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FflonkVerifier.sol
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FflonkVerifier.sol
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// SPDX-License-Identifier: GPL-3.0
/*
Copyright 2021 0KIMS association.
This file is generated with [snarkJS](https://github.com/iden3/snarkjs).
snarkJS is a free software: you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
snarkJS is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
License for more details.
You should have received a copy of the GNU General Public License
along with snarkJS. If not, see <https://www.gnu.org/licenses/>.
*/
pragma solidity >=0.7.0 <0.9.0;
contract FflonkVerifier {
uint32 constant n = 16777216; // Domain size
// Verification Key data
uint256 constant k1 = 2; // Plonk k1 multiplicative factor to force distinct cosets of H
uint256 constant k2 = 3; // Plonk k2 multiplicative factor to force distinct cosets of H
// OMEGAS
// Omega, Omega^{1/3}
uint256 constant w1 = 5709868443893258075976348696661355716898495876243883251619397131511003808859;
uint256 constant wr = 18200100796661656210024324131237448517259556535315737226009542456080026430510;
// Omega_3, Omega_3^2
uint256 constant w3 = 21888242871839275217838484774961031246154997185409878258781734729429964517155;
uint256 constant w3_2 = 4407920970296243842393367215006156084916469457145843978461;
// Omega_4, Omega_4^2, Omega_4^3
uint256 constant w4 = 21888242871839275217838484774961031246007050428528088939761107053157389710902;
uint256 constant w4_2 = 21888242871839275222246405745257275088548364400416034343698204186575808495616;
uint256 constant w4_3 = 4407920970296243842541313971887945403937097133418418784715;
// Omega_8, Omega_8^2, Omega_8^3, Omega_8^4, Omega_8^5, Omega_8^6, Omega_8^7
uint256 constant w8_1 = 19540430494807482326159819597004422086093766032135589407132600596362845576832;
uint256 constant w8_2 = 21888242871839275217838484774961031246007050428528088939761107053157389710902;
uint256 constant w8_3 = 13274704216607947843011480449124596415239537050559949017414504948711435969894;
uint256 constant w8_4 = 21888242871839275222246405745257275088548364400416034343698204186575808495616;
uint256 constant w8_5 = 2347812377031792896086586148252853002454598368280444936565603590212962918785;
uint256 constant w8_6 = 4407920970296243842541313971887945403937097133418418784715;
uint256 constant w8_7 = 8613538655231327379234925296132678673308827349856085326283699237864372525723;
// Verifier preprocessed input C_0(x)·[1]_1
uint256 constant C0x = 10330861150616913541207360924312278787207684679419670167996002791305440821704;
uint256 constant C0y = 14244524222232642134204580826860575549922689127759263461569792153452315036880;
// Verifier preprocessed input x·[1]_2
uint256 constant X2x1 = 21831381940315734285607113342023901060522397560371972897001948545212302161822;
uint256 constant X2x2 = 17231025384763736816414546592865244497437017442647097510447326538965263639101;
uint256 constant X2y1 = 2388026358213174446665280700919698872609886601280537296205114254867301080648;
uint256 constant X2y2 = 11507326595632554467052522095592665270651932854513688777769618397986436103170;
// Scalar field size
uint256 constant q = 21888242871839275222246405745257275088548364400416034343698204186575808495617;
// Base field size
uint256 constant qf = 21888242871839275222246405745257275088696311157297823662689037894645226208583;
// [1]_1
uint256 constant G1x = 1;
uint256 constant G1y = 2;
// [1]_2
uint256 constant G2x1 = 10857046999023057135944570762232829481370756359578518086990519993285655852781;
uint256 constant G2x2 = 11559732032986387107991004021392285783925812861821192530917403151452391805634;
uint256 constant G2y1 = 8495653923123431417604973247489272438418190587263600148770280649306958101930;
uint256 constant G2y2 = 4082367875863433681332203403145435568316851327593401208105741076214120093531;
// Proof calldata
// Byte offset of every parameter of the calldata
// Polynomial commitments
uint16 constant pC1 = 4 + 0; // [C1]_1
uint16 constant pC2 = 4 + 32*2; // [C2]_1
uint16 constant pW1 = 4 + 32*4; // [W]_1
uint16 constant pW2 = 4 + 32*6; // [W']_1
// Opening evaluations
