1[i] = new(bn256.G1).ScalarBaseMult(sPow) 35 | tauG2[i] = new(bn256.G2).ScalarBaseMult(sPow) 36 | } 37 | 38 | return &TrustedSetup{tauG1, tauG2}, nil 39 | } 40 | 41 | // Commit generates the commitment to the polynomial p(x) 42 | func Commit(ts *TrustedSetup, p []*big.Int) *bn256.G1 { 43 | c := evaluateG1(ts, p) 44 | return c 45 | } 46 | 47 | func evaluateG1(ts *TrustedSetup, p []*big.Int) *bn256.G1 { 48 | c := new(bn256.G1).ScalarMult(ts.Tau1[0], p[0]) 49 | for i := 1; i < len(p); i++ { 50 | sp := new(bn256.G1).ScalarMult(ts.Tau1[i], p[i]) 51 | c = new(bn256.G1).Add(c, sp) 52 | } 53 | return c 54 | } 55 | 56 | //nolint:deadcode,unused 57 | func evaluateG2(ts *TrustedSetup, p []*big.Int) *bn256.G2 { 58 | c := new(bn256.G2).ScalarMult(ts.Tau2[0], p[0]) 59 | for i := 1; i < len(p); i++ { 60 | sp := new(bn256.G2).ScalarMult(ts.Tau2[i], p[i]) 61 | c = new(bn256.G2).Add(c, sp) 62 | } 63 | return c 64 | } 65 | 66 | // EvaluationProof generates the evaluation proof 67 | func EvaluationProof(ts *TrustedSetup, p []*big.Int, z, y *big.Int) (*bn256.G1, error) { 68 | n := polynomialSub(p, []*big.Int{y}) // p-y 69 | // n := p // we can omit y (p(z)) 70 | d := []*big.Int{fNeg(z), big.NewInt(1)} // x-z 71 | q, rem := polynomialDiv(n, d) 72 | if compareBigIntArray(rem, arrayOfZeroes(len(rem))) { 73 | return nil, 74 | fmt.Errorf("remainder should be 0, instead is %d", rem) 75 | } 76 | 77 | // proof: e = [q(t)]₁ 78 | e := evaluateG1(ts, q) 79 | return e, nil 80 | } 81 | 82 | // Verify computes the KZG commitment verification 83 | func Verify(ts *TrustedSetup, c, proof *bn256.G1, z, y *big.Int) bool { 84 | s2 := ts.Tau2[1] // [t]₂ = sG ∈ 𝔾₂ = Tau2[1] 85 | zG2Neg := new(bn256.G2).Neg( 86 | new(bn256.G2).ScalarBaseMult(z)) // [z]₂ = zG ∈ 𝔾₂ 87 | // [t]₂ - [z]₂ 88 | sz := new(bn256.G2).Add(s2, zG2Neg) 89 | 90 | yG1Neg := new(bn256.G1).Neg( 91 | new(bn256.G1).ScalarBaseMult(y)) // [y]₁ = yG ∈ 𝔾₁ 92 | // c - [y]₁ 93 | cy := new(bn256.G1).Add(c, yG1Neg) 94 | 95 | h := new(bn256.G2).ScalarBaseMult(big.NewInt(1)) // H ∈ 𝔾₂ 96 | 97 | // e(proof, [t]₂ - [z]₂) == e(c - [y]₁, H) 98 | e1 := bn256.Pair(proof, sz) 99 | e2 := bn256.Pair(cy, h) 100 | return e1.String() == e2.String() 101 | } 102 | 103 | // 104 | // Batch proofs 105 | // 106 | 107 | // EvaluationBatchProof generates the evalutation proof for the given list of points 108 | func EvaluationBatchProof(ts *TrustedSetup, p []*big.Int, zs, ys []*big.Int) (*bn256.G1, error) { 109 | if len(zs) != len(ys) { 110 | return nil, fmt.Errorf("len(zs)!=len(ys), %d!=%d", len(zs), len(ys)) 111 | } 112 | if len(p) <= len(zs)+1 { 113 | return nil, fmt.Errorf("polynomial p(x) can not be of degree"+ 114 | " equal or smaller than the number of given points+1."+ 115 | " Polynomial p(x) degree: %d, number of points: %d", 116 | len(p), len(zs)) 117 | } 118 | 119 | // z(x) = (x-z0)(x-z1)...(x-zn) 120 | z := zeroPolynomial(zs) 121 | 122 | // I(x) = Lagrange interpolation through (z0, y0), (z1, y1), ... 123 | i, err := LagrangeInterpolation(zs, ys) 124 | if err != nil { 125 | return nil, err 126 | } 127 | 128 | // q(x) = ( p(x) - I(x) ) / z(x) 129 | pMinusI := polynomialSub(p, i) 130 | q, rem := polynomialDiv(pMinusI, z) 131 | if compareBigIntArray(rem, arrayOfZeroes(len(rem))) { 132 | return nil, 133 | fmt.Errorf("remainder should be 0, instead is %d", rem) 134 | } 135 | 136 | // proof: e = [q(t)]₁ 137 | e := evaluateG1(ts, q) 138 | return e, nil 139 | } 140 | 141 | // VerifyBatchProof computes the KZG batch proof commitment verification 142 | func VerifyBatchProof(ts *TrustedSetup, c, proof *bn256.G1, zs, ys []*big.Int) bool { 143 | // [z(s)]₂ 144 | z := zeroPolynomial(zs) 145 | zG2 := evaluateG2(ts, z) // [z(t)]₂ = z(t) G ∈ 𝔾₂ 146 | 147 | // I(x) = Lagrange interpolation through (z0, y0), (z1, y1), ... 