Shamir-Straus trick for pair and quartet
Signed-off-by: Uncle Fatso <uncle.fatso@ghostchain.io>
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30
README.md
30
README.md
@ -38,29 +38,39 @@ For full background and protocol details see the [project wiki](https://git.ghos
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+==================================================================================================+
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| Deployment Cost | Deployment Size | | | | |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| 2011993 | 9094 | | | | |
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| 2624055 | 11923 | | | | |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| | | | | | |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| Function Name | Min | Avg | Median | Max | # Calls |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| addJacobian | 1800 | 1800 | 1800 | 1800 | 88 |
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| addJacobian | 1896 | 1896 | 1896 | 1896 | 88 |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| addProjective | 1015 | 1015 | 1015 | 1015 | 44 |
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| addProjective | 1110 | 1110 | 1110 | 1110 | 44 |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| addProjectiveMixed | 967 | 967 | 967 | 967 | 44 |
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| addProjectiveMixed | 1084 | 1084 | 1084 | 1084 | 44 |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| doubleJacobian | 794 | 794 | 794 | 794 | 45 |
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| doubleJacobian | 889 | 889 | 889 | 889 | 45 |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| doubleProjective | 546 | 546 | 546 | 546 | 45 |
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| doubleProjective | 641 | 641 | 641 | 641 | 45 |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| mulEcTriplet | 205548 | 1140310 | 1546125 | 1958623 | 43 |
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| isOnCurve | 280 | 280 | 280 | 280 | 262 |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| mulProjectiveTriplet | 6436 | 193298 | 334976 | 385511 | 43 |
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| mulEcPair | 103258 | 731053 | 1058263 | 1269945 | 44 |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| toAffineJacobian | 40275 | 47740 | 47859 | 53863 | 133 |
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| mulEcQuartet | 308151 | 1561360 | 1982690 | 2672830 | 42 |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| toAffineProjective | 11056 | 13787 | 13904 | 16105 | 176 |
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| mulEcTriplet | 205666 | 1140527 | 1546387 | 1958933 | 43 |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| mulProjectivePair | 4265 | 178529 | 311013 | 338160 | 44 |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| mulProjectiveQuartet | 12001 | 206222 | 345559 | 396103 | 42 |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| mulProjectiveTriplet | 6622 | 199854 | 346604 | 397492 | 43 |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| toAffineJacobian | 40300 | 47765 | 47884 | 53888 | 133 |
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|----------------------------------------+-----------------+---------+---------+---------+---------|
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| toAffineProjective | 11129 | 13876 | 13864 | 16178 | 262 |
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╰----------------------------------------+-----------------+---------+---------+---------+---------╯
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```
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@ -81,6 +81,63 @@ contract MathTester {
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return EllipticCurveProjective.mulAddProjectiveTriplet(x1, y1, k1, x2, y2, k2, x3, y3, k3);
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}
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function mulEcPair(
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uint256 x1,
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uint256 y1,
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uint256 k1,
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uint256 x2,
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uint256 y2,
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uint256 k2
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) public pure returns(uint256, uint256) {
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(x1, y1) = EllipticCurve.ecMul(k1, x1, y1, A, P);
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(x2, y2) = EllipticCurve.ecMul(k2, x2, y2, A, P);
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(x1, y1) = EllipticCurve.ecAdd(x1, y1, x2, y2, A, P);
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return (x1, y1);
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}
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function mulProjectivePair(
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uint256 x1,
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uint256 y1,
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uint256 k1,
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uint256 x2,
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uint256 y2,
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uint256 k2
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) public pure returns(uint256, uint256, uint256) {
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return EllipticCurveProjective.mulAddProjectivePair(x1, y1, k1, x2, y2, k2);
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}
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function mulEcQuartet(
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uint256 x1, uint256 y1, uint256 k1,
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uint256 x2, uint256 y2, uint256 k2,
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uint256 x3, uint256 y3, uint256 k3,
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uint256 x4, uint256 y4, uint256 k4
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) public pure returns(uint256, uint256) {
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(x1, y1) = EllipticCurve.ecMul(k1, x1, y1, A, P);
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(x2, y2) = EllipticCurve.ecMul(k2, x2, y2, A, P);
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(x3, y3) = EllipticCurve.ecMul(k3, x3, y3, A, P);
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(x4, y4) = EllipticCurve.ecMul(k4, x4, y4, A, P);
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(x1, y1) = EllipticCurve.ecAdd(x1, y1, x2, y2, A, P);
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(x3, y3) = EllipticCurve.ecAdd(x3, y3, x4, y4, A, P);
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(x1, y1) = EllipticCurve.ecAdd(x1, y1, x3, y3, A, P);
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return (x1, y1);
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}
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function mulProjectiveQuartet(
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uint256 x1, uint256 y1, uint256 k1,
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uint256 x2, uint256 y2, uint256 k2,
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uint256 x3, uint256 y3, uint256 k3,
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uint256 x4, uint256 y4, uint256 k4