uint16 constant pEval_ql = 4 + 32*8; // q_L(xi)
uint16 constant pEval_qr = 4 + 32*9; // q_R(xi)
uint16 constant pEval_qm = 4 + 32*10; // q_M(xi)
uint16 constant pEval_qo = 4 + 32*11; // q_O(xi)
uint16 constant pEval_qc = 4 + 32*12; // q_C(xi)
uint16 constant pEval_s1 = 4 + 32*13; // S_{sigma_1}(xi)
uint16 constant pEval_s2 = 4 + 32*14; // S_{sigma_2}(xi)
uint16 constant pEval_s3 = 4 + 32*15; // S_{sigma_3}(xi)
uint16 constant pEval_a = 4 + 32*16; // a(xi)
uint16 constant pEval_b = 4 + 32*17; // b(xi)
uint16 constant pEval_c = 4 + 32*18; // c(xi)
uint16 constant pEval_z = 4 + 32*19; // z(xi)
uint16 constant pEval_zw = 4 + 32*20; // z_omega(xi)
uint16 constant pEval_t1w = 4 + 32*21; // T_1(xi omega)
uint16 constant pEval_t2w = 4 + 32*22; // T_2(xi omega)
uint16 constant pEval_inv = 4 + 32*23; // inv(batch) sent by the prover to avoid any inverse calculation to save gas,
// we check the correctness of the inv(batch) by computing batch
// and checking inv(batch) * batch == 1
// Memory data
// Challenges
uint16 constant pAlpha = 0; // alpha challenge
uint16 constant pBeta = 32; // beta challenge
uint16 constant pGamma = 64; // gamma challenge
uint16 constant pY = 96; // y challenge
uint16 constant pXiSeed = 128; // xi seed, from this value we compute xi = xiSeed^24
uint16 constant pXiSeed2 = 160; // (xi seed)^2
uint16 constant pXi = 192; // xi challenge
// Roots
// S_0 = roots_8(xi) = { h_0, h_0w_8, h_0w_8^2, h_0w_8^3, h_0w_8^4, h_0w_8^5, h_0w_8^6, h_0w_8^7 }
uint16 constant pH0w8_0 = 224;
uint16 constant pH0w8_1 = 256;
uint16 constant pH0w8_2 = 288;
uint16 constant pH0w8_3 = 320;
uint16 constant pH0w8_4 = 352;
uint16 constant pH0w8_5 = 384;
uint16 constant pH0w8_6 = 416;
uint16 constant pH0w8_7 = 448;
// S_1 = roots_4(xi) = { h_1, h_1w_4, h_1w_4^2, h_1w_4^3 }
uint16 constant pH1w4_0 = 480;
uint16 constant pH1w4_1 = 512;
uint16 constant pH1w4_2 = 544;
uint16 constant pH1w4_3 = 576;
// S_2 = roots_3(xi) U roots_3(xi omega)
// roots_3(xi) = { h_2, h_2w_3, h_2w_3^2 }
uint16 constant pH2w3_0 = 608;
uint16 constant pH2w3_1 = 640;
uint16 constant pH2w3_2 = 672;
// roots_3(xi omega) = { h_3, h_3w_3, h_3w_3^2 }
uint16 constant pH3w3_0 = 704;
uint16 constant pH3w3_1 = 736;
uint16 constant pH3w3_2 = 768;
uint16 constant pPi = 800; // PI(xi)
uint16 constant pR0 = 832; // r0(y)
uint16 constant pR1 = 864; // r1(y)
uint16 constant pR2 = 896; // r2(y)
uint16 constant pF = 928; // [F]_1, 64 bytes
uint16 constant pE = 992; // [E]_1, 64 bytes
uint16 constant pJ = 1056; // [J]_1, 64 bytes
uint16 constant pZh = 1184; // Z_H(xi)
// From this point we write all the variables that must be computed using the Montgomery batch inversion
uint16 constant pZhInv = 1216; // 1/Z_H(xi)
uint16 constant pDenH1 = 1248; // 1/( (y-h_1w_4) (y-h_1w_4^2) (y-h_1w_4^3) (y-h_1w_4^4) )
uint16 constant pDenH2 = 1280; // 1/( (y-h_2w_3) (y-h_2w_3^2) (y-h_2w_3^3) (y-h_3w_3) (y-h_3w_3^2) (y-h_3w_3^3) )
uint16 constant pLiS0Inv = 1312; // Reserve 8 * 32 bytes to compute r_0(X)
uint16 constant pLiS1Inv = 1568; // Reserve 4 * 32 bytes to compute r_1(X)
uint16 constant pLiS2Inv = 1696; // Reserve 6 * 32 bytes to compute r_2(X)
// Lagrange evaluations
uint16 constant pEval_l1 = 1888;
uint16 constant lastMem = 1920;
function verifyProof(bytes32[24] calldata proof, uint256[1] calldata pubSignals) public view returns (bool) {
assembly {
// Computes the inverse of an array of values
// See https://vitalik.ca/general/2018/07/21/starks_part_3.html in section where explain fields operations
// To save the inverse to be computed on chain the prover sends the inverse as an evaluation in commits.eval_inv
function inverseArray(pMem) {
let pAux := mload(0x40) // Point to the next free position
let acc := mload(add(pMem,pZhInv)) // Read the first element
