ghostchain/ghost-exodus-draft
5
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Compare commits

base: ghostchain:4ec272ca4a8bd8a3d4791ef3ae8183cf481f5765
Branches Tags
ghostchain:main
..
compare: ghostchain:92a8ce4b440c2059ae7feed6ab59a742cb197da7
Branches Tags
ghostchain:main

No commits in common. "4ec272ca4a8bd8a3d4791ef3ae8183cf481f5765" and "92a8ce4b440c2059ae7feed6ab59a742cb197da7" have entirely different histories.
4ec272ca4a ... 92a8ce4b44

3 changed files with 155 additions and 88 deletions
Show Stats Expand all files Collapse all files

31
README.md
View File

@ -38,7 +38,7 @@ For full background and protocol details see the [project wiki](https://git.ghos
+==================================================================================================+ +==================================================================================================+
| Deployment Cost | Deployment Size | | | | | | Deployment Cost | Deployment Size | | | | |
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
| 3269040 | 14905 | | | | | | 3247879 | 14807 | | | | |
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
| | | | | | | | | | | | | |
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
@ -54,19 +54,19 @@ For full background and protocol details see the [project wiki](https://git.ghos
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
| addProjective | 1252 | 1252 | 1252 | 1252 | 44 | | addProjective | 1252 | 1252 | 1252 | 1252 | 44 |
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
| doubleGhost | 699 | 699 | 699 | 699 | 45 | | doubleGhost | 732 | 732 | 732 | 732 | 45 |
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
| doubleJacobian | 1101 | 1101 | 1101 | 1101 | 45 | | doubleJacobian | 1101 | 1101 | 1101 | 1101 | 45 |
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
| doubleProjective | 677 | 677 | 677 | 677 | 45 | | doubleProjective | 710 | 710 | 710 | 710 | 45 |
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
| isOnCurve | 296 | 296 | 296 | 296 | 306 | | isOnCurve | 335 | 335 | 335 | 335 | 306 |
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
| isOnCurveGhost | 824 | 824 | 824 | 824 | 306 | | isOnCurveGhost | 863 | 863 | 863 | 863 | 306 |
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
| mulPairEc | 103705 | 731500 | 1058710 | 1270392 | 44 | | mulPairEc | 103705 | 731500 | 1058710 | 1270392 | 44 |
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
| mulPairGhost | 4185 | 174517 | 303073 | 330416 | 44 | | mulPairGhost | 4241 | 178420 | 310213 | 337584 | 44 |
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
| mulPairProjective | 5275 | 182401 | 327577 | 361377 | 44 | | mulPairProjective | 5275 | 182401 | 327577 | 361377 | 44 |
