You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
271 lines
9.9 KiB
271 lines
9.9 KiB
/**
|
|
* @author Privacy and Scaling Explorations team - pse.dev
|
|
* @dev Contains utility functions for ops in BN254; in G_1 mostly.
|
|
* @notice Forked from https://github.com/weijiekoh/libkzg/tree/master.
|
|
* Among others, a few of the changes we did on this fork were:
|
|
* - Templating the pragma version
|
|
* - Removing type wrappers and use uints instead
|
|
* - Performing changes on arg types
|
|
* - Update some of the `require` statements
|
|
* - Use the bn254 scalar field instead of checking for overflow on the babyjub prime
|
|
* - In batch checking, we compute auxiliary polynomials and their commitments at the same time.
|
|
*/
|
|
contract KZG10Verifier {
|
|
|
|
// prime of field F_p over which y^2 = x^3 + 3 is defined
|
|
uint256 public constant BN254_PRIME_FIELD =
|
|
21888242871839275222246405745257275088696311157297823662689037894645226208583;
|
|
uint256 public constant BN254_SCALAR_FIELD =
|
|
21888242871839275222246405745257275088548364400416034343698204186575808495617;
|
|
|
|
/**
|
|
* @notice Performs scalar multiplication in G_1.
|
|
* @param p G_1 point to multiply
|
|
* @param s Scalar to multiply by
|
|
* @return r G_1 point p multiplied by scalar s
|
|
*/
|
|
function mulScalar(uint256[2] memory p, uint256 s) internal view returns (uint256[2] memory r) {
|
|
uint256[3] memory input;
|
|
input[0] = p[0];
|
|
input[1] = p[1];
|
|
input[2] = s;
|
|
bool success;
|
|
assembly {
|
|
success := staticcall(sub(gas(), 2000), 7, input, 0x60, r, 0x40)
|
|
switch success
|
|
case 0 { invalid() }
|
|
}
|
|
require(success, "bn254: scalar mul failed");
|
|
}
|
|
|
|
/**
|
|
* @notice Negates a point in G_1.
|
|
* @param p G_1 point to negate
|
|
* @return uint256[2] G_1 point -p
|
|
*/
|
|
function negate(uint256[2] memory p) internal pure returns (uint256[2] memory) {
|
|
if (p[0] == 0 && p[1] == 0) {
|
|
return p;
|
|
}
|
|
return [p[0], BN254_PRIME_FIELD - (p[1] % BN254_PRIME_FIELD)];
|
|
}
|
|
|
|
/**
|
|
* @notice Adds two points in G_1.
|
|
* @param p1 G_1 point 1
|
|
* @param p2 G_1 point 2
|
|
* @return r G_1 point p1 + p2
|
|
*/
|
|
function add(uint256[2] memory p1, uint256[2] memory p2) internal view returns (uint256[2] memory r) {
|
|
bool success;
|
|
uint256[4] memory input = [p1[0], p1[1], p2[0], p2[1]];
|
|
assembly {
|
|
success := staticcall(sub(gas(), 2000), 6, input, 0x80, r, 0x40)
|
|
switch success
|
|
case 0 { invalid() }
|
|
}
|
|
|
|
require(success, "bn254: point add failed");
|
|
}
|
|
|
|
/**
|
|
* @notice Computes the pairing check e(p1, p2) * e(p3, p4) == 1
|
|
* @dev Note that G_2 points a*i + b are encoded as two elements of F_p, (a, b)
|
|
* @param a_1 G_1 point 1
|
|
* @param a_2 G_2 point 1
|
|
* @param b_1 G_1 point 2
|
|
* @param b_2 G_2 point 2
|
|
* @return result true if pairing check is successful
|
|
*/
|
|
function pairing(uint256[2] memory a_1, uint256[2][2] memory a_2, uint256[2] memory b_1, uint256[2][2] memory b_2)
|
|
internal
|
|
view
|
|
returns (bool result)