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, pDenH1)), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, pDenH2)), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, pLiS0Inv)), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, add(pLiS0Inv, 32))), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, add(pLiS0Inv, 64))), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, add(pLiS0Inv, 96))), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, add(pLiS0Inv, 128))), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, add(pLiS0Inv, 160))), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, add(pLiS0Inv, 192))), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, add(pLiS0Inv, 224))), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, pLiS1Inv)), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, add(pLiS1Inv, 32))), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, add(pLiS1Inv, 64))), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, add(pLiS1Inv, 96))), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, pLiS2Inv)), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, add(pLiS2Inv, 32))), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, add(pLiS2Inv, 64))), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, add(pLiS2Inv, 96))), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, add(pLiS2Inv, 128))), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, add(pLiS2Inv, 160))), q)
mstore(pAux, acc)
pAux := add(pAux, 32)
acc := mulmod(acc, mload(add(pMem, pEval_l1)), q)
mstore(pAux, acc)
let inv := calldataload(pEval_inv)
// Before using the inverse sent by the prover the verifier checks inv(batch) * batch === 1
if iszero(eq(1, mulmod(acc, inv, q))) {
mstore(0, 0)
return(0,0x20)
}
acc := inv
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, pEval_l1)), q)
mstore(add(pMem, pEval_l1), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, add(pLiS2Inv, 160))), q)
mstore(add(pMem, add(pLiS2Inv, 160)), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, add(pLiS2Inv, 128))), q)
mstore(add(pMem, add(pLiS2Inv, 128)), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, add(pLiS2Inv, 96))), q)
mstore(add(pMem, add(pLiS2Inv, 96)), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, add(pLiS2Inv, 64))), q)
mstore(add(pMem, add(pLiS2Inv, 64)), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, add(pLiS2Inv, 32))), q)
mstore(add(pMem, add(pLiS2Inv, 32)), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, pLiS2Inv)), q)
mstore(add(pMem, pLiS2Inv), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, add(pLiS1Inv, 96))), q)
mstore(add(pMem, add(pLiS1Inv, 96)), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, add(pLiS1Inv, 64))), q)
mstore(add(pMem, add(pLiS1Inv, 64)), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, add(pLiS1Inv, 32))), q)
mstore(add(pMem, add(pLiS1Inv, 32)), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, pLiS1Inv)), q)
mstore(add(pMem, pLiS1Inv), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, add(pLiS0Inv, 224))), q)
mstore(add(pMem, add(pLiS0Inv, 224)), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, add(pLiS0Inv, 192))), q)
mstore(add(pMem, add(pLiS0Inv, 192)), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, add(pLiS0Inv, 160))), q)
mstore(add(pMem, add(pLiS0Inv, 160)), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, add(pLiS0Inv, 128))), q)
mstore(add(pMem, add(pLiS0Inv, 128)), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, add(pLiS0Inv, 96))), q)
mstore(add(pMem, add(pLiS0Inv, 96)), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, add(pLiS0Inv, 64))), q)
mstore(add(pMem, add(pLiS0Inv, 64)), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, add(pLiS0Inv, 32))), q)
mstore(add(pMem, add(pLiS0Inv, 32)), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, pLiS0Inv)), q)
mstore(add(pMem, pLiS0Inv), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, pDenH2)), q)
mstore(add(pMem, pDenH2), inv)
pAux := sub(pAux, 32)
inv := mulmod(acc, mload(pAux), q)
acc := mulmod(acc, mload(add(pMem, pDenH1)), q)
mstore(add(pMem, pDenH1), inv)
mstore(add(pMem, pZhInv), acc)
}
function checkField(v) {
if iszero(lt(v, q)) {
mstore(0, 0)
return(0, 0x20)
}
}
function checkPointBelongsToBN128Curve(p) {
let x := calldataload(p)
let y := calldataload(add(p, 32))
// Check that the point is on the curve
// y^2 = x^3 + 3
let x3_3 := addmod(mulmod(x, mulmod(x, x, qf), qf), 3, qf)
let y2 := mulmod(y, y, qf)
if iszero(eq(x3_3, y2)) {
mstore(0, 0)
return(0, 0x20)
}
}
// Validate all the evaluations sent by the prover ∈ F
function checkInput() {
