From 072ba4491780b14019cfbf2d1103bbacfd2f8798 Mon Sep 17 00:00:00 2001 From: Kevin Charm Date: Sun, 11 Aug 2024 16:52:14 +0200 Subject: [PATCH] chicken soup --- .github/workflows/ci.yml | 22 ++ .gitignore | 2 + .prettierignore | 8 + .prettierrc | 45 +++ LICENSE | 21 ++ README.md | 11 + broth16.jpg | Bin 0 -> 389153 bytes package.json | 28 ++ src/groth16.ts | 232 +++++++++++++ src/poly.ts | 197 +++++++++++ src/qap.ts | 107 ++++++ test/groth16.spec.ts | 86 +++++ test/poly.spec.ts | 63 ++++ tsconfig.json | 12 + yarn.lock | 693 +++++++++++++++++++++++++++++++++++++++ 15 files changed, 1527 insertions(+) create mode 100644 .github/workflows/ci.yml create mode 100644 .gitignore create mode 100644 .prettierignore create mode 100644 .prettierrc create mode 100644 LICENSE create mode 100644 README.md create mode 100644 broth16.jpg create mode 100644 package.json create mode 100644 src/groth16.ts create mode 100644 src/poly.ts create mode 100644 src/qap.ts create mode 100644 test/groth16.spec.ts create mode 100644 test/poly.spec.ts create mode 100644 tsconfig.json create mode 100644 yarn.lock diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml new file mode 100644 index 0000000..1d80513 --- /dev/null +++ b/.github/workflows/ci.yml @@ -0,0 +1,22 @@ +name: CI +on: [push] + +jobs: + test: + name: Test + runs-on: ubuntu-latest + steps: + - name: checkout repo + uses: actions/checkout@v3 + + - name: install node + uses: actions/setup-node@v3 + with: + cache: yarn + node-version: 18 + + - name: install node deps + run: yarn install --frozen-lockfile + + - name: test + run: yarn test diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..f06235c --- /dev/null +++ b/.gitignore @@ -0,0 +1,2 @@ +node_modules +dist diff --git a/.prettierignore b/.prettierignore new file mode 100644 index 0000000..b9f3fe1 --- /dev/null +++ b/.prettierignore @@ -0,0 +1,8 @@ +node_modules +artifacts +cache +coverage* +gasReporterOutput.json +typechain +typechain-types +LICENSE diff --git a/.prettierrc b/.prettierrc new file mode 100644 index 0000000..996cd8a --- /dev/null +++ b/.prettierrc @@ -0,0 +1,45 @@ +{ + "overrides": [ + { + "files": ["*.ts"], + "options": { + "parser": "typescript", + "tabWidth": 4, + "semi": false, + "singleQuote": true, + "printWidth": 100 + } + }, + { + "files": ["*.json"], + "options": { + "parser": "json", + "tabWidth": 2 + } + }, + { + "files": ["*.html"], + "options": { + "parser": "html", + "tabWidth": 4 + } + }, + { + "files": ["*.css"], + "options": { + "parser": "css", + "tabWidth": 4 + } + }, + { + "files": ["*.md"], + "options": { + "parser": "markdown", + "tabWidth": 4, + "semi": false, + "singleQuote": true, + "printWidth": 100 + } + } + ] + } diff --git a/LICENSE b/LICENSE new file mode 100644 index 0000000..53a1247 --- /dev/null +++ b/LICENSE @@ -0,0 +1,21 @@ +MIT License + +Copyright (c) 2024 Kevin Charm + +Permission is hereby granted, free of charge, to any person obtaining a copy +of this software and associated documentation files (the "Software"), to deal +in the Software without restriction, including without limitation the rights +to use, copy, modify, merge, publish, distribute, sublicense, and/or sell +copies of the Software, and to permit persons to whom the Software is +furnished to do so, subject to the following conditions: + +The above copyright notice and this permission notice shall be included in all +copies or substantial portions