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
@ -76,7 +76,7 @@ For full background and protocol details see the [project wiki](https://git.ghos
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
| mulSingleEc | 2093 | 327159 | 408874 | 600125 | 44 | | mulSingleEc | 2093 | 327159 | 408874 | 600125 | 44 |
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
| mulSingleGhost | 1775 | 109646 | 143531 | 212019 | 44 | | mulSingleGhost | 1803 | 113386 | 148753 | 219187 | 44 |
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
| mulSingleProjective | 1916 | 114289 | 149838 | 220860 | 44 | | mulSingleProjective | 1916 | 114289 | 149838 | 220860 | 44 |
|----------------------------------------+-----------------+---------+---------+---------+---------| |----------------------------------------+-----------------+---------+---------+---------+---------|
@ -96,26 +96,23 @@ For full background and protocol details see the [project wiki](https://git.ghos
+============================================================================================================+ +============================================================================================================+
| Deployment Cost | Deployment Size | | | | | | Deployment Cost | Deployment Size | | | | |
|-----------------------------------------------------+-----------------+--------+--------+--------+---------| |-----------------------------------------------------+-----------------+--------+--------+--------+---------|
| 1687628 | 7589 | | | | | | 1871747 | 8463 | | | | |
|-----------------------------------------------------+-----------------+--------+--------+--------+---------| |-----------------------------------------------------+-----------------+--------+--------+--------+---------|
| | | | | | | | | | | | | |
|-----------------------------------------------------+-----------------+--------+--------+--------+---------| |-----------------------------------------------------+-----------------+--------+--------+--------+---------|
| Function Name | Min | Avg | Median | Max | # Calls | | Function Name | Min | Avg | Median | Max | # Calls |
|-----------------------------------------------------+-----------------+--------+--------+--------+---------| |-----------------------------------------------------+-----------------+--------+--------+--------+---------|
| setPubkey | 111516 | 111529 | 111528 | 111540 | 9 | | setPubkey | 111552 | 111565 | 111564 | 111576 | 9 |
|-----------------------------------------------------+-----------------+--------+--------+--------+---------| |-----------------------------------------------------+-----------------+--------+--------+--------+---------|