// Check proof commitments fullfill bn128 curve equation Y^2 = X^3 + 3
checkPointBelongsToBN128Curve(pC1)
checkPointBelongsToBN128Curve(pC2)
checkPointBelongsToBN128Curve(pW1)
checkPointBelongsToBN128Curve(pW2)
checkField(calldataload(pEval_ql))
checkField(calldataload(pEval_qr))
checkField(calldataload(pEval_qm))
checkField(calldataload(pEval_qo))
checkField(calldataload(pEval_qc))
checkField(calldataload(pEval_s1))
checkField(calldataload(pEval_s2))
checkField(calldataload(pEval_s3))
checkField(calldataload(pEval_a))
checkField(calldataload(pEval_b))
checkField(calldataload(pEval_c))
checkField(calldataload(pEval_z))
checkField(calldataload(pEval_zw))
checkField(calldataload(pEval_t1w))
checkField(calldataload(pEval_t2w))
checkField(calldataload(pEval_inv))
// Points are checked in the point operations precompiled smart contracts
}
function computeChallenges(pMem, pPublic) {
// Compute challenge.beta & challenge.gamma
mstore(add(pMem, 1920 ), C0x)
mstore(add(pMem, 1952 ), C0y)
mstore(add(pMem, 1984), calldataload(pPublic))
mstore(add(pMem, 2016 ), calldataload(pC1))
mstore(add(pMem, 2048 ), calldataload(add(pC1, 32)))
mstore(add(pMem, pBeta), mod(keccak256(add(pMem, lastMem), 160), q))
mstore(add(pMem, pGamma), mod(keccak256(add(pMem, pBeta), 32), q))
// Get xiSeed & xiSeed2
mstore(add(pMem, lastMem), mload(add(pMem, pGamma)))
mstore(add(pMem, 1952), calldataload(pC2))
mstore(add(pMem, 1984), calldataload(add(pC2, 32)))
let xiSeed := mod(keccak256(add(pMem, lastMem), 96), q)
mstore(add(pMem, pXiSeed), xiSeed)
mstore(add(pMem, pXiSeed2), mulmod(xiSeed, xiSeed, q))
// Compute roots.S0.h0w8
mstore(add(pMem, pH0w8_0), mulmod(mload(add(pMem, pXiSeed2)), mload(add(pMem, pXiSeed)), q))
mstore(add(pMem, pH0w8_1), mulmod(mload(add(pMem, pH0w8_0)), w8_1, q))
mstore(add(pMem, pH0w8_2), mulmod(mload(add(pMem, pH0w8_0)), w8_2, q))
mstore(add(pMem, pH0w8_3), mulmod(mload(add(pMem, pH0w8_0)), w8_3, q))
mstore(add(pMem, pH0w8_4), mulmod(mload(add(pMem, pH0w8_0)), w8_4, q))
mstore(add(pMem, pH0w8_5), mulmod(mload(add(pMem, pH0w8_0)), w8_5, q))
mstore(add(pMem, pH0w8_6), mulmod(mload(add(pMem, pH0w8_0)), w8_6, q))
mstore(add(pMem, pH0w8_7), mulmod(mload(add(pMem, pH0w8_0)), w8_7, q))
// Compute roots.S1.h1w4
mstore(add(pMem, pH1w4_0), mulmod(mload(add(pMem, pH0w8_0)), mload(add(pMem, pH0w8_0)), q))
mstore(add(pMem, pH1w4_1), mulmod(mload(add(pMem, pH1w4_0)), w4, q))
mstore(add(pMem, pH1w4_2), mulmod(mload(add(pMem, pH1w4_0)), w4_2, q))
mstore(add(pMem, pH1w4_3), mulmod(mload(add(pMem, pH1w4_0)), w4_3, q))
// Compute roots.S2.h2w3
mstore(add(pMem, pH2w3_0), mulmod(mload(add(pMem, pH1w4_0)), mload(add(pMem, pXiSeed2)), q))
mstore(add(pMem, pH2w3_1), mulmod(mload(add(pMem, pH2w3_0)), w3, q))
mstore(add(pMem, pH2w3_2), mulmod(mload(add(pMem, pH2w3_0)), w3_2, q))
// Compute roots.S2.h2w3
mstore(add(pMem, pH3w3_0), mulmod(mload(add(pMem, pH2w3_0)), wr, q))
mstore(add(pMem, pH3w3_1), mulmod(mload(add(pMem, pH3w3_0)), w3, q))
mstore(add(pMem, pH3w3_2), mulmod(mload(add(pMem, pH3w3_0)), w3_2, q))
let xin := mulmod(mulmod(mload(add(pMem, pH2w3_0)), mload(add(pMem, pH2w3_0)), q), mload(add(pMem, pH2w3_0)), q)
mstore(add(pMem, pXi), xin)
// Compute xi^n
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mulmod(xin, xin, q)
xin:= mod(add(sub(xin, 1), q), q)
mstore(add(pMem, pZh), xin)
mstore(add(pMem, pZhInv), xin) // We will invert later together with lagrange pols
// Compute challenge.alpha
mstore(add(pMem, lastMem), xiSeed)
calldatacopy(add(pMem, 1952), pEval_ql, 480)
mstore(add(pMem, pAlpha), mod(keccak256(add(pMem, lastMem), 512), q))
// Compute challenge.y
mstore(add(pMem, lastMem), mload(add(pMem, pAlpha)))
mstore(add(pMem, 1952 ), calldataload(pW1))
mstore(add(pMem, 1984 ), calldataload(add(pW1, 32)))
mstore(add(pMem, pY), mod(keccak256(add(pMem, lastMem), 96), q))
}
function computeLiS0(pMem) {
let root0 := mload(add(pMem, pH0w8_0))
let y := mload(add(pMem, pY))
let den1 := 1
den1 := mulmod(den1, root0, q)
den1 := mulmod(den1, root0, q)
den1 := mulmod(den1, root0, q)
den1 := mulmod(den1, root0, q)
den1 := mulmod(den1, root0, q)
den1 := mulmod(den1, root0, q)
den1 := mulmod(8, den1, q)
let den2 := mload(add(pMem, add(pH0w8_0, mul(mod(mul(7, 0), 8), 32))))