of the Software. + +THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE +SOFTWARE. diff --git a/README.md b/README.md new file mode 100644 index 0000000..48c834b --- /dev/null +++ b/README.md @@ -0,0 +1,11 @@ +# broth16 + +![](./broth16.jpg) + +This is an exercise to learn how Groth16 works. + +## Useful reference material + +- J. Groth, [On the Size of Pairing-based Non-interactive Arguments](https://eprint.iacr.org/2016/260.pdf) +- V. Buterin, [Quadratic Arithmetic Programs: from Zero to Hero](https://medium.com/@VitalikButerin/quadratic-arithmetic-programs-from-zero-to-hero-f6d558cea649) +- N. 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za@4eaejmN9qzDm10E4ZFkoI@b%ICXCC33J1s2@&739(sl*LPc3bub!-9eBK2F0goXdB@Ep!-0?YXWEOW_k~{g63NSp04D)L$zY)-JJy<>nW1q!dd{wG8M-(ODEE Liq86S{=18" + }, + "packageManager": "yarn@1.22.22", + "scripts": { + "prepare": "yarn tsc", + "test": "NODE_OPTIONS=--experimental-global-webcrypto ./node_modules/.bin/mocha -r ts-node/register test/**/*.spec.ts" + }, + "author": "Kevin Charm ", + "license": "MIT", + "devDependencies": { + "@types/chai": "^4.3.16", + "@types/mocha": "^10.0.7", + "@types/node": "^18", + "chai": "^4", + "mocha": "^10.6.0", + "prettier": "^3.3.2", + "ts-node": "^10.9.2", + "typescript": "^5.5.3" + }, + "dependencies": { + "@kevincharm/blstoise": "^1.0.1" + } +} diff --git a/src/groth16.ts b/src/groth16.ts new file mode 100644 index 0000000..67563b8 --- /dev/null +++ b/src/groth16.ts @@ -0,0 +1,232 @@ +import { Fr, PointG1, PointG2, validatePairing } from '@kevincharm/blstoise' +import { randomFr } from '@kevincharm/blstoise/dist/src/utils' +import { addPolys, evalPoly } from './poly' +import { QAP, computeSolutionPoly, computeDivisorPoly, tPoly } from './qap' + +/** + * Combined phase 1 & 2 trusted setup outputs for Groth16 + */ +export interface TrustedSetup { + power: number + tauG1: PointG1[] + tauG2: PointG2[] + alphaG1: PointG1 + betaG1: PointG1 + betaG2: PointG2 + gammaG2: PointG2 + deltaG1: PointG1 + deltaG2: PointG2 + cTauPublicG1: PointG1[] + cTauPrivateG1: PointG1[] + tTauInvDeltaG1: PointG1[] +} + +/** + * Perform a circuit-specific trusted setup for Groth16 proofs. + * @param qap Polynomials representing the quadratic arithmetic programme + */ +export function setup({ l, u, v, w }: QAP): TrustedSetup { + const degree = u[0].length + const tau = randomFr() // toxic waste + const alpha = randomFr() // toxic waste + const beta = randomFr() // toxic waste + const gamma = randomFr() // toxic waste + const delta = randomFr() // toxic waste + + const G1 = PointG1.generator() + const G2 = PointG2.generator() + const tauG1 = Array.from({ length: degree }, (_, i) => G1.mul(tau.exp(BigInt(i)).value)) + const tauG2 = Array.from({ length: degree }, (_, i) => G2.mul(tau.exp(BigInt(i)).value)) + const alphaG1 = G1.mul(alpha.value) + const betaG1 = G1.mul(beta.value) + const betaG2 = G2.mul(beta.value) + + const gammaG2 = G2.mul(gamma.value) + const deltaG1 = G1.mul(delta.value) + const deltaG2 = G2.mul(delta.value) + + // foreach row (polynomial) in QAP.u \i -> beta*u_i(x) + const betaU = u.map((uPoly) => uPoly.map((x) => x.mul(beta))) + // foreach row (polynomial) in QAP.v \i -> alpha*v_i(x) + const alphaV = v.map((vPoly) => vPoly.map((x) => x.mul(alpha))) + // From Groth16, 3.2 NIZK arguments for quadratic arithmetic programs + // This is the combined sum term from C and AB (for both public and private