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271 lines
9.9 KiB
271 lines
9.9 KiB
/**
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* @author Privacy and Scaling Explorations team - pse.dev
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* @dev Contains utility functions for ops in BN254; in G_1 mostly.
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* @notice Forked from https://github.com/weijiekoh/libkzg.
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* Among others, a few of the changes we did on this fork were:
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* - Templating the pragma version
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* - Removing type wrappers and use uints instead
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* - Performing changes on arg types
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* - Update some of the `require` statements
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* - Use the bn254 scalar field instead of checking for overflow on the babyjub prime
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* - In batch checking, we compute auxiliary polynomials and their commitments at the same time.
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*/
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contract KZG10Verifier {
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// prime of field F_p over which y^2 = x^3 + 3 is defined
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uint256 public constant BN254_PRIME_FIELD =
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21888242871839275222246405745257275088696311157297823662689037894645226208583;
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uint256 public constant BN254_SCALAR_FIELD =
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21888242871839275222246405745257275088548364400416034343698204186575808495617;
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/**
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* @notice Performs scalar multiplication in G_1.
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* @param p G_1 point to multiply
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* @param s Scalar to multiply by
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* @return r G_1 point p multiplied by scalar s
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*/
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function mulScalar(uint256[2] memory p, uint256 s) internal view returns (uint256[2] memory r) {
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uint256[3] memory input;
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input[0] = p[0];
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input[1] = p[1];
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input[2] = s;
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bool success;
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assembly {
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success := staticcall(sub(gas(), 2000), 7, input, 0x60, r, 0x40)
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switch success
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case 0 { invalid() }
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}
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require(success, "bn254: scalar mul failed");
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}
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/**
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* @notice Negates a point in G_1.
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* @param p G_1 point to negate
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* @return uint256[2] G_1 point -p
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*/
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function negate(uint256[2] memory p) internal pure returns (uint256[2] memory) {
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if (p[0] == 0 && p[1] == 0) {
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return p;
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}
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return [p[0], BN254_PRIME_FIELD - (p[1] % BN254_PRIME_FIELD)];
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}
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/**
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* @notice Adds two points in G_1.
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* @param p1 G_1 point 1
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* @param p2 G_1 point 2
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* @return r G_1 point p1 + p2
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*/
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function add(uint256[2] memory p1, uint256[2] memory p2) internal view returns (uint256[2] memory r) {
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bool success;
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uint256[4] memory input = [p1[0], p1[1], p2[0], p2[1]];
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assembly {
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success := staticcall(sub(gas(), 2000), 6, input, 0x80, r, 0x40)
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switch success
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case 0 { invalid() }
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}
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require(success, "bn254: point add failed");
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}
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/**
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* @notice Computes the pairing check e(p1, p2) * e(p3, p4) == 1
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* @dev Note that G_2 points a*i + b are encoded as two elements of F_p, (a, b)
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* @param a_1 G_1 point 1
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* @param a_2 G_2 point 1
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* @param b_1 G_1 point 2
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* @param b_2 G_2 point 2
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* @return result true if pairing check is successful
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*/
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function pairing(uint256[2] memory a_1, uint256[2][2] memory a_2, uint256[2] memory b_1, uint256[2][2] memory b_2)
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internal
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view
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returns (bool result)
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{
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uint256[12] memory input = [
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a_1[0],
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a_1[1],
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a_2[0][1], // imaginary part first
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a_2[0][0],
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a_2[1][1], // imaginary part first
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a_2[1][0],
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b_1[0],
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b_1[1],
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b_2[0][1], // imaginary part first
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b_2[0][0],
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b_2[1][1], // imaginary part first
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b_2[1][0]
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];
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uint256[1] memory out;
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bool success;
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assembly {
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success := staticcall(sub(gas(), 2000), 8, input, 0x180, out, 0x20)
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switch success
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case 0 { invalid() }
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}
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require(success, "bn254: pairing failed");
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return out[0] == 1;
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}
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uint256[2] G_1 = [
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{{ g1.0[0] }},
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{{ g1.0[1] }}
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];
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uint256[2][2] G_2 = [
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[
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{{ g2.0[0][0] }},
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{{ g2.0[0][1] }}
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],
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[
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{{ g2.0[1][0] }},
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{{ g2.0[1][1] }}
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]
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];
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uint256[2][2] VK = [
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[
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{{ vk.0[0][0] }},
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{{ vk.0[0][1] }}
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],
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[
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{{ vk.0[1][0] }},
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{{ vk.0[1][1] }}
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]
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];
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uint256[2][{{ g1_crs_len }}] G1_CRS = [
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{%- for (i, point) in g1_crs.iter().enumerate() %}
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[
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{{ point.0[0] }},
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{{ point.0[1] }}
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{% if loop.last -%}
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]
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{%- else -%}
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],
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{%- endif -%}
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{% endfor -%}
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];
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/**
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* @notice Verifies a single point evaluation proof. Function name follows `ark-poly`.
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* @dev To avoid ops in G_2, we slightly tweak how the verification is done.
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* @param c G_1 point commitment to polynomial.
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* @param pi G_1 point proof.
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* @param x Value to prove evaluation of polynomial at.
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* @param y Evaluation poly(x).
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* @return result Indicates if KZG proof is correct.
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*/
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function check(uint256[2] calldata c, uint256[2] calldata pi, uint256 x, uint256 y)
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public
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view
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returns (bool result)
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{
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//
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// we want to:
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// 1. avoid gas intensive ops in G2
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// 2. format the pairing check in line with what the evm opcode expects.