let den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH0w8_0, mul(0, 32))))), q), q)
mstore(add(pMem, add(pLiS0Inv, 0)), mulmod(den1, mulmod(den2, den3, q), q))
den2 := mload(add(pMem, add(pH0w8_0, mul(mod(mul(7, 1), 8), 32))))
den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH0w8_0, mul(1, 32))))), q), q)
mstore(add(pMem, add(pLiS0Inv, 32)), mulmod(den1, mulmod(den2, den3, q), q))
den2 := mload(add(pMem, add(pH0w8_0, mul(mod(mul(7, 2), 8), 32))))
den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH0w8_0, mul(2, 32))))), q), q)
mstore(add(pMem, add(pLiS0Inv, 64)), mulmod(den1, mulmod(den2, den3, q), q))
den2 := mload(add(pMem, add(pH0w8_0, mul(mod(mul(7, 3), 8), 32))))
den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH0w8_0, mul(3, 32))))), q), q)
mstore(add(pMem, add(pLiS0Inv, 96)), mulmod(den1, mulmod(den2, den3, q), q))
den2 := mload(add(pMem, add(pH0w8_0, mul(mod(mul(7, 4), 8), 32))))
den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH0w8_0, mul(4, 32))))), q), q)
mstore(add(pMem, add(pLiS0Inv, 128)), mulmod(den1, mulmod(den2, den3, q), q))
den2 := mload(add(pMem, add(pH0w8_0, mul(mod(mul(7, 5), 8), 32))))
den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH0w8_0, mul(5, 32))))), q), q)
mstore(add(pMem, add(pLiS0Inv, 160)), mulmod(den1, mulmod(den2, den3, q), q))
den2 := mload(add(pMem, add(pH0w8_0, mul(mod(mul(7, 6), 8), 32))))
den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH0w8_0, mul(6, 32))))), q), q)
mstore(add(pMem, add(pLiS0Inv, 192)), mulmod(den1, mulmod(den2, den3, q), q))
den2 := mload(add(pMem, add(pH0w8_0, mul(mod(mul(7, 7), 8), 32))))
den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH0w8_0, mul(7, 32))))), q), q)
mstore(add(pMem, add(pLiS0Inv, 224)), mulmod(den1, mulmod(den2, den3, q), q))
}
function computeLiS1(pMem) {
let root0 := mload(add(pMem, pH1w4_0))
let y := mload(add(pMem, pY))
let den1 := 1
den1 := mulmod(den1, root0, q)
den1 := mulmod(den1, root0, q)
den1 := mulmod(4, den1, q)
let den2 := mload(add(pMem, add(pH1w4_0, mul(mod(mul(3, 0), 4), 32))))
let den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH1w4_0, mul(0, 32))))), q), q)
mstore(add(pMem, add(pLiS1Inv, 0)), mulmod(den1, mulmod(den2, den3, q), q))
den2 := mload(add(pMem, add(pH1w4_0, mul(mod(mul(3, 1), 4), 32))))
den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH1w4_0, mul(1, 32))))), q), q)
mstore(add(pMem, add(pLiS1Inv, 32)), mulmod(den1, mulmod(den2, den3, q), q))
den2 := mload(add(pMem, add(pH1w4_0, mul(mod(mul(3, 2), 4), 32))))
den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH1w4_0, mul(2, 32))))), q), q)
mstore(add(pMem, add(pLiS1Inv, 64)), mulmod(den1, mulmod(den2, den3, q), q))
den2 := mload(add(pMem, add(pH1w4_0, mul(mod(mul(3, 3), 4), 32))))
den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH1w4_0, mul(3, 32))))), q), q)
mstore(add(pMem, add(pLiS1Inv, 96)), mulmod(den1, mulmod(den2, den3, q), q))
}
function computeLiS2(pMem) {
let y := mload(add(pMem, pY))
let den1 := mulmod(mulmod(3,mload(add(pMem, pH2w3_0)),q), addmod(mload(add(pMem, pXi)) ,mod(sub(q, mulmod(mload(add(pMem, pXi)), w1 ,q)), q), q), q)
let den2 := mload(add(pMem, add(pH2w3_0, mul(mod(mul(2, 0), 3), 32))))
let den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH2w3_0, mul(0, 32))))), q), q)
mstore(add(pMem, add(pLiS2Inv, 0)), mulmod(den1, mulmod(den2, den3, q), q))
den2 := mload(add(pMem, add(pH2w3_0, mul(mod(mul(2, 1), 3), 32))))
den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH2w3_0, mul(1, 32))))), q), q)
mstore(add(pMem, add(pLiS2Inv, 32)), mulmod(den1, mulmod(den2, den3, q), q))
den2 := mload(add(pMem, add(pH2w3_0, mul(mod(mul(2, 2), 3), 32))))
den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH2w3_0, mul(2, 32))))), q), q)
mstore(add(pMem, add(pLiS2Inv, 64)), mulmod(den1, mulmod(den2, den3, q), q))
den1 := mulmod(mulmod(3,mload(add(pMem, pH3w3_0)),q), addmod(mulmod(mload(add(pMem, pXi)), w1 ,q),mod(sub(q, mload(add(pMem, pXi))), q), q), q)
den2 := mload(add(pMem, add(pH3w3_0, mul(mod(mul(2, 0), 3), 32))))
den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH3w3_0, mul(0, 32))))), q), q)
mstore(add(pMem, add(pLiS2Inv, 96)), mulmod(den1, mulmod(den2, den3, q), q))