148 | i, err := LagrangeInterpolation(zs, ys) 149 | if err != nil { 150 | return false 151 | } 152 | // [i(t)]₁ 153 | iG1 := evaluateG1(ts, i) // [i(t)]₁ = i(t) G ∈ 𝔾₁ 154 | 155 | // c - [i(t)]₁ 156 | iG1Neg := new(bn256.G1).Neg(iG1) 157 | ciG1 := new(bn256.G1).Add(c, iG1Neg) 158 | 159 | h := new(bn256.G2).ScalarBaseMult(big.NewInt(1)) // H ∈ 𝔾₂ 160 | 161 | // e(proof, [z(t)]₂) == e(c - [I(t)]₁, H) 162 | e1 := bn256.Pair(proof, zG2) 163 | e2 := bn256.Pair(ciG1, h) 164 | return e1.String() == e2.String() 165 | } 166 | -------------------------------------------------------------------------------- /kzg_test.go: -------------------------------------------------------------------------------- 1 | package kzg 2 | 3 | import ( 4 | "math/big" 5 | "testing" 6 | 7 | "github.com/stretchr/testify/assert" 8 | ) 9 | 10 | func TestSimpleFlow(t *testing.T) { 11 | // p(x) = x^3 + x + 5 12 | p := []*big.Int{ 13 | big.NewInt(5), 14 | big.NewInt(1), // x^1 15 | big.NewInt(0), // x^2 16 | big.NewInt(1), // x^3 17 | } 18 | assert.Equal(t, "x³ + x¹ + 5", PolynomialToString(p)) 19 | 20 | // TrustedSetup 21 | ts, err := NewTrustedSetup(len(p)) 22 | assert.Nil(t, err) 23 | 24 | // Commit 25 | c := Commit(ts, p) 26 | 27 | // p(z)=y --> p(3)=35 28 | z := big.NewInt(3) 29 | y := big.NewInt(35) 30 | 31 | // z & y: to prove an evaluation p(z)=y 32 | proof, err := EvaluationProof(ts, p, z, y) 33 | assert.Nil(t, err) 34 | 35 | v := Verify(ts, c, proof, z, y) 36 | assert.True(t, v) 37 | 38 | v = Verify(ts, c, proof, big.NewInt(4), y) 39 | assert.False(t, v) 40 | } 41 | 42 | func TestBatchProof(t *testing.T) { 43 | // p(x) = x^3 + x + 5 44 | p := []*big.Int{ 45 | big.NewInt(5), 46 | big.NewInt(1), // x^1 47 | big.NewInt(0), // x^2 48 | big.NewInt(1), // x^3 49 | big.NewInt(10), // x^4 50 | } 51 | assert.Equal(t, "10x⁴ + x³ + x¹ + 5", PolynomialToString(p)) 52 | 53 | // TrustedSetup 54 | ts, err := NewTrustedSetup(len(p)) 55 | assert.Nil(t, err) 56 | 57 | // Commit 58 | c := Commit(ts, p) 59 | 60 | // 1st point: p(z)=y --> p(3)=35 61 | z0 := big.NewInt(3) 62 | y0 := polynomialEval(p, z0) 63 | 64 | // 2nd point: p(10)=1015 65 | z1 := big.NewInt(10) 66 | y1 := polynomialEval(p, z1) 67 | 68 | // 3nd point: p(256)=16777477 69 | z2 := big.NewInt(256) 70 | y2 := polynomialEval(p, z2) 71 | 72 | zs := []*big.Int{z0, z1, z2} 73 | ys := []*big.Int{y0, y1, y2} 74 | 75 | // prove an evaluation of the multiple z_i & y_i 76 | proof, err := EvaluationBatchProof(ts, p, zs, ys) 77 | assert.Nil(t, err) 78 | 79 | // batch proof verification 80 | v := VerifyBatchProof(ts, c, proof, zs, ys) 81 | assert.True(t, v) 82 | 83 | // changing order of the points to be verified 84 | zs[0], zs[1], zs[2] = zs[1], zs[2], zs[0] 85 | ys[0], ys[1], ys[2] = ys[1], ys[2], ys[0] 86 | v = VerifyBatchProof(ts, c, proof, zs, ys) 87 | assert.True(t, v) 88 | 89 | // change a value of zs and check that verification fails 90 | zs[0] = big.NewInt(2) 91 | v = VerifyBatchProof(ts, c, proof, zs, ys) 92 | assert.False(t, v) 93 | 94 | // using a value that is not in the evaluation proof should generate a 95 | // proof that will not correctly be verified 96 | zs = []*big.Int{z0, z1, z2} 97 | ys = []*big.Int{y0, y1, y2} 98 | proof, err = EvaluationBatchProof(ts, p, zs, ys) 99 | assert.Nil(t, err) 100 | zs[2] = big.NewInt(2500) 101 | ys[2] = polynomialEval(p, zs[2]) 102 | v = VerifyBatchProof(ts, c, proof, zs, ys) 103 | assert.False(t, v) 104 | } 105 | --------------------------------------------------------------------------------