|
|
{
|
|
uint256[12] memory input = [
|
|
a_1[0],
|
|
a_1[1],
|
|
a_2[0][1], // imaginary part first
|
|
a_2[0][0],
|
|
a_2[1][1], // imaginary part first
|
|
a_2[1][0],
|
|
b_1[0],
|
|
b_1[1],
|
|
b_2[0][1], // imaginary part first
|
|
b_2[0][0],
|
|
b_2[1][1], // imaginary part first
|
|
b_2[1][0]
|
|
];
|
|
|
|
uint256[1] memory out;
|
|
bool success;
|
|
|
|
assembly {
|
|
success := staticcall(sub(gas(), 2000), 8, input, 0x180, out, 0x20)
|
|
switch success
|
|
case 0 { invalid() }
|
|
}
|
|
|
|
require(success, "bn254: pairing failed");
|
|
|
|
return out[0] == 1;
|
|
}
|
|
|
|
uint256[2] G_1 = [
|
|
{{ g1.0[0] }},
|
|
{{ g1.0[1] }}
|
|
];
|
|
uint256[2][2] G_2 = [
|
|
[
|
|
{{ g2.0[0][0] }},
|
|
{{ g2.0[0][1] }}
|
|
],
|
|
[
|
|
{{ g2.0[1][0] }},
|
|
{{ g2.0[1][1] }}
|
|
]
|
|
];
|
|
uint256[2][2] VK = [
|
|
[
|
|
{{ vk.0[0][0] }},
|
|
{{ vk.0[0][1] }}
|
|
],
|
|
[
|
|
{{ vk.0[1][0] }},
|
|
{{ vk.0[1][1] }}
|
|
]
|
|
];
|
|
|
|
uint256[2][{{ g1_crs_len }}] G1_CRS = [
|
|
{%- for (i, point) in g1_crs.iter().enumerate() %}
|
|
[
|
|
{{ point.0[0] }},
|
|
{{ point.0[1] }}
|
|
{% if loop.last -%}
|
|
]
|
|
{%- else -%}
|
|
],
|
|
{%- endif -%}
|
|
{% endfor -%}
|
|
];
|
|
|
|
/**
|
|
* @notice Verifies a single point evaluation proof. Function name follows `ark-poly`.
|
|
* @dev To avoid ops in G_2, we slightly tweak how the verification is done.
|
|
* @param c G_1 point commitment to polynomial.
|
|
* @param pi G_1 point proof.
|
|
* @param x Value to prove evaluation of polynomial at.
|
|
* @param y Evaluation poly(x).
|
|
* @return result Indicates if KZG proof is correct.
|
|
*/
|
|
function check(uint256[2] calldata c, uint256[2] calldata pi, uint256 x, uint256 y)
|
|
public
|
|
view
|
|
returns (bool result)
|
|
{
|
|
//
|
|
// we want to:
|
|
// 1. avoid gas intensive ops in G2
|
|
// 2. format the pairing check in line with what the evm opcode expects.
|
|
//
|
|
// we can do this by tweaking the KZG check to be:
|
|
//
|
|
// e(pi, vk - x * g2) = e(c - y * g1, g2) [initial check]
|
|
// e(pi, vk - x * g2) * e(c - y * g1, g2)^{-1} = 1
|
|
// e(pi, vk - x * g2) * e(-c + y * g1, g2) = 1 [bilinearity of pairing for all subsequent steps]
|
|
// e(pi, vk) * e(pi, -x * g2) * e(-c + y * g1, g2) = 1
|
|
// e(pi, vk) * e(-x * pi, g2) * e(-c + y * g1, g2) = 1
|
|
// e(pi, vk) * e(x * -pi - c + y * g1, g2) = 1 [done]
|
|
// |_ rhs_pairing _|
|
|
//
|
|
uint256[2] memory rhs_pairing =
|
|
add(mulScalar(negate(pi), x), add(negate(c), mulScalar(G_1, y)));
|
|
return pairing(pi, VK, rhs_pairing, G_2);
|
|
}
|
|
|
|
function evalPolyAt(uint256[] memory _coefficients, uint256 _index) public pure returns (uint256) {
|
|
uint256 m = BN254_SCALAR_FIELD;
|
|
uint256 result = 0;
|
|
uint256 powerOfX = 1;
|
|
|
|
for (uint256 i = 0; i < _coefficients.length; i++) {
|
|
uint256 coeff = _coefficients[i];
|
|
assembly {
|
|
result := addmod(result, mulmod(powerOfX, coeff, m), m)
|
|
powerOfX := mulmod(powerOfX, _index, m)
|
|
}
|
|
}
|
|
return result;
|
|
}
|
|
|
|
/**
|
|
* @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).