den2 := mload(add(pMem, add(pH3w3_0, mul(mod(mul(2, 1), 3), 32))))
den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH3w3_0, mul(1, 32))))), q), q)
mstore(add(pMem, add(pLiS2Inv, 128)), mulmod(den1, mulmod(den2, den3, q), q))
den2 := mload(add(pMem, add(pH3w3_0, mul(mod(mul(2, 2), 3), 32))))
den3 := addmod(y, mod(sub(q, mload(add(pMem, add(pH3w3_0, mul(2, 32))))), q), q)
mstore(add(pMem, add(pLiS2Inv, 160)), mulmod(den1, mulmod(den2, den3, q), q))
}
// Prepare all the denominators that must be inverted, placed them in consecutive memory addresses
function computeInversions(pMem) {
// 1/ZH(xi) used in steps 8 and 9 of the verifier to multiply by 1/Z_H(xi)
// Value computed during computeChallenges function and stores in pMem+pZhInv
// 1/((y - h1) (y - h1w4) (y - h1w4_2) (y - h1w4_3))
// used in steps 10 and 11 of the verifier
let y := mload(add(pMem, pY))
let w := addmod(y, mod(sub(q, mload(add(pMem, pH1w4_0))), q), q)
w := mulmod(w, addmod(y, mod(sub(q, mload(add(pMem, pH1w4_1))), q), q), q)
w := mulmod(w, addmod(y, mod(sub(q, mload(add(pMem, pH1w4_2))), q), q), q)
w := mulmod(w, addmod(y, mod(sub(q, mload(add(pMem, pH1w4_3))), q), q), q)
mstore(add(pMem, pDenH1), w)
// 1/((y - h2) (y - h2w3) (y - h2w3_2) (y - h3) (y - h3w3) (y - h3w3_2))
w := addmod(y, mod(sub(q, mload(add(pMem, pH2w3_0))), q), q)
w := mulmod(w, addmod(y, mod(sub(q, mload(add(pMem, pH2w3_1))), q), q), q)
w := mulmod(w, addmod(y, mod(sub(q, mload(add(pMem, pH2w3_2))), q), q), q)
w := mulmod(w, addmod(y, mod(sub(q, mload(add(pMem, pH3w3_0))), q), q), q)
w := mulmod(w, addmod(y, mod(sub(q, mload(add(pMem, pH3w3_1))), q), q), q)
w := mulmod(w, addmod(y, mod(sub(q, mload(add(pMem, pH3w3_2))), q), q), q)
mstore(add(pMem, pDenH2), w)
// Denominator needed in the verifier when computing L_i^{S0}(X)
computeLiS0(pMem)
// Denominator needed in the verifier when computing L_i^{S1}(X)
computeLiS1(pMem)
// Denominator needed in the verifier when computing L_i^{S2}(X)
computeLiS2(pMem)
// L_i where i from 1 to num public inputs, needed in step 6 and 7 of the verifier to compute L_1(xi) and PI(xi)
w := 1
let xi := mload(add(pMem, pXi))
mstore(add(pMem, pEval_l1), mulmod(n, mod(add(sub(xi, w), q), q), q))
// Execute Montgomery batched inversions of the previous prepared values
inverseArray(pMem) }
// Compute Lagrange polynomial evaluation L_i(xi)
function computeLagrange(pMem) {
let zh := mload(add(pMem, pZh))
let w := 1
mstore(add(pMem, pEval_l1 ), mulmod(mload(add(pMem, pEval_l1 )), zh, q))
}
// Compute public input polynomial evaluation PI(xi)
function computePi(pMem, pPub) {
let pi := 0
pi := mod(add(sub(pi, mulmod(mload(add(pMem, pEval_l1)), calldataload(pPub), q)), q), q)
mstore(add(pMem, pPi), pi)
}
// Compute r0(y) by interpolating the polynomial r0(X) using 8 points (x,y)
// where x = {h9, h0w8, h0w8^2, h0w8^3, h0w8^4, h0w8^5, h0w8^6, h0w8^7}
// and y = {C0(h0), C0(h0w8), C0(h0w8^2), C0(h0w8^3), C0(h0w8^4), C0(h0w8^5), C0(h0w8^6), C0(h0w8^7)}
// and computing C0(xi)
function computeR0(pMem) {
let num := 1
let y := mload(add(pMem, pY))
num := mulmod(num, y, q)
num := mulmod(num, y, q)
num := mulmod(num, y, q)
num := mulmod(num, y, q)
num := mulmod(num, y, q)
num := mulmod(num, y, q)
num := mulmod(num, y, q)
num := mulmod(num, y, q)
num := addmod(num, mod(sub(q, mload(add(pMem, pXi))), q), q)
let res
let h0w80
let c0Value
let h0w8i
// Compute c0Value = ql + (h0w8i) qr + (h0w8i)^2 qo + (h0w8i)^3 qm + (h0w8i)^4 qc +
// + (h0w8i)^5 S1 + (h0w8i)^6 S2 + (h0w8i)^7 S3
h0w80 := mload(add(pMem, pH0w8_0))
c0Value := addmod(calldataload(pEval_ql), mulmod(calldataload(pEval_qr), h0w80, q), q)
h0w8i := mulmod(h0w80, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qo), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qm), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qc), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s1), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s2), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s3), h0w8i, q), q)
res := addmod(res, mulmod(c0Value, mulmod(num, mload(add(pMem, add(pLiS0Inv, 0))), q), q), q)
// Compute c0Value = ql + (h0w8i) qr + (h0w8i)^2 qo + (h0w8i)^3 qm + (h0w8i)^4 qc +
// + (h0w8i)^5 S1 + (h0w8i)^6 S2 + (h0w8i)^7 S3