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) public pure returns (uint256, uint256, uint256) {
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return EllipticCurveProjective.mulAddProjectiveQuartet(
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x1, y1, k1,
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x2, y2, k2,
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x3, y3, k3,
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x4, y4, k4
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);
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}
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function toAffineJacobian(uint256 x, uint256 y, uint256 z) public pure returns (uint256, uint256) {
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return EllipticCurve.toAffine(x, y, z, P);
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}
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@ -10,23 +10,24 @@ pragma solidity ^0.8.0;
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library EllipticCurveProjective {
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// Constants are taken from https://en.bitcoin.it/wiki/Secp256k1
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uint256 private constant A = 0;
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uint256 private constant B = 7;
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uint256 private constant P = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F;
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uint256 private constant N = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141;
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uint256 public constant B = 7;
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uint256 public constant P = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F;
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uint256 public constant N = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141;
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uint256 private constant B3 = 21;
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function findMaxBitLength(
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uint256 k1,
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uint256 k2,
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uint256 k3
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uint256 k3,
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uint256 k4
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) internal pure returns (uint256 bits) {
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assembly {
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// Find maximum of the three scalars
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let max := k1
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if gt(k2, max) { max := k2 }
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if gt(k3, max) { max := k3 }
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if gt(k4, max) { max := k4 }
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// if (v >> 128 != 0) { v >>= 128; bits += 128; }
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if gt(max, 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF) {
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@ -71,6 +72,61 @@ library EllipticCurveProjective {
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}
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}
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function mulAddProjectivePair(
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uint256 x1, uint256 y1, uint256 k1,
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uint256 x2, uint256 y2, uint256 k2
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) internal pure returns (uint256 x3, uint256 y3, uint256 z3) {
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// We implement the Straus-Shamir trick described in
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// Trading Inversions for Multiplications in Elliptic Curve Cryptography.
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// (https://eprint.iacr.org/2003/257.pdf Page 7).
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uint256 bits = findMaxBitLength(k1, k2, 0, 0);
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uint256[4] memory precomputedXs;
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uint256[4] memory precomputedYs;
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uint256[4] memory precomputedZs;
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precomputedXs[1] = x2; precomputedYs[1] = y2; precomputedZs[1] = 1; // 01: P2
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precomputedXs[2] = x1; precomputedYs[2] = y1; precomputedZs[2] = 1; // 10: P1
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(x3, y3, z3) = projectiveAddMixed(x1, y1, 1, x2, y2); // 11: P1+P2
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precomputedXs[3] = x3;
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precomputedYs[3] = y3;
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precomputedZs[3] = z3;
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x3 = 0;
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y3 = 1;
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z3 = 0;
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for (; bits > 0;) {
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unchecked { --bits; }
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(x3, y3, z3) = projectiveDouble(x3, y3, z3);
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uint8 mask;
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assembly {
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mask := or(
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shl(1, and(shr(bits, k1), 1)),
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and(shr(bits, k2), 1)
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)
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}
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if (mask == 0) {
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continue;
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}
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if (mask == 3) {
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(x3, y3, z3) = projectiveAdd(
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x3, y3, z3, precomputedXs[mask], precomputedYs[mask], precomputedZs[mask]
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);
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} else {
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(x3, y3, z3) = projectiveAddMixed(
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x3, y3, z3, precomputedXs[mask], precomputedYs[mask]
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);
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}
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}
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}
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function mulAddProjectiveTriplet(
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uint256 x1, uint256 y1, uint256 k1,
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uint256 x2, uint256 y2, uint256 k2,
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@ -80,7 +136,7 @@ library EllipticCurveProjective {
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// Trading Inversions for Multiplications in Elliptic Curve Cryptography.