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//
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// we can do this by tweaking the KZG check to be:
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//
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// e(pi, vk - x * g2) = e(c - y * g1, g2) [initial check]
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// e(pi, vk - x * g2) * e(c - y * g1, g2)^{-1} = 1
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// e(pi, vk - x * g2) * e(-c + y * g1, g2) = 1 [bilinearity of pairing for all subsequent steps]
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// e(pi, vk) * e(pi, -x * g2) * e(-c + y * g1, g2) = 1
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// e(pi, vk) * e(-x * pi, g2) * e(-c + y * g1, g2) = 1
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// e(pi, vk) * e(x * -pi - c + y * g1, g2) = 1 [done]
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// |_ rhs_pairing _|
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//
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uint256[2] memory rhs_pairing =
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add(mulScalar(negate(pi), x), add(negate(c), mulScalar(G_1, y)));
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return pairing(pi, VK, rhs_pairing, G_2);
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}
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function evalPolyAt(uint256[] memory _coefficients, uint256 _index) public pure returns (uint256) {
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uint256 m = BN254_SCALAR_FIELD;
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uint256 result = 0;
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uint256 powerOfX = 1;
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for (uint256 i = 0; i < _coefficients.length; i++) {
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uint256 coeff = _coefficients[i];
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assembly {
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result := addmod(result, mulmod(powerOfX, coeff, m), m)
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powerOfX := mulmod(powerOfX, _index, m)
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}
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}
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return result;
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}
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/**
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* @notice Ensures that z(x) == 0 and l(x) == y for all x in x_vals and y in y_vals. It returns the commitment to z(x) and l(x).
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* @param z_coeffs coefficients of the zero polynomial z(x) = (x - x_1)(x - x_2)...(x - x_n).
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* @param l_coeffs coefficients of the lagrange polynomial l(x).
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* @param x_vals x values to evaluate the polynomials at.
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* @param y_vals y values to which l(x) should evaluate to.
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* @return uint256[2] commitment to z(x).
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* @return uint256[2] commitment to l(x).
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*/
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function checkAndCommitAuxPolys(
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uint256[] memory z_coeffs,
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uint256[] memory l_coeffs,
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uint256[] memory x_vals,
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uint256[] memory y_vals
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) public view returns (uint256[2] memory, uint256[2] memory) {
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// z(x) is of degree len(x_vals), it is a product of linear polynomials (x - x_i)
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// l(x) is of degree len(x_vals) - 1
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uint256[2] memory z_commit;
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uint256[2] memory l_commit;
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for (uint256 i = 0; i < x_vals.length; i++) {
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z_commit = add(z_commit, mulScalar(G1_CRS[i], z_coeffs[i])); // update commitment to z(x)
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l_commit = add(l_commit, mulScalar(G1_CRS[i], l_coeffs[i])); // update commitment to l(x)
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uint256 eval_z = evalPolyAt(z_coeffs, x_vals[i]);
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uint256 eval_l = evalPolyAt(l_coeffs, x_vals[i]);
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require(eval_z == 0, "checkAndCommitAuxPolys: wrong zero poly");
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require(eval_l == y_vals[i], "checkAndCommitAuxPolys: wrong lagrange poly");
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}
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// z(x) has len(x_vals) + 1 coeffs, we add to the commitment the last coeff of z(x)
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z_commit = add(z_commit, mulScalar(G1_CRS[z_coeffs.length - 1], z_coeffs[z_coeffs.length - 1]));
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return (z_commit, l_commit);
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}
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/**
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* @notice Verifies a batch of point evaluation proofs. Function name follows `ark-poly`.
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* @dev To avoid ops in G_2, we slightly tweak how the verification is done.
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* @param c G1 point commitment to polynomial.
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* @param pi G2 point proof.
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* @param x_vals Values to prove evaluation of polynomial at.
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* @param y_vals Evaluation poly(x).
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* @param l_coeffs Coefficients of the lagrange polynomial.
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* @param z_coeffs Coefficients of the zero polynomial z(x) = (x - x_1)(x - x_2)...(x - x_n).
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* @return result Indicates if KZG proof is correct.
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*/
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function batchCheck(
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uint256[2] calldata c,
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uint256[2][2] calldata pi,
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uint256[] calldata x_vals,
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uint256[] calldata y_vals,
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uint256[] calldata l_coeffs,
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uint256[] calldata z_coeffs
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) public view returns (bool result) {
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//
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// we want to:
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// 1. avoid gas intensive ops in G2
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// 2. format the pairing check in line with what the evm opcode expects.
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//
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// we can do this by tweaking the KZG check to be:
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//
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// e(z(r) * g1, pi) * e(g1, l(r) * g2) = e(c, g2) [initial check]
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// e(z(r) * g1, pi) * e(l(r) * g1, g2) * e(c, g2)^{-1} = 1 [bilinearity of pairing]
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// e(z(r) * g1, pi) * e(l(r) * g1 - c, g2) = 1 [done]
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//
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(uint256[2] memory z_commit, uint256[2] memory l_commit) =
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checkAndCommitAuxPolys(z_coeffs, l_coeffs, x_vals, y_vals);
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uint256[2] memory neg_commit = negate(c);
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return pairing(z_commit, pi, add(l_commit, neg_commit), G_2);
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}
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}
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