|
|
* @param z_coeffs coefficients of the zero polynomial z(x) = (x - x_1)(x - x_2)...(x - x_n).
|
|
* @param l_coeffs coefficients of the lagrange polynomial l(x).
|
|
* @param x_vals x values to evaluate the polynomials at.
|
|
* @param y_vals y values to which l(x) should evaluate to.
|
|
* @return uint256[2] commitment to z(x).
|
|
* @return uint256[2] commitment to l(x).
|
|
*/
|
|
function checkAndCommitAuxPolys(
|
|
uint256[] memory z_coeffs,
|
|
uint256[] memory l_coeffs,
|
|
uint256[] memory x_vals,
|
|
uint256[] memory y_vals
|
|
) public view returns (uint256[2] memory, uint256[2] memory) {
|
|
// z(x) is of degree len(x_vals), it is a product of linear polynomials (x - x_i)
|
|
// l(x) is of degree len(x_vals) - 1
|
|
uint256[2] memory z_commit;
|
|
uint256[2] memory l_commit;
|
|
for (uint256 i = 0; i < x_vals.length; i++) {
|
|
z_commit = add(z_commit, mulScalar(G1_CRS[i], z_coeffs[i])); // update commitment to z(x)
|
|
l_commit = add(l_commit, mulScalar(G1_CRS[i], l_coeffs[i])); // update commitment to l(x)
|
|
|
|
uint256 eval_z = evalPolyAt(z_coeffs, x_vals[i]);
|
|
uint256 eval_l = evalPolyAt(l_coeffs, x_vals[i]);
|
|
|
|
require(eval_z == 0, "checkAndCommitAuxPolys: wrong zero poly");
|
|
require(eval_l == y_vals[i], "checkAndCommitAuxPolys: wrong lagrange poly");
|
|
}
|
|
// z(x) has len(x_vals) + 1 coeffs, we add to the commitment the last coeff of z(x)
|
|
z_commit = add(z_commit, mulScalar(G1_CRS[z_coeffs.length - 1], z_coeffs[z_coeffs.length - 1]));
|
|
|
|
return (z_commit, l_commit);
|
|
}
|
|
|
|
/**
|
|
* @notice Verifies a batch of point evaluation proofs. Function name follows `ark-poly`.
|
|
* @dev To avoid ops in G_2, we slightly tweak how the verification is done.
|
|
* @param c G1 point commitment to polynomial.
|
|
* @param pi G2 point proof.
|
|
* @param x_vals Values to prove evaluation of polynomial at.
|
|
* @param y_vals Evaluation poly(x).
|
|
* @param l_coeffs Coefficients of the lagrange polynomial.
|
|
* @param z_coeffs Coefficients of the zero polynomial z(x) = (x - x_1)(x - x_2)...(x - x_n).
|
|
* @return result Indicates if KZG proof is correct.
|
|
*/
|
|
function batchCheck(
|
|
uint256[2] calldata c,
|
|
uint256[2][2] calldata pi,
|
|
uint256[] calldata x_vals,
|
|
uint256[] calldata y_vals,
|
|
uint256[] calldata l_coeffs,
|
|
uint256[] calldata z_coeffs
|
|
) public view returns (bool result) {
|
|
//
|
|
// we want to:
|
|
// 1. avoid gas intensive ops in G2
|
|
// 2. format the pairing check in line with what the evm opcode expects.
|
|
//
|
|
// we can do this by tweaking the KZG check to be:
|
|
//
|
|
// e(z(r) * g1, pi) * e(g1, l(r) * g2) = e(c, g2) [initial check]
|
|
// e(z(r) * g1, pi) * e(l(r) * g1, g2) * e(c, g2)^{-1} = 1 [bilinearity of pairing]
|
|
// e(z(r) * g1, pi) * e(l(r) * g1 - c, g2) = 1 [done]
|
|
//
|
|
(uint256[2] memory z_commit, uint256[2] memory l_commit) =
|
|
checkAndCommitAuxPolys(z_coeffs, l_coeffs, x_vals, y_vals);
|
|
uint256[2] memory neg_commit = negate(c);
|
|
return pairing(z_commit, pi, add(l_commit, neg_commit), G_2);
|
|
}
|
|
}
|