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// (https://eprint.iacr.org/2003/257.pdf Page 7).
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uint256 bits = findMaxBitLength(k1, k2, k3);
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uint256 bits = findMaxBitLength(k1, k2, k3, 0);
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uint256[8] memory precomputedXs;
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uint256[8] memory precomputedYs;
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@ -143,6 +199,118 @@ library EllipticCurveProjective {
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}
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}
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function mulAddProjectiveQuartet(
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uint256 x1, uint256 y1, uint256 k1,
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uint256 x2, uint256 y2, uint256 k2,
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uint256 x3, uint256 y3, uint256 k3,
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uint256 x4, uint256 y4, uint256 k4
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) internal pure returns (uint256 x5, uint256 y5, uint256 z5) {
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// We implement the Straus-Shamir trick described in
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// Trading Inversions for Multiplications in Elliptic Curve Cryptography.
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// (https://eprint.iacr.org/2003/257.pdf Page 7).
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uint256 bits = findMaxBitLength(k1, k2, k3, k4);
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uint256[16] memory precomputedXs;
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uint256[16] memory precomputedYs;
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uint256[16] memory precomputedZs;
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precomputedXs[1] = x4; precomputedYs[1] = y4; precomputedZs[1] = 1; // 0001: P4
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precomputedXs[2] = x3; precomputedYs[2] = y3; precomputedZs[2] = 1; // 0010: P3
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precomputedXs[4] = x2; precomputedYs[4] = y2; precomputedZs[4] = 1; // 0100: P2
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precomputedXs[8] = x1; precomputedYs[8] = y1; precomputedZs[8] = 1; // 1000: P1
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(precomputedXs[3], precomputedYs[3], precomputedZs[3]) = projectiveAddMixed(x4, y4, 1, x3, y3); // 0011: P4+P3
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(precomputedXs[6], precomputedYs[6], precomputedZs[6]) = projectiveAddMixed(x3, y3, 1, x2, y2); // 0110: P3+P2
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(precomputedXs[12], precomputedYs[12], precomputedZs[12]) = projectiveAddMixed(x2, y2, 1, x1, y1); // 1100: P2+P1
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(precomputedXs[9], precomputedYs[9], precomputedZs[9]) = projectiveAddMixed(x4, y4, 1, x1, y1); // 1001: P4+P1
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(precomputedXs[10], precomputedYs[10], precomputedZs[10]) = projectiveAddMixed(x1, y1, 1, x3, y3); // 1010: P1+P3
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(precomputedXs[5], precomputedYs[5], precomputedZs[5]) = projectiveAddMixed(x4, y4, 1, x2, y2); // 0101: P4+P2
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(precomputedXs[7], precomputedYs[7], precomputedZs[7]) = projectiveAddMixed(
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precomputedXs[3],
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precomputedYs[3],
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precomputedZs[3],
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x2,
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y2
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); // 0111: P4+P3+P2
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(precomputedXs[11], precomputedYs[11], precomputedZs[11]) = projectiveAddMixed(
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precomputedXs[3],
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precomputedYs[3],
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precomputedZs[3],
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x1,
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y1
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); // 1011: P4+P3+P1
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(precomputedXs[13], precomputedYs[13], precomputedZs[13]) = projectiveAddMixed(
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precomputedXs[12],
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precomputedYs[12],
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precomputedZs[12],
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x4,
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y4
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); // 1101: P4+P2+P1