| verify | 538818 | 609541 | 578174 | 884038 | 9 | | verify | 611046 | 680989 | 648023 | 955747 | 9 |
╰-----------------------------------------------------+-----------------+--------+--------+--------+---------╯ ╰-----------------------------------------------------+-----------------+--------+--------+--------+---------╯
``` ```
The `verify` function consumes a maximum of ~~955,747~~ 884,038 gas units when processing 1,000 signers with 300 missing signatures. Assuming an ETH price of $4,200 and gas cost of 2 gwei: The `verify` function consumes a maximum of 955,747 gas units when processing 1,000 signers with 300 missing signatures. Assuming an ETH price of $4,200 and gas cost of 2 gwei:
* ~~test_verify_5_of_5_works() (gas: 735686 = $6.18) - Baseline (100%)~~ * test_verify_5_of_5_works() (gas: 735686 = $6.18) - Baseline (100%)
* ~~test_verify_67_of_100() (gas: 812318 = $6.82) - 10% increase~~ * test_verify_67_of_100() (gas: 812318 = $6.82) - 10% increase
* ~~test_verify_700_of_1000() (gas: 1118143 = $9.39) - 51% increase~~ * test_verify_700_of_1000() (gas: 1118143 = $9.39) - 51% increase
* test_verify_5_of_5_works() (gas: 663441 = $5.57) - Baseline (100%)
* test_verify_67_of_100() (gas: 740823 = $6.22) - 11% increase
* test_verify_700_of_1000() (gas: 1046398 = $8.78) - 57% increase
The gas cost scales reasonably across different numbers of signers, making the solution practical for real-world use cases. The gas cost scales reasonably across different numbers of signers, making the solution practical for real-world use cases.

94
src/Verifier.sol
View File

@ -27,8 +27,6 @@ abstract contract GhostVerifier {
bytes calldata proof, bytes calldata proof,
bytes calldata missedIndexes bytes calldata missedIndexes
) internal view returns (uint256 res) { ) internal view returns (uint256 res) {
require(s < GhostEllipticCurves.N); // scalar check
uint256 px = pubkeyX; uint256 px = pubkeyX;
uint256 py = pubkeyY; uint256 py = pubkeyY;
uint256 e; uint256 e;
@ -36,22 +34,21 @@ abstract contract GhostVerifier {
uint256 ry; uint256 ry;
{ {
uint256[] memory rNonces = new uint256[](4);
uint256[] memory coefficients = new uint256[](2); uint256[] memory coefficients = new uint256[](2);
{ _reconstructNonces(rNonces, nonces);
uint256[] memory rNonces = new uint256[](4); _computeCoefficients(px, m, rNonces, coefficients);
_reconstructNonces(rNonces, nonces);
_computeCoefficients(px, m, rNonces, coefficients);
(rx, ry) = _aggregateNonce(rNonces, coefficients);
}
(rx, ry) = _aggregateNonce(rNonces, coefficients);
e = _computeChallenge(bytes32(rx), bytes32(px), m); e = _computeChallenge(bytes32(rx), bytes32(px), m);
require(e < GhostEllipticCurves.N); // scalar check