inputs) + // evaluated at tau: + // C = \sum_{i=0}^{m} \beta u_i(\tau) + \alpha v_i(\tau) + w_i(\tau) + const cG1: PointG1[] = [] + for (let i = 0; i < w.length; i++) { + const c = G1.mul(evalPoly(addPolys(addPolys(betaU[i], alphaV[i]), w[i]), tau).value) + cG1.push(c) + } + + // Public inputs - scale sum term by 1/gamma + // (1/gamma) * [C]_1 + const invGamma = gamma.inv() + const cTauPublicG1 = cG1.slice(0, l + 1).map((p) => p.mul(invGamma.value)) + + // Private inputs - scale sum term by 1/delta + // (1/delta) * [C]_1 + const invDelta = delta.inv() + const cTauPrivateG1 = cG1.slice(l + 1).map((p) => p.mul(invDelta.value)) + + // We need this later to compute h(tau)t(tau) in the proving step. + // See: https://github.com/nikkolasg/playsnark/blob/c406bfdeefbc60345696bfad9105568e91f04c13/groth16.go#L180 + // [tau^i * t(tau) / delta]_1 + const tx = tPoly(degree) + const tTau = evalPoly(tx, tau) + const tTauInvDeltaG1 = tauG1.map((ptau) => ptau.mul(tTau.value).mul(invDelta.value)) + + return { + power: degree, + tauG1, + tauG2, + alphaG1, + betaG1, + betaG2, + gammaG2, + deltaG1, + deltaG2, + cTauPublicG1, + cTauPrivateG1, + tTauInvDeltaG1, + } +} + +/** + * Verification key for Groth16 proofs; + * which is a subset of the circuit-specific trusted setup + */ +export interface VerificationKey { + alphaG1: PointG1 + betaG2: PointG2 + gammaG2: PointG2 + deltaG2: PointG2 + cTauPublicG1: PointG1[] +} + +/** Jens Groth's succinct noninteractive argument of knowledge */ +export type Proof = [PointG1, PointG1, PointG2] + +/** + * Compute a succinct Groth16 proof for a given programme and a witness vector. + * @param setup Circuit-specific trusted setup + * @param qap Polynomials representing the quadratic arithmetic programme + * @param witness Witness vector (i.e. a solution to the programme) + * @param shouldCheckProof Whether to check that the proof verifies before returning + * @returns A succinct proof that the witness satisfies the QAP + */ +export function prove( + setup: TrustedSetup, + qap: QAP, + witness: Fr[], + shouldCheckProof?: boolean, +): { proof: Proof; vk: VerificationKey } { + const r = randomFr() // toxic waste + const s = randomFr() // toxic waste + + // We fix the first witness to 1 (used for constants) + const preparedWitness = [Fr.one(), ...witness] + // Compute polynomials a_i u_i(x) + const { au, av, ht } = computeSolutionPoly(preparedWitness, qap) + + // A = [\alpha + \sum_{i=0}^{m} a_i u_i(\tau) + r*delta]_1 + const sumA = au.map(({ value }, i) => setup.tauG1[i].mul(value)).reduce((acc, p) => acc.add(p)) + const A = setup.alphaG1.add(sumA).add(setup.deltaG1.mul(r.value)) + // B = [\beta + \sum_{i=0}^{m} a_i v_i(\tau) + s*delta]_1 + const sumBG1 = av + .map(({ value }, i) => setup.tauG1[i].mul(value)) + .reduce((acc, p) => acc.add(p)) + const BG1 = setup.betaG1.add(sumBG1).add(setup.deltaG1.mul(s.value)) + // B = [\beta + \sum_{i=0}^{m} a_i v_i(\tau) + s*delta]_2 + const sumBG2 = av + .map(({ value }, i) => setup.tauG2[i].mul(value)) + .reduce((acc, p) => acc.add(p)) + const BG2 = setup.betaG2.add(sumBG2).add(setup.deltaG2.mul(s.value)) + + // We need the +h(\tau)*t(\tau) part of C + const h = computeDivisorPoly(ht, tPoly(qap.u[0].length)) + const htTauG1 = h + .map((x, i) => setup.tTauInvDeltaG1[i].mul(x.value)) + .reduce((acc, p) => acc.add(p)) + // Private inputs (+ intermediate constraints) part of C + // \sum_{i=l+1}^{m} a_i(\beta u_i(\tau) + \alpha v_i(\tau) + w_i(\tau) + const sumC = preparedWitness + .slice(qap.l + 1) + .map(({ value }, i) => setup.cTauPrivateG1[i].mul(value)) + .reduce((acc, p) => acc.add(p)) + // -(r * s * \delta) + const negRsDelta = setup.deltaG1.mul(r.mul(s).value).neg() + // Full C term evaluated in G1 + // C = \frac{1}{\delta} (\sum_{i=l+1}^{m} a_i(\beta u_i(\tau) + \alpha v_i(\tau) + w_i(\tau)) + h(\tau)t(\tau)) + // + As + rB - rs\delta + const C = sumC.add(htTauG1).add(A.mul(s.value)).add(BG1.mul(r.value)).add(negRsDelta) + // Proof triple, as described by Groth, 3.2 NIZK arguments for quadratic arithmetic programs + const proof: [PointG1, PointG1, PointG2] = [A, C, BG2] + + // This is the public inputs part + // \frac{1}{\gamma} (\sum_{i=0}^{l} a_i(\beta u_i(\tau) + \alpha v_i(\tau) + w_i(\tau))) + const publicInputsSumG1 = setup.cTauPublicG1 + .map((p, i) => p.mul(preparedWitness[i].value)) + .reduce((acc, p) => acc.add(p)) + + if (shouldCheckProof) { + // Check that proof verifies + const ps = [A.neg(), setup.alphaG1, publicInputsSumG1, C] + const qs = [BG2, setup.betaG2, setup.gammaG2, setup.deltaG2] + if (!validatePairing(ps, qs)) { + throw new Error('Pairing check failed') + } + } + + return { + proof, + vk: { + alphaG1: setup.alphaG1, + betaG2: setup.betaG2, + gammaG2: setup.gammaG2, + deltaG2: setup.deltaG2, + cTauPublicG1: setup.cTauPublicG1, + }, + } +} + +/** + * Verify a Groth16 proof for a given public input vector. + * + * @param proof Groth16 proof + * @param publicInputs Public input vector + * @param vk Circuit-specific trusted setup + * @returns True if the proof is valid + */ +export function verify(proof: Proof, publicInputs: Fr[], vk: VerificationKey): boolean { + const [A, C, BG2] = proof + const [alphaG1, betaG2, gammaG2, deltaG2] = [vk.alphaG1, vk.betaG2, vk.gammaG2, vk.deltaG2] + + // We fix the first public input to 1 (used for constants) + const preparedPublicInputs = [Fr.one(), ...publicInputs] + if (preparedPublicInputs.length !== vk.cTauPublicG1.length) { + throw new Error( + `Invalid number of public inputs: expected ${vk.cTauPublicG1.length - 1}, got ${publicInputs.length}`, + ) + } + // \sum_{i=0}^{l} a_i * [\frac{\beta u_i(\tau) + \alpha v_i(\tau) + w_i(\tau)}{\gamma}]_1 + const publicInputsSumG1 = vk.cTauPublicG1 + .map((p, i) => p.mul(preparedPublicInputs[i].value)) + .reduce((acc, p) => acc.add(p)) + + // Now from the Groth16 paper, 3.2 NIZK arguments for quadratic arithmetic programs. + // + // Parse proof = ([A]_1, [C]_1, [B]_2) + // + // Accept the proof iff + // [A]_1 * [B]_2 + // = [\alpha]_1 * [\beta]_2 + // + (\sum_{i=0}^{l} a_i * [\frac{\beta u_i(\tau) + \alpha v_i(\tau) + w_i(\tau)}{\gamma}]_1) * [\gamma]_2 + // + [C]_1 * [\delta]_2 + // Turn this into a pairing check: + // Given asymmetric pairing e(P,Q) :: G1 x G2 -> GT + // e([-A]_1, [B]_2) * e([\alpha]_1, [\beta]_2) * e([publicInputsSum]_1, [\gamma]_2) * e([C]_1, [\delta]_2) = 1 + const ps = [A.neg(), alphaG1, publicInputsSumG1, C] + const qs = [BG2, betaG2, gammaG2, deltaG2] + return validatePairing(ps, qs) +} diff --git a/src/poly.ts b/src/poly.ts new file mode 100644 index 0000000..08d10e1 --- /dev/null +++ b/src/poly.ts @@ -0,0 +1,197 @@ +import { Fr } from '@kevincharm/blstoise' + +// Most of these functions come from Vitalik's QAP implementation: +// https://github.com/ethereum/research/blob/master/zksnark/qap_creator.py + +/** + * Helper to convert a list of coefficients to a list of polynomial + * curve scalars coefficients. + * + * @param coeffs Coefficients + * @returns Polynomial in F_r + */ +export function toPoly(coeffs: (number | bigint)[]): Fr[] { + return coeffs.map((c) => new Fr(BigInt(c))) +} + +/** + * Create a zero poly + * + * @param degree Degree of polynomial + * @returns Polynomial with all coefficients zero + */ +export function zeroPoly(degree: number): Fr[] { + return Array.from({ length: degree }, () => Fr.zero()) +} + +/** + * Add two polynomials + * + * @param lhs Addend + * @param rhs Addend + * @returns Sum + */ +export function addPolys(lhs: Fr[], rhs: Fr[]): Fr[] { + const length = Math.max(lhs.length, rhs.length) + const result: Fr[] = Array.from({ length }, () => Fr.zero()) + for (let i = 0; i < lhs.length; i++) { + result[i] = result[i].add(lhs[i]) + } + for (let i = 0; i < rhs.length; i++) { + result[i] = result[i].add(rhs[i]) + } + return result +} + +/** + * Subtract two polynomials + * + * @param lhs Minuend + * @param rhs Subtrahend + * @returns Difference + */ +export function subPolys(lhs: Fr[], rhs: Fr[]): Fr[] { + const length = Math.max(lhs.length, rhs.length) + const result: Fr[] = Array.from({ length }, (_, i) => (i < lhs.length ? lhs[i] : Fr.zero())) + for (let i = 0; i < rhs.length; i++) { + result[i] = result[i].sub(rhs[i]) + } + return result +} + +/** + * Polynomial multiplication + * + * @param lhs Multiplicand + * @param rhs Multiplier + * @returns Multiplication + */ +export function mulPolys(lhs: Fr[], rhs: Fr[]): Fr[] { + const length = lhs.length + rhs.length - 1 + const result: Fr[] = Array.from({ length }, () => Fr.zero()) + for (let i = 0; i < lhs.length; i++) { + for (let j = 0; j < rhs.length; j++) { + result[i + j] = result[i + j].add(lhs[i].mul(rhs[j])) + } + } + return result +} + +/** + * Polynomial division + * + * @param lhs Dividend + * @param rhs Divisor + * @returns Quotient and remainder + */ +export function divPolys(lhs: Fr[], rhs: Fr[]): [quotient: Fr[], remainder: Fr[]] { + const length = lhs.length - rhs.length + 1 + const result = Array.from({ length }, () => Fr.zero()) + let remainder = lhs // ref + while (remainder.length >= rhs.length) { + const leadingFactor = remainder.at(-1)!.mul(rhs.at(-1)!.inv()) + const pos = remainder.length - rhs.length + result[pos] = leadingFactor + const spent = [...zeroPoly(pos), leadingFactor] + remainder = subPolys(remainder, mulPolys(rhs, spent)).slice(0, -1) + } + return [result, remainder] +} + +/** + * Evaluate a polynomial p(x) + * + * @param p Polynomial + * @param x Position + * @returns p(x) + */ +export function evalPoly(p: Fr[], x: Fr): Fr { + return p.reduce((acc, coeff, i) => acc.add(coeff.mul(x.exp(BigInt(i)))), Fr.zero()) +} + +/** + * Compute a polynomial p(x) s.t. p(x) = 0 foreach x in xs + * Borrowed from https://github.com/GuildOfWeavers/galois/blob/master/lib/PrimeField.ts#L992 + * + * @param xs + * @returns + */ +function zpoly(xs: Fr[]): Fr[] { + const result: Fr[] = Array.from({ length: xs.length + 1 }, () => Fr.zero()) + result[result.length - 1] = Fr.one() + + for (let i = 0, p = result.length - 2; i < xs.length; i++, p--) { + result[p] = Fr.zero() + for (let j = p; j < result.length - 1; j++) { + result[j] = result[j].sub(result[j + 1].mul(xs[i])) + } + } + return result +} + +/** + * Lagrange interpolation + * Borrowed from https://github.com/GuildOfWeavers/galois/blob/f3d9cfbf2fe7857f3840bdba3406e2ba9ea548c7/lib/PrimeField.ts#L808 + * + * @param xs + * @param ys + * @returns Interpolated polynomial fitting f(xs) = ys + */ +export function interpolate(xs: Fr[], ys: Fr[]): Fr[] { + if (xs.length !== ys.length) { + throw new Error('length mismatch') + } + + const root = zpoly(xs) + const divisor = [Fr.zero(), Fr.one()] + const numerators: Fr[][] = Array.from({ length: xs.length }, () => []) + for (let i = 0; i < xs.length; i++) { + divisor[0] = xs[i].neg() + ;[numerators[i]] = divPolys(root, divisor) + } + + const denominators: Fr[] = Array.from({ length: xs.length }, () => Fr.zero()) + for (let i = 0; i < xs.length; i++) { + denominators[i] = evalPoly(numerators[i], xs[i]) + } + const invDenValues = denominators.map((d) => d.inv()) + + const r = Array.from({ length: xs.length }, () => Fr.zero()) + for (let i = 0; i < xs.length; i++) { + const ySlice = ys[i].mul(invDenValues[i]) + for (let j = 0; j < xs.length; j++) { + if (!numerators[i][j].equals(Fr.zero()) && !ys[i].equals(Fr.zero())) { + r[j] = r[j].add(numerators[i][j].mul(ySlice)) + } + } + } + return r +} + +/** + * Transpose a matrix + * Matrix must be correctly formed! + * + * e.g. + * [[1, 2, 3], + * [4, 5, 6]] + * => + * [[1, 4], + * [2, 5], + * [3, 6]] + * + * @param matrix + * @returns Transposed matrix + */ +export function transpose(matrix: T[][]) { + // Assumes an m x n matrix is correctly formed + const m = matrix.length // rows + const n = matrix[0].length // cols + const out: T[][] = Array.from({ length: n }, () => Array.from({ length: m }, () => null!)) + for (let i = 0; i < n; i++) { + for (let j = 0; j < m; j++) { + out[i][j] = matrix[j][i] + } + } + return out +} diff --git a/src/qap.ts b/src/qap.ts new file mode 100644 index 0000000..1cef903 --- /dev/null +++ b/src/qap.ts @@ -0,0 +1,107 @@ +import { Fr } from '@kevincharm/blstoise' +import { transpose, interpolate, addPolys, mulPolys, subPolys, evalPoly, divPolys } from './poly' + +export type FrMatrix = Fr[][] + +/** + * Z is minimal polynomial t(x) = (x - 1)(x - 2)...(x - n) + * s.t. it is equal to zero at all points that correspond to our gates. + * Any polynomial that is equal to zero at all of these points must be + * a multiple of Z. + * If a polynomial is a multiple of Z, then its evaluation at any of those + * points will be zero. We will exploit this equivalence. + * Generate t(x) = (x - 1)(x - 2)...(x - n) + * + * @param degree The degree of the polynomial + */ +export function tPoly(degree: number): Fr[] { + let t = [Fr.one()] + for (let i = 1n; i <= degree; i++) { + t = mulPolys(t, [new Fr(i).neg(), Fr.one()]) + } + return t +} + +/** + * Quadratic arithmetic programme + */ +export interface QAP { + /** Number of public inputs in the witness vector */ + l: number + /** Lagrange basis for R1CS "A" gates */ + u: FrMatrix + /** Lagrange basis for R1CS "B" gates */ + v: FrMatrix + /** Lagrange basis for R1CS "C" gates */ + w: FrMatrix +} + +/** + * Create a quadratic arithmetic programme from an R1CS matrix triple (A, B, C) + */ +export function fromR1cstoQap(l: number, A: FrMatrix, B: FrMatrix, C: FrMatrix): QAP { + const A_t = transpose(A) + const B_t = transpose(B) + const C_t = transpose(C) + + // These are the x values or "positions" where each element of the transposed + // matrices are evaluated at. + const xs = Array.from({ length: A_t[0].length }, (_, i) => new Fr(BigInt(i + 1))) + + // Each element is a polynomial evaluation at its respective index. + const Upolys = A_t.map((row) => interpolate(xs, row)) + const Vpolys = B_t.map((row) => interpolate(xs, row)) + const Wpolys = C_t.map((row) => interpolate(xs, row)) + + return { + l, + u: Upolys, + v: Vpolys, + w: Wpolys, + } +} + +/** + * Compute solution polynomials given a witness vector and a QAP. + * + * @param witness witness vector + * @param qap Quadratic arithmetic programme + * @returns Solution polynomials + */ +export function computeSolutionPoly(witness: Fr[], { u, v, w }: QAP) { + const au = u.reduce((acc, row, i) => addPolys(acc, mulPolys([witness[i]], row)), []) + const av = v.reduce((acc, row, i) => addPolys(acc, mulPolys([witness[i]], row)), []) + const aw = w.reduce((acc, row, i) => addPolys(acc, mulPolys([witness[i]], row)), []) + // h(x)*t(x) + const ht = subPolys(mulPolys(au, av), aw) + + for (let i = 1n; i <= u[0].length; i++) { + if (!evalPoly(ht, new Fr(i)).equals(Fr.zero())) { + throw new Error(`Solution polynomial does not evaluate to zero at ${i}`) + } + } + + return { + au, + av, + ht, + } +} + +/** + * Divide the "solution polynomial" by the minimal polynomial t(x), and verify + * that the remainder is zero. + * The output here is h(x). + * + * @param ht h(x)t(x) from the solution polynomial + * @param t minimal polynomial t(x) + */ +export function computeDivisorPoly(ht: Fr[], t: Fr[]): Fr[] { + const [h, remainder] = divPolys(ht, t) + for (const x in remainder) { + if (!remainder[x].equals(Fr.zero())) { + throw new Error(`Remainder is not zero at ${x}`) + } + } + return h +} diff --git a/test/groth16.spec.ts b/test/groth16.spec.ts new file mode 100644 index 0000000..b57cca3 --- /dev/null +++ b/test/groth16.spec.ts @@ -0,0 +1,86 @@ +import { expect } from 'chai' +import { setup, prove, verify } from '../src/groth16' +import { FrMatrix, fromR1cstoQap } from '../src/qap' +import { Fr } from '@kevincharm/blstoise' + +function prepareMatrix(matrix: number[][]): FrMatrix { + return matrix.map((row) => row.map((x) => new Fr(BigInt(x)))) +} + +// Vitalik's famous example from https://medium.com/@VitalikButerin/quadratic-arithmetic-programs-from-zero-to-hero-f6d558cea649 +// We want to prove that we know the solution to a cubic equation: +// x^3 + x + 5 = 35 (where the answer x = 3). +// We define the equation as follows: +// y = x**3 +// output = y + x + 5 +// +// We flatten the equation to quadratic ones: +// t1 = x * x +// y = t1 * x +// t2 = y + x +// output = t2 + 5 +// +// Note that t1 and t2 are *intermediate variables* that were created in order to decompose the original +// equation into a sequence of (at most) quadratic equations. +// +// Now we convert the gates (i.e. the multiplications and additions) into R1CS. +// In R1CS, we have groups of 3 vectors (a,b,c), and the solution to an R1CS programme is a witness s vector +// that satisfies the equation s.a * s.b = s.c (. = dot product) +// +// Mapping: +// [1, x, output, t1, y, t2] +// The solution vector will consist of assignments for all of the above variables in that order. +// +// (a,b,c) for the 1st gate (t1 = x * x): +// a = [0, 1, 0, 0, 0, 0] +// b = [0, 1, 0, 0, 0, 0] +// c = [0, 0, 0, 1, 0, 0] +// +// (a,b,c) for the 2nd gate (y = t1 * x): +// a = [0, 0, 0, 1, 0, 0] +// b = [0, 1, 0, 0, 0, 0] +// c = [0, 0, 0, 0, 1, 0] +// +// (a,b,c) for the 3rd gate (t2 = y + x): +// a = [0, 1, 0, 0, 1, 0] <-- "y + x" +// b = [1, 0, 0, 0, 0, 0] <-- "addition check" of s.a +// c = [0, 0, 0, 0, 0, 1] +// +// (a,b,c) for the 4th gate (output = t2 + 5): +// a = [5, 0, 0, 0, 0, 1] +// b = [1, 0, 0, 0, 0, 0] +// c = [0, 0, 1, 0, 0, 0] + +const R1CS = { + A: prepareMatrix([ + [0, 1, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0], + [0, 1, 0, 0, 1, 0], + [5, 0, 0, 0, 0, 1], + ]), + B: prepareMatrix([ + [0, 1, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0], + [1, 0, 0, 0, 0, 0], + [1, 0, 0, 0, 0, 0], + ]), + C: prepareMatrix([ + [0, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 1, 0], + [0, 0, 0, 0, 0, 1], + [0, 0, 1, 0, 0, 0], + ]), +} +const witness = [3, 35, 9, 27, 30].map((x) => new Fr(BigInt(x))) + +describe('groth16', () => { + it('compute and verify proof', () => { + const l = 1 + const qap = fromR1cstoQap(l, R1CS.A, R1CS.B, R1CS.C) + const trustedSetup = setup(qap) + const { proof, vk } = prove(trustedSetup, qap, witness) + + const publicInputs = witness.slice(0, l) + expect(verify(proof, publicInputs, vk)).to.equal(true) + }).timeout(10000) +}) diff --git a/test/poly.spec.ts b/test/poly.spec.ts new file mode 100644 index 0000000..92d1121 --- /dev/null +++ b/test/poly.spec.ts @@ -0,0 +1,63 @@ +import { expect } from 'chai' +import { addPolys, divPolys, interpolate, mulPolys, subPolys, toPoly, transpose } from '../src/poly' +import { Fr } from '@kevincharm/blstoise' + +function normalise(poly: (Fr | number | bigint)[]): string[] { + return poly.map((x) => { + if (x instanceof Fr) { + return String(x.value) + } else { + return String(x) + } + }) +} + +describe('poly', () => { + it('add', () => { + const lhs = toPoly([1, 2, 3]) + const rhs = toPoly([3, 2, 1]) + expect(normalise(addPolys(lhs, rhs))).to.deep.eq(normalise([4, 4, 4])) + }) + + it('sub', () => { + const lhs = toPoly([0, 1, 2]) + const rhs = toPoly([1, 1, 1]) + expect(normalise(subPolys(lhs, rhs))).to.deep.eq( + normalise([Fr.zero().sub(new Fr(1n)).value, 0, 1]), + ) + }) + + it('mul', () => { + const lhs = toPoly([1, 2, 3]) + const rhs = toPoly([7, 8, 9]) + expect(normalise(mulPolys(lhs, rhs))).to.deep.eq( + normalise([1 * 7, 1 * 8 + 2 * 7, 1 * 9 + 2 * 8 + 3 * 7, 2 * 9 + 3 * 8, 3 * 9]), + ) + }) + + it('div', () => { + const lhs = toPoly([1, 4]) + const rhs = toPoly([1, 2]) + const [quotient, remainder] = divPolys(lhs, rhs) + expect(normalise(quotient)).to.deep.eq(normalise([2])) + expect(normalise(remainder)).to.deep.eq(normalise([Fr.one().sub(new Fr(2n))])) + }) + + it('lagrange interpolation', () => { + const l = interpolate(toPoly([1, 2, 3]), toPoly([4, 5, 6])) + expect(normalise(l)).to.deep.eq(normalise([3, 1, 0])) + }) + + it('matrix transpose', () => { + const transposed = transpose([ + [1, 2, 3], + [4, 5, 6], + [7, 8, 9], + ]) + expect(transposed).to.deep.eq([ + [1, 4, 7], + [2, 5, 8], + [3, 6, 9], + ]) + }) +}) diff --git a/tsconfig.json b/tsconfig.json new file mode 100644 index 0000000..d4882b8 --- /dev/null +++ b/tsconfig.json @@ -0,0 +1,12 @@ +{ + "compilerOptions": { + "target": "es2020", + "module": "NodeNext", + "moduleResolution": "NodeNext", + "strict": true, + "esModuleInterop": true, + "outDir": "dist", + "declaration": true + }, + "include": ["./src", "./test"] +} diff --git a/yarn.lock b/yarn.lock new file mode 100644 index 0000000..f33d913 --- /dev/null +++ b/yarn.lock @@ -0,0 +1,693 @@ +# THIS IS AN AUTOGENERATED FILE. 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