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(precomputedXs[14], precomputedYs[14], precomputedZs[14]) = projectiveAddMixed(
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precomputedXs[12],
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precomputedYs[12],
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precomputedZs[12],
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x3,
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y3
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); // 1110: P3+P2+P1
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(precomputedXs[15], precomputedYs[15], precomputedZs[15]) = projectiveAddMixed(
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precomputedXs[14],
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precomputedYs[14],
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precomputedZs[14],
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x4,
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y4
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); // 1111: P4+P3+P2+P1
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x5 = 0;
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y5 = 1;
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z5 = 0;
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for (; bits > 0;) {
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unchecked { --bits; }
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(x5, y5, z5) = projectiveDouble(x5, y5, z5);
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uint8 mask;
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uint16 bitmask;
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assembly {
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mask := or(
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shl(3, and(shr(bits, k1), 1)),
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or(
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shl(2, and(shr(bits, k2), 1)),
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or(
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shl(1, and(shr(bits, k3), 1)),
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and(shr(bits, k4), 1)
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)
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)
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)
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// bitmask for 1,2,4, 8 is 278 = 0b0000000100110110
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bitmask := and(shl(278, mask), 1)
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}
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if (mask == 0) {
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continue;
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}
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if (bitmask == 1) {
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(x5, y5, z5) = projectiveAddMixed(
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x5, y5, z5, precomputedXs[mask], precomputedYs[mask]
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);
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} else {
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(x5, y5, z5) = projectiveAdd(
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x5, y5, z5, precomputedXs[mask], precomputedYs[mask], precomputedZs[mask]
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);
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}
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}
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}
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function projectiveAddMixed(
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uint256 X1,
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uint256 Y1,
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@ -197,8 +365,6 @@ library EllipticCurveProjective {
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// Y3 = (Y1Y2 + 3bZ1Z2)(Y1Y2 − 3bZ1Z2) + 9bX1X2(X1Z2 + X2Z1),
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// Z3 = (Y1Z2 + Y2Z1)(Y1Y2 + 3bZ1Z2) + 3X1X2(X1Y2 + X2Y1),
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// TODO: Can we optimize it even more?
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uint256 t0 = mulmod(X1, X2, P); // 1. t0 ← X1 · X2 => (X1·X2)
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uint256 t1 = mulmod(Y1, Y2, P); // 2. t1 ← Y1 · Y2 => (Y1·Y2)
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uint256 t2 = mulmod(Z1, Z2, P); // 3. t2 ← Z1 · Z2 => (Z1·Z2)
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@ -292,4 +458,14 @@ library EllipticCurveProjective {
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_x = mulmod(x, t0, P);
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_y = mulmod(y, t0, P);
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}
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function isOnCurve(uint256 x, uint256 y) internal pure returns (bool) {
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// TODO: check if something is zero