(rx, ry) = _restoreAdaptiveNonce(nonces, coefficients); (rx, ry) = _restoreAdaptiveNonce(nonces, coefficients);
} }
require(signersHash == sha256(proof)); // check proof correctness {
(px, py) = _aggregatePubkey(px, py, proof, missedIndexes); require(signersHash == sha256(proof)); // check proof correctness
(px, py) = _aggregatePubkey(px, py, proof, missedIndexes);
}
unchecked { unchecked {
s = GhostEllipticCurves.N - mulmod(s, px, GhostEllipticCurves.N); s = GhostEllipticCurves.N - mulmod(s, px, GhostEllipticCurves.N);
@ -151,6 +148,81 @@ abstract contract GhostVerifier {
return (r1x, r1y); return (r1x, r1y);
} }
function _computeAggregationCoefficients(
uint16 length,
bytes32[] memory ais,
bytes calldata proof
) internal pure {
uint16 i = length;
for (; i > 0;) {
unchecked { --i; }
uint256 pix;
uint16 l;
uint16 r;
assembly {
let base := add(proof.offset, mul(i, 128))
pix := calldataload(base)
l := add(shl(1, i), 1)
r := add(l, 1)
}
ais[i] = bytes32(_computeCoefficientKeyAgg(
l < length ? ais[l] : bytes32(0x0),
r < length ? ais[r] : bytes32(0x0),
bytes32(pix)
));
}
}
function _checkAggregationCorrectness(
uint256 xx, uint256 yy, uint256 ai,
bytes calldata proof
) internal pure returns (uint256 res) {
uint256 px;
uint256 py;
uint256 pz;
uint16 i;
for (; i < 3;) {
uint256 x;
uint256 y;
assembly {
let base := add(proof.offset, mul(i, 128))
let j := mul(gt(i, 0), 64)
x := calldataload(add(base, j))
y := calldataload(add(base, add(32, j)))
}
if (i == 0) {
(px, py, pz) = GhostEllipticCurves.mulAddAffineSingle(x, y, ai);
} else {
(px, py, pz) = GhostEllipticCurves.projectiveAddMixed(px, py, pz, x, y);
}
unchecked { ++i; }
}
(px, py) = GhostEllipticCurves.toAffine(px, py, pz);
uint256 hix;
assembly {
let base := proof.offset
hix := calldataload(add(base, 64))
let hiy := calldataload(add(base, 96))
if iszero(and(eq(xx, px), eq(xx, hix))) {
revert(0, 0)
}
if iszero(and(eq(yy, py), eq(yy, hiy))) {
revert(0, 0)
}
}
}
function _aggregatePubkey( function _aggregatePubkey(
uint256 px, uint256 py, uint256 px, uint256 py,
bytes calldata proof, bytes calldata proof,

118
src/libraries/GhostEllipticCurves.sol
View File

@ -147,20 +147,18 @@ library GhostEllipticCurves {
// Y3 = (Y^2 − 9bZ^2)(Y^2 + 3bZ^2) + 24bY^2Z^2, // Y3 = (Y^2 − 9bZ^2)(Y^2 + 3bZ^2) + 24bY^2Z^2,
// Z3 = 8Y^3Z // Z3 = 8Y^3Z
assembly { uint256 t0 = mulmod(Y, Y, P); // 1. t0 ← Y · Y => (Y²)
let t0 := mulmod(Y, Y, P) // 1. t0 ← Y · Y => (Y²) Z3 = mulmod(8, t0, P); // 2. Z3 ← t0 + t0 => (8Y²)
Z3 := mulmod(8, t0, P) // 2. Z3 ← t0 + t0 => (8Y²) uint256 t1 = mulmod(Y, Z, P); // 3. t1 ← Y · Z => (YZ)
let t1 := mulmod(Y, Z, P) // 3. t1 ← Y · Z => (YZ) uint256 t2 = mulmod(Z, Z, P); // 4. t2 ← Z · Z => (Z²)