h0w80 := mload(add(pMem, pH0w8_1))
c0Value := addmod(calldataload(pEval_ql), mulmod(calldataload(pEval_qr), h0w80, q), q)
h0w8i := mulmod(h0w80, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qo), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qm), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qc), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s1), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s2), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s3), h0w8i, q), q)
res := addmod(res, mulmod(c0Value, mulmod(num, mload(add(pMem, add(pLiS0Inv, 32))), q), q), q)
// Compute c0Value = ql + (h0w8i) qr + (h0w8i)^2 qo + (h0w8i)^3 qm + (h0w8i)^4 qc +
// + (h0w8i)^5 S1 + (h0w8i)^6 S2 + (h0w8i)^7 S3
h0w80 := mload(add(pMem, pH0w8_2))
c0Value := addmod(calldataload(pEval_ql), mulmod(calldataload(pEval_qr), h0w80, q), q)
h0w8i := mulmod(h0w80, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qo), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qm), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qc), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s1), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s2), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s3), h0w8i, q), q)
res := addmod(res, mulmod(c0Value, mulmod(num, mload(add(pMem, add(pLiS0Inv, 64))), q), q), q)
// Compute c0Value = ql + (h0w8i) qr + (h0w8i)^2 qo + (h0w8i)^3 qm + (h0w8i)^4 qc +
// + (h0w8i)^5 S1 + (h0w8i)^6 S2 + (h0w8i)^7 S3
h0w80 := mload(add(pMem, pH0w8_3))
c0Value := addmod(calldataload(pEval_ql), mulmod(calldataload(pEval_qr), h0w80, q), q)
h0w8i := mulmod(h0w80, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qo), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qm), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qc), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s1), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s2), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s3), h0w8i, q), q)
res := addmod(res, mulmod(c0Value, mulmod(num, mload(add(pMem, add(pLiS0Inv, 96))), q), q), q)
// Compute c0Value = ql + (h0w8i) qr + (h0w8i)^2 qo + (h0w8i)^3 qm + (h0w8i)^4 qc +
// + (h0w8i)^5 S1 + (h0w8i)^6 S2 + (h0w8i)^7 S3
h0w80 := mload(add(pMem, pH0w8_4))
c0Value := addmod(calldataload(pEval_ql), mulmod(calldataload(pEval_qr), h0w80, q), q)
h0w8i := mulmod(h0w80, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qo), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qm), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qc), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s1), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s2), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s3), h0w8i, q), q)
res := addmod(res, mulmod(c0Value, mulmod(num, mload(add(pMem, add(pLiS0Inv, 128))), q), q), q)
// Compute c0Value = ql + (h0w8i) qr + (h0w8i)^2 qo + (h0w8i)^3 qm + (h0w8i)^4 qc +
// + (h0w8i)^5 S1 + (h0w8i)^6 S2 + (h0w8i)^7 S3
h0w80 := mload(add(pMem, pH0w8_5))
c0Value := addmod(calldataload(pEval_ql), mulmod(calldataload(pEval_qr), h0w80, q), q)
h0w8i := mulmod(h0w80, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qo), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qm), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qc), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s1), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s2), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s3), h0w8i, q), q)
res := addmod(res, mulmod(c0Value, mulmod(num, mload(add(pMem, add(pLiS0Inv, 160))), q), q), q)
// Compute c0Value = ql + (h0w8i) qr + (h0w8i)^2 qo + (h0w8i)^3 qm + (h0w8i)^4 qc +
// + (h0w8i)^5 S1 + (h0w8i)^6 S2 + (h0w8i)^7 S3
h0w80 := mload(add(pMem, pH0w8_6))
c0Value := addmod(calldataload(pEval_ql), mulmod(calldataload(pEval_qr), h0w80, q), q)
h0w8i := mulmod(h0w80, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qo), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qm), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qc), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s1), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s2), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s3), h0w8i, q), q)
res := addmod(res, mulmod(c0Value, mulmod(num, mload(add(pMem, add(pLiS0Inv, 192))), q), q), q)