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uint lhs = mulmod(y, y, P); // y^2
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uint rhs = mulmod(mulmod(x, x, P), x, P); // x^3
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rhs = addmod(rhs, B, P); // x^3 + 7
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return lhs == rhs;
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}
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}
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@ -22,6 +22,78 @@ contract MathTesterTest is Test {
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points = abi.decode(data, (Point[]));
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}
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function test_quartet() public view {
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uint256 len = points.length - 3;
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for (uint256 i; i < len;) {
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(uint256 x_p, uint256 y_p, uint256 z_p) = math.mulProjectiveQuartet(
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uint256(points[i].x),
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uint256(points[i].y),
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uint256(points[i].k),
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uint256(points[i+1].x),
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uint256(points[i+1].y),
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uint256(points[i+1].k),
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uint256(points[i+2].x),
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uint256(points[i+2].y),
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uint256(points[i+2].k),
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uint256(points[i+3].x),
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uint256(points[i+3].y),
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uint256(points[i+3].k)
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);
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(x_p, y_p) = math.toAffineProjective(x_p, y_p, z_p);
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(uint256 x_j, uint256 y_j) = math.mulEcQuartet(
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uint256(points[i].x),
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uint256(points[i].y),
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uint256(points[i].k),
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uint256(points[i+1].x),
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uint256(points[i+1].y),
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||||
uint256(points[i+1].k),
|
||||
uint256(points[i+2].x),
|
||||
uint256(points[i+2].y),
|
||||
uint256(points[i+2].k),
|
||||
uint256(points[i+3].x),
|
||||
uint256(points[i+3].y),
|
||||
uint256(points[i+3].k)
|
||||
);
|
||||
|
||||
assertEq(x_p, x_j);
|
||||
assertEq(y_p, y_j);
|
||||
assertEq(math.isOnCurve(x_p, y_p), true);
|
||||
|
||||
unchecked { ++i; }
|
||||
}
|
||||
}
|
||||
|
||||
function test_pair() public view {
|
||||
uint256 len = points.length - 1;
|
||||
for (uint256 i; i < len;) {
|
||||
(uint256 x_p, uint256 y_p, uint256 z_p) = math.mulProjectivePair(
|
||||
uint256(points[i].x),
|
||||
uint256(points[i].y),
|
||||
uint256(points[i].k),
|
||||
uint256(points[i+1].x),
|
||||
uint256(points[i+1].y),
|
||||
uint256(points[i+1].k)
|
||||
);
|
||||
(x_p, y_p) = math.toAffineProjective(x_p, y_p, z_p);
|
||||
|
||||
(uint256 x_j, uint256 y_j) = math.mulEcPair(
|
||||
uint256(points[i].x),
|
||||
uint256(points[i].y),
|
||||
uint256(points[i].k),
|
||||
uint256(points[i+1].x),
|
||||
uint256(points[i+1].y),
|
||||
uint256(points[i+1].k)
|
||||
);
|
||||
|
||||
assertEq(x_p, x_j);
|
||||
assertEq(y_p, y_j);
|
||||
assertEq(math.isOnCurve(x_p, y_p), true);
|
||||
|
||||
unchecked { ++i; }
|
||||
}
|
||||
}
|
||||
|
||||
function test_triplet() public view {
|
||||
uint256 len = points.length - 2;
|
||||
for (uint256 i; i < len;) {
|
||||
@ -49,10 +121,10 @@ contract MathTesterTest is Test {
|
||||
uint256(points[i+2].y),
|
||||
uint256(points[i+2].k)
|
||||
);
|
||||
// (x_j, y_j) = math.toAffineJacobian(x_j, y_j, z_j);
|
||||
|
||||
assertEq(x_p, x_j);
|
||||
assertEq(y_p, y_j);
|
||||
assertEq(math.isOnCurve(x_p, y_p), true);
|
||||
|
||||
unchecked { ++i; }
|
||||
}
|
||||
@ -79,6 +151,7 @@ contract MathTesterTest is Test {
|
||||
|
||||
assertEq(x_p, x_j);
|
||||
assertEq(y_p, y_j);
|
||||
assertEq(math.isOnCurve(x_p, y_p), true);
|
||||
|
||||
unchecked { ++i; }
|
||||
}
|
||||
@ -105,6 +178,7 @@ contract MathTesterTest is Test {
|
||||
|
||||
assertEq(x_p, x_j);
|
||||
assertEq(y_p, y_j);
|
||||
assertEq(math.isOnCurve(x_p, y_p), true);
|
||||
|
||||
unchecked { ++i; }
|
||||
}
|
||||
@ -127,6 +201,7 @@ contract MathTesterTest is Test {
|
||||
|
||||
assertEq(x_p, x_j);
|
||||
assertEq(y_p, y_j);
|
||||
assertEq(math.isOnCurve(x_p, y_p), true);
|
||||
|
||||
unchecked { ++i; }
|
||||
}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user