let t2 := mulmod(Z, Z, P) // 4. t2 ← Z · Z => (Z²) t2 = mulmod(21, t2, P); // 5. t2 ← b3 · t2 => (3bZ²)
t2 := mulmod(21, t2, P) // 5. t2 ← b3 · t2 => (3bZ²) X3 = mulmod(t2, Z3, P); // 6. X3 ← t2 · Z3 => (3bZ²8Y²)
X3 := mulmod(t2, Z3, P) // 6. X3 ← t2 · Z3 => (3bZ²8Y²) Y3 = addmod(t0, t2, P); // 7. Y3 ← t0 + t2 => (Y² + 3bZ²)
Y3 := addmod(t0, t2, P) // 7. Y3 ← t0 + t2 => (Y² + 3bZ²) Z3 = mulmod(t1, Z3, P); // 8. Z3 ← t1 · Z3 => (YZ · 8Y² = 8Y³Z)
Z3 := mulmod(t1, Z3, P) // 8. Z3 ← t1 · Z3 => (YZ · 8Y² = 8Y³Z) t1 = addmod(t0, P - mulmod(3, t2, P), P); // 9. t1 ← t0 - (3 · t2) => (Y² - 9bZ²)
t1 := addmod(t0, sub(P, mulmod(3, t2, P)), P) // 9. t1 ← t0 - (3 · t2) => (Y² - 9bZ²) Y3 = addmod(X3, mulmod(t1, Y3, P), P); // 10. Y3 ← t1 · (t1 · Y3) => ((Y² - 9bZ²) · (Y² + 3bZ²))
Y3 := addmod(X3, mulmod(t1, Y3, P), P) // 10. Y3 ← t1 · (t1 · Y3) => ((Y² - 9bZ²) · (Y² + 3bZ²)) X3 = mulmod(t1, mulmod(X, Y, P), P); // 11. X3 ← t1 · (X1 · Y1) => ((Y² - 9bZ²) · XY)
X3 := mulmod(t1, mulmod(X, Y, P), P) // 11. X3 ← t1 · (X1 · Y1) => ((Y² - 9bZ²) · XY) X3 = addmod(X3, X3, P); // 12. X3 ← X3 + X3 => ((Y² - 9bZ²) · 2XY)
X3 := addmod(X3, X3, P) // 12. X3 ← X3 + X3 => ((Y² - 9bZ²) · 2XY
}
} }
function projectiveAddMixed( function projectiveAddMixed(
@ -219,63 +217,66 @@ library GhostEllipticCurves {
// Y3 = (Y1Y2 + 3bZ1Z2)(Y1Y2 − 3bZ1Z2) + 9bX1X2(X1Z2 + X2Z1), // Y3 = (Y1Y2 + 3bZ1Z2)(Y1Y2 − 3bZ1Z2) + 9bX1X2(X1Z2 + X2Z1),
// Z3 = (Y1Z2 + Y2Z1)(Y1Y2 + 3bZ1Z2) + 3X1X2(X1Y2 + X2Y1), // Z3 = (Y1Z2 + Y2Z1)(Y1Y2 + 3bZ1Z2) + 3X1X2(X1Y2 + X2Y1),
uint256 t0 = mulmod(X1, X2, P); // 1. t0 ← X1 · X2 => (X1·X2) uint256 t0 = mulmod(X1, X2, P); // 1. t0 ← X1 · X2 => (X1·X2)
uint256 t1 = mulmod(Y1, Y2, P); // 2. t1 ← Y1 · Y2 => (Y1·Y2) uint256 t1 = mulmod(Y1, Y2, P); // 2. t1 ← Y1 · Y2 => (Y1·Y2)
uint256 t2 = mulmod(Z1, Z2, P); // 3. t2 ← Z1 · Z2 => (Z1·Z2) uint256 t2 = mulmod(Z1, Z2, P); // 3. t2 ← Z1 · Z2 => (Z1·Z2)
uint256 t3 = addmod(X1, Y1, P); // 4. t3 ← X1 + Y1 => (X1 + Y1) uint256 t3 = addmod(X1, Y1, P); // 4. t3 ← X1 + Y1 => (X1 + Y1)
uint256 t4 = addmod(X2, Y2, P); // 5. t4 ← X2 + Y2 => (X2 + Y2) uint256 t4 = addmod(X2, Y2, P); // 5. t4 ← X2 + Y2 => (X2 + Y2)
t3 = mulmod(t3, t4, P); // 6. t3 ← t3 · t4 => ((X1 + Y1) · (X2 + Y2)) t3 = mulmod(t3, t4, P); // 6. t3 ← t3 · t4 => ((X1 + Y1) · (X2 + Y2))
t4 = addmod(t0, t1, P); // 7. t4 ← t0 + t1 => (X1·X2 + Y1·Y2) t4 = addmod(t0, t1, P); // 7. t4 ← t0 + t1 => (X1·X2 + Y1·Y2)
t3 = addmod(t3, P - t4, P); // 8. t3 ← t3 - t4 => ((X1 + Y1)·(X2 + Y2) - X1·X2 - Y1·Y2) t3 = addmod(t3, P - t4, P); // 8. t3 ← t3 - t4 => ((X1 + Y1)·(X2 + Y2) - X1·X2 - Y1·Y2)
t4 = addmod(Y1, Z1, P); // 9. t4 ← Y1 + Z1 => (Y1 + Z1) t4 = addmod(Y1, Z1, P); // 9. t4 ← Y1 + Z1 => (Y1 + Z1)
X3 = addmod(Y2, Z2, P); // 10. X3 ← Y2 + Z2 => (Y2 + Z2) X3 = addmod(Y2, Z2, P); // 10. X3 ← Y2 + Z2 => (Y2 + Z2)
t4 = mulmod(t4, X3, P); // 11. t4 ← t4 · X3 => ((Y1 + Z1) · (Y2 + Z2)) t4 = mulmod(t4, X3, P); // 11. t4 ← t4 · X3 => ((Y1 + Z1) · (Y2 + Z2))
X3 = addmod(t1, t2, P); // 12. X3 ← t1 + t2 => (Y1·Y2 + Z1·Z2) X3 = addmod(t1, t2, P); // 12. X3 ← t1 + t2 => (Y1·Y2 + Z1·Z2)
t4 = addmod(t4, P - X3, P); // 13. t4 ← t4 - X3 => ((Y1 + Z1)·(Y2 + Z2) - Y1·Y2 - Z1·Z2) t4 = addmod(t4, P - X3, P); // 13. t4 ← t4 - X3 => ((Y1 + Z1)·(Y2 + Z2) - Y1·Y2 - Z1·Z2)
X3 = addmod(X1, Z1, P); // 14. X3 ← X1 + Z1 => (X1 + Z1) X3 = addmod(X1, Z1, P); // 14. X3 ← X1 + Z1 => (X1 + Z1)
Y3 = addmod(X2, Z2, P); // 15. Y3 ← X2 + Z2 => (X2 + Z2) Y3 = addmod(X2, Z2, P); // 15. Y3 ← X2 + Z2 => (X2 + Z2)
X3 = mulmod(X3, Y3, P); // 16. X3 ← X3 · Y3 => ((X1 + Z1) · (X2 + Z2)) X3 = mulmod(X3, Y3, P); // 16. X3 ← X3 · Y3 => ((X1 + Z1) · (X2 + Z2))
Y3 = addmod(t0, t2, P); // 17. Y3 ← t0 + t2 => (X1·X2 + Z1·Z2) Y3 = addmod(t0, t2, P); // 17. Y3 ← t0 + t2 => (X1·X2 + Z1·Z2)
Y3 = addmod(X3, P - Y3, P); // 18. Y3 ← X3 - Y3 => ((X1 + Z1)·(X2 + Z2) - X1·X2 - Z1·Z2) Y3 = addmod(X3, P - Y3, P); // 18. Y3 ← X3 - Y3 => ((X1 + Z1)·(X2 + Z2) - X1·X2 - Z1·Z2)
X3 = addmod(t0, t0, P); // 19. X3 ← t0 + t0 => (2·X1·X2) X3 = addmod(t0, t0, P); // 19. X3 ← t0 + t0 => (2·X1·X2)
t0 = addmod(X3, t0, P); // 20. t0 ← X3 + t0 => (3·X1·X2) t0 = addmod(X3, t0, P); // 20. t0 ← X3 + t0 => (3·X1·X2)
t2 = mulmod(21, t2, P); // 21. t2 ← B3 · t2 => 3b · Z1·Z2 t2 = mulmod(21, t2, P); // 21. t2 ← B3 · t2 => 3b · Z1·Z2
Z3 = addmod(t1, t2, P); // 22. Z3 ← t1 + t2 => Y1·Y2 + 3·b·Z1·Z2 Z3 = addmod(t1, t2, P); // 22. Z3 ← t1 + t2 => Y1·Y2 + 3·b·Z1·Z2
t1 = addmod(t1, P - t2, P); // 23. t1 ← t1 - t2 => Y1·Y2 - 3·b·Z1·Z2 t1 = addmod(t1, P - t2, P); // 23. t1 ← t1 - t2 => Y1·Y2 - 3·b·Z1·Z2
Y3 = mulmod(21, Y3, P); // 24. Y3 ← B3 · Y3 => 3b · ((X1+Z1)(X2+Z2) - X1·X2 - Z1·Z2) Y3 = mulmod(21, Y3, P); // 24. Y3 ← B3 · Y3 => 3b · ((X1+Z1)(X2+Z2) - X1·X2 - Z1·Z2)
X3 = mulmod(t4, Y3, P); // 25. X3 ← t4 · Y3 => 3b·((Y1+Z1)(Y2+Z2)-Y1·Y2-Z1·Z2) · ((X1+Z1)(X2+Z2)-X1·X2-Z1·Z2) X3 = mulmod(t4, Y3, P); // 25. X3 ← t4 · Y3 => 3b·((Y1+Z1)(Y2+Z2)-Y1·Y2-Z1·Z2) · ((X1+Z1)(X2+Z2)-X1·X2-Z1·Z2)
t2 = mulmod(t3, t1, P); // 26. t2 ← t3 · t1 => ((X1+Y1)(X2+Y2)-X1·X2-Y1·Y2) · (Y1·Y2 - 3·b·Z1·Z2) t2 = mulmod(t3, t1, P); // 26. t2 ← t3 · t1 => ((X1+Y1)(X2+Y2)-X1·X2-Y1·Y2) · (Y1·Y2 - 3·b·Z1·Z2)
X3 = addmod(t2, P - X3, P); // 27. X3 ← t2 - X3 => (X1Y2+X2Y1)(Y1·Y2-3·b·Z1·Z2) - 3b(Y1·Z2+Y2·Z1)(X1·Z2+X2·Z1) X3 = addmod(t2, P - X3, P); // 27. X3 ← t2 - X3 => (X1Y2+X2Y1)(Y1·Y2-3·b·Z1·Z2) - 3b(Y1·Z2+Y2·Z1)(X1·Z2+X2·Z1)
