g2IsInSubGroup uses that the naive metod the verify that point Q is from G2 subgroup. I can be optimized by using the Frobenius endomorphism. According to the blogpost and quoted there paper (Example 3.1.2) and keeping all computations in Jacobian coordinates we cen reduce significantly the cost of this step.
Recommendation: Implement g2IsInSubGroup accordingly to mentioned paper.
diff --git a/precompiles/EcPairing.yul b/precompiles/EcPairing.yul
index d9e08c5..b64e1f2 100644
--- a/precompiles/EcPairing.yul
+++ b/precompiles/EcPairing.yul
@@ -66,6 +66,13 @@ object "EcPairing" {
ret := 111032442853175714102588374283752698368366046808579839647964533820976443843465
}
+ /// @notice Constant function for the alt_bn128 curve seed (parameter `x`).
+ /// @dev See https://eips.ethereum.org/EIPS/eip-196 for further details.
+ /// @return ret The alt_bn128 curve seed.
+ function X() -> ret {
+ ret := 4965661367192848881
+ }
+
/// @notice Constant function for decimal representation of the NAF for the Millers Loop.
/// @dev Millers loop uses to iterate the NAF representation of the value t = 6x^2 + 1. Where x = 4965661367192848881 is a parameter of the BN 256 curve.
/// @dev For details of the x parameter: https://hackmd.io/@jpw/bn254#Barreto-Naehrig-curves.
@@ -474,6 +481,40 @@ object "EcPairing" {
ny0, ny1 := fp2Neg(y0, y1)
}
+ /// @notice Constant function for the alt_bn128 returning `(xi)^ ((N - 1) // 2)`. Where `xi` is D-type twist param.
+ /// @dev See https://eprint.iacr.org/2022/352.pdf (2 Preliminaries) for further details.
+ /// @return ret Twisted curve `xi2 = (xi)^ ((N - 1) // 2)` value in Montgomery form.
+ function xi2() -> xi0, xi1 {
+ xi0 := intoMontgomeryForm(2821565182194536844548159561693502659359617185244120367078079554186484126554)
+ xi1 := intoMontgomeryForm(3505843767911556378687030309984248845540243509899259641013678093033130930403)
+ }
+
+ /// @notice Constant function for the alt_bn128 returning `(xi)^ ((N - 1) // 3)`. Where `xi` is D-type twist param.
+ /// @dev See https://eprint.iacr.org/2022/352.pdf (2 Preliminaries) for further details.
+ /// @return ret Twisted curve `xi3 = (xi)^ ((N - 1) // 3)` value in Montgomery form.
+ function xi3() -> xi0, xi1 {
+ xi0 := intoMontgomeryForm(21575463638280843010398324269430826099269044274347216827212613867836435027261)
+ xi1 := intoMontgomeryForm(10307601595873709700152284273816112264069230130616436755625194854815875713954)
+ }
+
+ /// @notice Frobenius endomophism used to G2 sub group check for twisted curve.
+ /// @dev For more datail see https://eprint.iacr.org/2022/348.pdf
+ /// @param xp0, xp1 The x coordinate of the point on twisted curve.
+ /// @param yp0, yp1 The y coordinate of the point on twisted curve.
+ /// @param zp0, zp1 The z coordinate of the point on twisted curve.
+ /// @return Point on twisted curve transformed by the phi endomorphism
+ function endomorphism(xp0, xp1, yp0, yp1, zp0, zp1) -> xr0, xr1, yr0, yr1, zr0, zr1 {
+ let xp0_c, xp1_c := fp2Conjugate(xp0, xp1)
+ let yp0_c, yp1_c := fp2Conjugate(yp0, yp1)
+
+ let xi2_0, xi2_1 := xi2()
+ let xi3_0, xi3_1 := xi3()
+
+ xr0, xr1 := fp2Mul(xp0_c, xp1_c, xi3_0, xi3_1)
+ yr0, yr1 := fp2Mul(yp0_c, yp1_c, xi2_0, xi2_1)
+ zr0, zr1 := fp2Conjugate(zp0, zp1)
+ }
+
/// @notice Check if a G2 point in jacobian coordinates is in the subgroup of the twisted curve.
/// @dev The coordinates are encoded in Montgomery form.
/// @param xp0, xp1 The x coordinate of the point.
@@ -481,6 +522,44 @@ object "EcPairing" {
/// @param zp0, zp1 The z coordinate of the point.