// Compute c0Value = ql + (h0w8i) qr + (h0w8i)^2 qo + (h0w8i)^3 qm + (h0w8i)^4 qc +
// + (h0w8i)^5 S1 + (h0w8i)^6 S2 + (h0w8i)^7 S3
h0w80 := mload(add(pMem, pH0w8_7))
c0Value := addmod(calldataload(pEval_ql), mulmod(calldataload(pEval_qr), h0w80, q), q)
h0w8i := mulmod(h0w80, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qo), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qm), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_qc), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s1), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s2), h0w8i, q), q)
h0w8i := mulmod(h0w8i, h0w80, q)
c0Value := addmod(c0Value, mulmod(calldataload(pEval_s3), h0w8i, q), q)
res := addmod(res, mulmod(c0Value, mulmod(num, mload(add(pMem, add(pLiS0Inv, 224))), q), q), q)
mstore(add(pMem, pR0), res)
}
// Compute r1(y) by interpolating the polynomial r1(X) using 4 points (x,y)
// where x = {h1, h1w4, h1w4^2, h1w4^3}
// and y = {C1(h1), C1(h1w4), C1(h1w4^2), C1(h1w4^3)}
// and computing T0(xi)
function computeR1(pMem) {
let num := 1
let y := mload(add(pMem, pY))
num := mulmod(num, y, q)
num := mulmod(num, y, q)
num := mulmod(num, y, q)
num := mulmod(num, y, q)
num := addmod(num, mod(sub(q, mload(add(pMem, pXi))), q), q)
let t0
let evalA := calldataload(pEval_a)
let evalB := calldataload(pEval_b)
let evalC := calldataload(pEval_c)
t0 := mulmod(calldataload(pEval_ql), evalA, q)
t0 := addmod(t0, mulmod(calldataload(pEval_qr), evalB, q) ,q)
t0 := addmod(t0, mulmod(calldataload(pEval_qm), mulmod(evalA, evalB, q), q) ,q)
t0 := addmod(t0, mulmod(calldataload(pEval_qo), evalC, q) ,q)
t0 := addmod(t0, calldataload(pEval_qc) ,q)
t0 := addmod(t0, mload(add(pMem, pPi)), q)
t0 := mulmod(t0, mload(add(pMem, pZhInv)), q)
let res
let c1Value
let h1w4
let square
c1Value := evalA
h1w4 := mload(add(pMem, pH1w4_0))
c1Value := addmod(c1Value, mulmod(h1w4, evalB, q), q)
square := mulmod(h1w4, h1w4, q)
c1Value := addmod(c1Value, mulmod(square, evalC, q), q)
c1Value := addmod(c1Value, mulmod(mulmod(square, h1w4, q), t0, q), q)
res := addmod(res, mulmod(c1Value, mulmod(num, mload(add(pMem, add(pLiS1Inv, mul(0, 32)))), q), q), q)
c1Value := evalA
h1w4 := mload(add(pMem, pH1w4_1))
c1Value := addmod(c1Value, mulmod(h1w4, evalB, q), q)
square := mulmod(h1w4, h1w4, q)
c1Value := addmod(c1Value, mulmod(square, evalC, q), q)
c1Value := addmod(c1Value, mulmod(mulmod(square, h1w4, q), t0, q), q)
res := addmod(res, mulmod(c1Value, mulmod(num, mload(add(pMem, add(pLiS1Inv, mul(1, 32)))), q), q), q)
c1Value := evalA
h1w4 := mload(add(pMem, pH1w4_2))
c1Value := addmod(c1Value, mulmod(h1w4, evalB, q), q)
square := mulmod(h1w4, h1w4, q)
c1Value := addmod(c1Value, mulmod(square, evalC, q), q)
c1Value := addmod(c1Value, mulmod(mulmod(square, h1w4, q), t0, q), q)
res := addmod(res, mulmod(c1Value, mulmod(num, mload(add(pMem, add(pLiS1Inv, mul(2, 32)))), q), q), q)
c1Value := evalA
h1w4 := mload(add(pMem, pH1w4_3))
c1Value := addmod(c1Value, mulmod(h1w4, evalB, q), q)
square := mulmod(h1w4, h1w4, q)
c1Value := addmod(c1Value, mulmod(square, evalC, q), q)
c1Value := addmod(c1Value, mulmod(mulmod(square, h1w4, q), t0, q), q)
res := addmod(res, mulmod(c1Value, mulmod(num, mload(add(pMem, add(pLiS1Inv, mul(3, 32)))), q), q), q)
mstore(add(pMem, pR1), res)
}
// Compute r2(y) by interpolating the polynomial r2(X) using 6 points (x,y)
// where x = {[h2, h2w3, h2w3^2], [h3, h3w3, h3w3^2]}
// and y = {[C2(h2), C2(h2w3), C2(h2w3^2)], [C2(h3), C2(h3w3), C2(h3w3^2)]}
// and computing T1(xi) and T2(xi)
function computeR2(pMem) {
let y := mload(add(pMem, pY))
let num := 1
num := mulmod(y, num, q)
num := mulmod(y, num, q)
num := mulmod(y, num, q)
num := mulmod(y, num, q)
num := mulmod(y, num, q)
num := mulmod(y, num, q)
let num2 := 1
num2 := mulmod(y, num2, q)
num2 := mulmod(y, num2, q)
num2 := mulmod(y, num2, q)
num2 := mulmod(num2, addmod(mulmod(mload(add(pMem, pXi)), w1 ,q), mload(add(pMem, pXi)), q), q)
num := addmod(num, mod(sub(q, num2), q), q)
num2 := mulmod(mulmod(mload(add(pMem, pXi)), w1 ,q), mload(add(pMem, pXi)), q)
num := addmod(num, num2, q)
let t1
let t2
let betaXi := mulmod(mload(add(pMem, pBeta)), mload(add(pMem, pXi)), q)
let gamma := mload(add(pMem, pGamma))