Y3 = mulmod(Y3, t0, P); // 28. Y3 ← Y3 · t0 => 3·b·((X1+Z1)(X2+Z2) - X1·X2 - Z1·Z2) · 3·X1·X2 = 9·b·X1·X2·(X1·Z2+X2·Z1) Y3 = mulmod(Y3, t0, P); // 28. Y3 ← Y3 · t0 => 3·b·((X1+Z1)(X2+Z2) - X1·X2 - Z1·Z2) · 3·X1·X2 = 9·b·X1·X2·(X1·Z2+X2·Z1)
t1 = mulmod(t1, Z3, P); // 29. t1 ← t1 · Z3 => (Y1·Y2-3·b·Z1·Z2) · (Y1·Y2+3·b·Z1·Z2) t1 = mulmod(t1, Z3, P); // 29. t1 ← t1 · Z3 => (Y1·Y2-3·b·Z1·Z2) · (Y1·Y2+3·b·Z1·Z2)
Y3 = addmod(t1, Y3, P); // 30. Y3 ← t1 + Y3 => (Y1Y2+3·b·Z1·Z2)(Y1·Y2-3·b·Z1·Z2) + 9b·X1·X2·(X1·Z2+X2·Z1) Y3 = addmod(t1, Y3, P); // 30. Y3 ← t1 + Y3 => (Y1Y2+3·b·Z1·Z2)(Y1·Y2-3·b·Z1·Z2) + 9b·X1·X2·(X1·Z2+X2·Z1)
t0 = mulmod(t0, t3, P); // 31. t0 ← t0 · t3 => (3·X1·X2) · ((X1+Y1)(X2+Y2)-X1·X2-Y1·Y2) t0 = mulmod(t0, t3, P); // 31. t0 ← t0 · t3 => (3·X1·X2) · ((X1+Y1)(X2+Y2)-X1·X2-Y1·Y2)
Z3 = mulmod(Z3, t4, P); // 32. Z3 ← Z3 · t4 => (Y1·Y2+3·b·Z1·Z2) · ((Y1+Z1)(Y2+Z2)-Y1·Y2-Z1·Z2) Z3 = mulmod(Z3, t4, P); // 32. Z3 ← Z3 · t4 => (Y1·Y2+3·b·Z1·Z2) · ((Y1+Z1)(Y2+Z2)-Y1·Y2-Z1·Z2)
Z3 = addmod(Z3, t0, P); // 33. Z3 ← Z3 + t0 => (Y1·Z2+Y2·Z1)·(Y1·Y2+3·b·Z1·Z2) + 3·X1·X2·(X1·Y2+X2·Y1) Z3 = addmod(Z3, t0, P); // 33. Z3 ← Z3 + t0 => (Y1·Z2+Y2·Z1)·(Y1·Y2+3·b·Z1·Z2) + 3·X1·X2·(X1·Y2+X2·Y1)
} }
function isOnCurve(uint256 x, uint256 y) internal pure returns (bool result) { function isOnCurve(uint256 x, uint256 y) internal pure returns (bool) {
assembly { uint lhs = mulmod(y, y, P); // y^2
let lhs := mulmod(y, y, P) uint rhs = mulmod(mulmod(x, x, P), x, P); // x^3
let rhs := mulmod(mulmod(x, x, P), x, P) rhs = addmod(rhs, B, P); // x^3 + 7
rhs := addmod(rhs, B, P) return lhs == rhs;
result := eq(lhs, rhs)
}
} }
function toAffine( function toAffine(
uint256 x, uint256 x,
uint256 y, uint256 y,
uint256 z uint256 z
) internal pure returns (uint256 x_, uint256 y_) { ) internal pure returns (uint256, uint256) {
uint256 t0 = modInverse(z);
x = mulmod(x, t0, P);
y = mulmod(y, t0, P);
return (x, y);
}
function modInverse(uint256 r1) internal pure returns (uint256 t0) {
// Extended Euclidean algorithm (iterative, assembly) // Extended Euclidean algorithm (iterative, assembly)
// Typically lower average gas than the modexp precompile for single inversions, // Typically lower average gas than the modexp precompile for single inversions,
// but execution time (and gas) depends on input values — not constant-time. // but execution time (and gas) depends on input values — not constant-time.
assembly { assembly {
let t0 := 0
let t1 := 1 let t1 := 1
let r0 := P let r0 := P
let r1 := z
for {} r1 {} { for {} r1 {} {
let q := div(r0, r1) let q := div(r0, r1)
@ -294,9 +295,6 @@ library GhostEllipticCurves {
if slt(t0, 0) { if slt(t0, 0) {
t0 := add(t0, P) t0 := add(t0, P)
} }
x_ := mulmod(x, t0, P)
y_ := mulmod(y, t0, P)
} }
} }
} }