/// @return ret True if the point is in the subgroup, false otherwise.
function g2IsInSubGroup(xp0, xp1, yp0, yp1, zp0, zp1) -> ret {
+ // P * X
+ let px_xp0, px_xp1, px_yp0, px_yp1, px_zp0, px_zp1 := g2ScalarMul(xp0, xp1, yp0, yp1, zp0, zp1, X())
+ // P * (X + 1)
+ let px1_xp0, px1_xp1, px1_yp0, px1_yp1, px1_zp0, px1_zp1 := g2JacobianAdd(px_xp0, px_xp1, px_yp0, px_yp1, px_zp0, px_zp1, xp0, xp1, yp0, yp1, zp0, zp1)
+ // P * 2X
+ let p2x_xp0, p2x_xp1, p2x_yp0, p2x_yp1, p2x_zp0, p2x_zp1 := g2JacobianDouble(px_xp0, px_xp1, px_yp0, px_yp1, px_zp0, px_zp1)
+
+ // phi(P * X)
+ let e_px_xp0, e_px_xp1, e_px_yp0, e_px_yp1, e_px_zp0, e_px_zp1 := endomorphism(px_xp0, px_xp1, px_yp0, px_yp1, px_zp0, px_zp1)
+ // phi(phi(P * X))
+ let e2_px_xp0, e2_px_xp1, e2_px_yp0, e2_px_yp1, e2_px_zp0, e2_px_zp1 := endomorphism(e_px_xp0, e_px_xp1, e_px_yp0, e_px_yp1, e_px_zp0, e_px_zp1)
+
+ // phi(phi(phi(P * 2X)))
+ p2x_xp0, p2x_xp1, p2x_yp0, p2x_yp1, p2x_zp0, p2x_zp1 := endomorphism(p2x_xp0, p2x_xp1, p2x_yp0, p2x_yp1, p2x_zp0, p2x_zp1)
+ p2x_xp0, p2x_xp1, p2x_yp0, p2x_yp1, p2x_zp0, p2x_zp1 := endomorphism(p2x_xp0, p2x_xp1, p2x_yp0, p2x_yp1, p2x_zp0, p2x_zp1)
+ p2x_xp0, p2x_xp1, p2x_yp0, p2x_yp1, p2x_zp0, p2x_zp1 := endomorphism(p2x_xp0, p2x_xp1, p2x_yp0, p2x_yp1, p2x_zp0, p2x_zp1)
+
+ let l1x0, l1x2, l1y0, l1y2, l1z0, l1z2 := g2JacobianAdd(px1_xp0, px1_xp1, px1_yp0, px1_yp1, px1_zp0, px1_zp1, e_px_xp0, e_px_xp1, e_px_yp0, e_px_yp1, e_px_zp0, e_px_zp1)
+ l1x0, l1x2, l1y0, l1y2, l1z0, l1z2 := g2JacobianAdd(l1x0, l1x2, l1y0, l1y2, l1z0, l1z2, e2_px_xp0, e2_px_xp1, e2_px_yp0, e2_px_yp1, e2_px_zp0, e2_px_zp1)
+
+ let l1z0_square, l1z2_square := fp2Mul(l1z0, l1z2, l1z0, l1z2)
+ let p2x_zp0_square, p2x_zp1_square := fp2Mul(p2x_zp0, p2x_zp1, p2x_zp0, p2x_zp1)
+
+ let r00, r01 := fp2Mul(p2x_xp0, p2x_xp1, l1z0_square, l1z2_square)
+ let r10, r11 := fp2Mul(l1x0, l1x2, p2x_zp0_square, p2x_zp1_square)
+
+ let l1z0_cube, l1z2_cube := fp2Mul(l1z0_square, l1z2_square, l1z0, l1z2)
+ let p2x_zp0_cube, p2x_zp1_cube := fp2Mul(p2x_zp0_square, p2x_zp1_square, p2x_zp0, p2x_zp1)
+
+ let l00, l01 := fp2Mul(p2x_yp0, p2x_yp1, l1z0_cube, l1z2_cube)
+ let l10, l11 := fp2Mul(l1y0, l1y2, p2x_zp0_cube, p2x_zp1_cube)
+
+ let r1 := and(eq(r00, r10), eq(r01, r11))
+ let r2 := and(eq(l00, l10), eq(l01, l11))
+ ret := and(r1, r2)
+ }
+
+ function g2IsInSubGroupNaive(xp0, xp1, yp0, yp1, zp0, zp1) -> ret {
let xr0, xr1, yr0, yr1, zr0, zr1 := g2ScalarMul(xp0, xp1, yp0, yp1, zp0, zp1, TWISTED_SUBGROUP_ORDER())
ret := and(iszero(zr0), iszero(zr1))
}
Gas usage summary for ecpairing_two_point_match_3 unit test:
jrchatruc
commented
2 months ago
Context: EcPairing.yul#L457
Description:
g2IsInSubGroupuses that the naive metod the verify that pointQis fromG2subgroup. I can be optimized by using the Frobenius endomorphism. According to the blogpost and quoted there paper (Example 3.1.2) and keeping all computations in Jacobian coordinates we cen reduce significantly the cost of this step.Recommendation: Implement
g2IsInSubGroupaccordingly to mentioned paper.Gas usage summary for
ecpairing_two_point_match_3unit test:zkSync:
Spearbit: