verifier_fflonk.sol.ejs

// 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     = <%= 2**power %>; // Domain size

    // Verification Key data
    uint256 constant k1   = <%= k1 %>;   // Plonk k1 multiplicative factor to force distinct cosets of H
    uint256 constant k2   = <%= k2 %>;   // Plonk k2 multiplicative factor to force distinct cosets of H

    // OMEGAS
    // Omega, Omega^{1/3}
    uint256 constant w1   = <%= w  %>;
    uint256 constant wr   = <%= wr %>;
    // Omega_3, Omega_3^2
    uint256 constant w3   = <%= w3 %>;
    uint256 constant w3_2 = <%= w3_2 %>;
    // Omega_4, Omega_4^2, Omega_4^3
    uint256 constant w4   = <%= w4 %>;
    uint256 constant w4_2 = <%= w4_2 %>;
    uint256 constant w4_3 = <%= w4_3 %>;
    // Omega_8, Omega_8^2, Omega_8^3, Omega_8^4, Omega_8^5, Omega_8^6, Omega_8^7
    uint256 constant w8_1 = <%= w8 %>;
    uint256 constant w8_2 = <%= w8_2 %>;
    uint256 constant w8_3 = <%= w8_3 %>;
    uint256 constant w8_4 = <%= w8_4 %>;
    uint256 constant w8_5 = <%= w8_5 %>;
    uint256 constant w8_6 = <%= w8_6 %>;
    uint256 constant w8_7 = <%= w8_7 %>;

    // Verifier preprocessed input C_0(x)·[1]_1
    uint256 constant C0x  = <%= C0[0] %>;
    uint256 constant C0y  = <%= C0[1] %>;

    // Verifier preprocessed input x·[1]_2
    uint256 constant X2x1 = <%= X_2[0][0] %>;
    uint256 constant X2x2 = <%= X_2[0][1] %>;
    uint256 constant X2y1 = <%= X_2[1][0] %>;
    uint256 constant X2y2 = <%= X_2[1][1] %>;

    // 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 = <%= 224 + 32 %>;
    uint16 constant pH0w8_2 = <%= 224 + 32 * 2 %>;
    uint16 constant pH0w8_3 = <%= 224 + 32 * 3 %>;
    uint16 constant pH0w8_4 = <%= 224 + 32 * 4 %>;
    uint16 constant pH0w8_5 = <%= 224 + 32 * 5 %>;
    uint16 constant pH0w8_6 = <%= 224 + 32 * 6 %>;
    uint16 constant pH0w8_7 = <%= 224 + 32 * 7 %>;

    // S_1 = roots_4(xi) = { h_1, h_1w_4, h_1w_4^2, h_1w_4^3 }
    uint16 constant pH1w4_0 = <%= 224 + 32 * 8 %>;
    uint16 constant pH1w4_1 = <%= 224 + 32 * 9 %>;
    uint16 constant pH1w4_2 = <%= 224 + 32 * 10 %>;
    uint16 constant pH1w4_3 = <%= 224 + 32 * 11 %>;

    // 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 = <%= 224 + 32 * 12 %>;
    uint16 constant pH2w3_1 = <%= 224 + 32 * 13 %>;
    uint16 constant pH2w3_2 = <%= 224 + 32 * 14 %>;
    // roots_3(xi omega) = { h_3, h_3w_3, h_3w_3^2 }
    uint16 constant pH3w3_0 = <%= 224 + 32 * 15 %>;
    uint16 constant pH3w3_1 = <%= 224 + 32 * 16 %>;
    uint16 constant pH3w3_2 = <%= 224 + 32 * 17 %>;

    uint16 constant pPi     = <%= 224 + 32 * 18 %>; // PI(xi)
    uint16 constant pR0     = <%= 224 + 32 * 19 %>; // r0(y)
    uint16 constant pR1     = <%= 224 + 32 * 20 %>; // r1(y)
    uint16 constant pR2     = <%= 224 + 32 * 21 %>; // r2(y)

    uint16 constant pF      = <%= 224 + 32 * 22 %>;  // [F]_1, 64 bytes
    uint16 constant pE      = <%= 224 + 32 * 22 + 64 %>;  // [E]_1, 64 bytes
    uint16 constant pJ      = <%= 224 + 32 * 22 + 64 * 2 %>; // [J]_1, 64 bytes

    uint16 constant pZh     = <%= 224 + 32 * 22 + 64 * 4 %>; // Z_H(xi)
    // From this point we write all the variables that must be computed using the Montgomery batch inversion
    uint16 constant pZhInv  = <%= 224 + 32 * 23 + 64 * 4 %>; // 1/Z_H(xi)
    uint16 constant pDenH1  = <%= 224 + 32 * 24 + 64 * 4 %>; // 1/( (y-h_1w_4) (y-h_1w_4^2) (y-h_1w_4^3) (y-h_1w_4^4) )
    uint16 constant pDenH2  = <%= 224 + 32 * 25 + 64 * 4 %>; // 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 = <%= 224 + 32 * 26 + 64 * 4 %>; // Reserve 8 * 32 bytes to compute r_0(X)
    uint16 constant pLiS1Inv = <%= 224 + 32 * 34 + 64 * 4 %>; // Reserve 4 * 32 bytes to compute r_1(X)
    uint16 constant pLiS2Inv = <%= 224 + 32 * 38 + 64 * 4 %>; // Reserve 6 * 32 bytes to compute r_2(X)
    // Lagrange evaluations
    <% for (let i = 1; i <= Math.max(nPublic, 1); i++) { %>
    uint16 constant pEval_l<%=i%> = <%= 224 + 32 * (43 + i) + 64 * 4 %>;
    <% } %>
    <% let pLastMem = 224 + 32 * (44 + Math.max(nPublic,1)) + 64 * 4 %>
    uint16 constant lastMem = <%= pLastMem %>;
     
<% const inversionArray = ["pZhInv", "pDenH1", "pDenH2", "pLiS0Inv", "add(pLiS0Inv, 32)", "add(pLiS0Inv, 64)", "add(pLiS0Inv, 96)",
    "add(pLiS0Inv, 128)", "add(pLiS0Inv, 160)", "add(pLiS0Inv, 192)", "add(pLiS0Inv, 224)", "pLiS1Inv", "add(pLiS1Inv, 32)", 
    "add(pLiS1Inv, 64)", "add(pLiS1Inv, 96)", "pLiS2Inv", "add(pLiS2Inv, 32)", "add(pLiS2Inv, 64)", "add(pLiS2Inv, 96)",
    "add(pLiS2Inv, 128)", "add(pLiS2Inv, 160)"] -%>
<% for (let i=1; i<=Math.max(nPublic, 1); i++) { -%>
<%      inversionArray.push(`pEval_l${i}`); -%>
<% } -%>

    function verifyProof(bytes32[24] calldata proof, uint256[<%- Math.max(nPublic, 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,<%- inversionArray[0] %>))       // Read the first element
                mstore(pAux, acc)

<%  for(let i = 1; i < inversionArray.length; ++i) { -%>
                pAux := add(pAux, 32)
                acc := mulmod(acc, mload(add(pMem, <%- inversionArray[i] %>)), 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

<%  for(let i = inversionArray.length - 1; i > 0; --i) { -%>
                pAux := sub(pAux, 32)
                inv := mulmod(acc, mload(pAux), q)
                acc := mulmod(acc, mload(add(pMem, <%- inversionArray[i] %>)), q)
                mstore(add(pMem, <%- inversionArray[i] %>), inv)
<% } -%>

                mstore(add(pMem, <%- inversionArray[0] %>), 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)
                }
            }  
            
            function checkProofData() {
                // Check proof commitments belong to the bn128 curve
                checkPointBelongsToBN128Curve(pC1)
                checkPointBelongsToBN128Curve(pC2)
                checkPointBelongsToBN128Curve(pW1)
                checkPointBelongsToBN128Curve(pW2)

                // Check proof commitments coordinates are in the field
                checkField(calldataload(pC1))
                checkField(calldataload(add(pC1, 32)))
                checkField(calldataload(pC2))
                checkField(calldataload(add(pC2, 32)))
                checkField(calldataload(pW1))
                checkField(calldataload(add(pW1, 32)))
                checkField(calldataload(pW2))
                checkField(calldataload(add(pW2, 32)))

                // Check proof evaluations are in the field
                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))
            }

            function computeChallenges(pMem, pPublic) {
                // Compute challenge.beta & challenge.gamma
                mstore(add(pMem, <%= pLastMem %> ), C0x)
                mstore(add(pMem, <%= pLastMem + 32 %> ), C0y)

                mstore(add(pMem, <%= pLastMem + 64 %>), calldataload(pPublic))
                <%for (let i=1; i<nPublic;i++) { %>
                mstore(add(pMem, <%= pLastMem + 64 + i * 32 %> ), calldataload(add(pPublic, <%= i * 32 %>)))
                <%}%>
                

                mstore(add(pMem, <%= pLastMem + nPublic * 32 + 64 %> ),  calldataload(pC1))
                mstore(add(pMem, <%= pLastMem + nPublic * 32 + 96 %> ),  calldataload(add(pC1, 32)))

                mstore(add(pMem, pBeta),  mod(keccak256(add(pMem, lastMem), <%= nPublic * 32 + 128 %>), 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, <%= pLastMem + 32 %>), calldataload(pC2))
                mstore(add(pMem, <%= pLastMem + 64 %>), 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
                <%for ( let i = 0; i < power; i++) { %>
                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, <%= pLastMem + 32 %>), 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, <%= pLastMem + 32 %> ),  calldataload(pW1))
                mstore(add(pMem, <%= pLastMem + 64 %> ),  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))
                <% for (let i=1; i<=Math.max(nPublic, 1); i++) { %>
                mstore(add(pMem, pEval_l<%=i%>), mulmod(n, mod(add(sub(xi, w), q), q), q))
                <% if (i<Math.max(nPublic, 1)) { %>
                w := mulmod(w, w1, 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
                <% for (let i=1; i<=Math.max(nPublic, 1); i++) { 
                    if (i===1) { %>
                    mstore(add(pMem, pEval_l1 ), mulmod(mload(add(pMem, pEval_l1 )), zh, q))
                    <% } else { %>
                    mstore(add(pMem, pEval_l<%=i%>), mulmod(w, mulmod(mload(add(pMem, pEval_l<%=i%>)), zh, q), q))
                    <% } 
                    if (i<Math.max(nPublic, 1)) { %>
                    w := mulmod(w, w1, 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)
                <% for (let i=1; i<nPublic; i++) { %>
                pi := mod(add(sub(pi, mulmod(mload(add(pMem, pEval_l<%= i + 1 %>)), calldataload(add(pPub, <%= 32 * i %>)), 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
<% for(let i = 0; i < 8; ++i) {  -%>
                    
                // 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_<%- i %>))
                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, <%- i * 32 %>))), 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
<% for(let i = 0; i < 4; ++i) {  -%>
                c1Value := evalA
                h1w4 := mload(add(pMem, pH1w4_<%-i%>))

                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(<%- i %>, 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))

                t2 := addmod(calldataload( pEval_a), addmod(betaXi, gamma, q) ,q)
                t2 := mulmod(t2,
                            addmod(calldataload( pEval_b),
                            addmod(mulmod(betaXi, k1, q), gamma, q) ,q), q)
                t2 := mulmod(t2,
                            addmod(calldataload( pEval_c),
                            addmod(mulmod(betaXi, k2, q), gamma, q) ,q), q)
                t2 := mulmod(t2, calldataload(pEval_z), q)

                //Let's use t1 as a temporal variable to save one local
                t1 := addmod(calldataload(pEval_a), addmod(mulmod(mload(add(pMem, pBeta)), calldataload(pEval_s1), q), gamma, q) ,q)
                t1 := mulmod(t1,
                      addmod(calldataload(pEval_b), addmod(mulmod(mload(add(pMem, pBeta)), calldataload(pEval_s2), q), gamma, q) ,q), q)
                t1 := mulmod(t1,
                      addmod(calldataload(pEval_c), addmod(mulmod(mload(add(pMem, pBeta)), calldataload(pEval_s3), q), gamma, q) ,q), q)
                t1 := mulmod(t1, calldataload(pEval_zw), q)

                t2:= addmod(t2, mod(sub(q, t1), q), q)
                t2 := mulmod(t2, mload(add(pMem, pZhInv)), q)

                // Compute T1(xi)
                t1 := sub(calldataload(pEval_z), 1)
                t1 := mulmod(t1, mload(add(pMem, pEval_l1)) ,q)
                t1 := mulmod(t1, mload(add(pMem, pZhInv)) ,q)

                // Let's use local variable gamma to save the result
                gamma:=0
                
                let hw
                let c2Value 

                hw := mload(add(pMem, pH2w3_0))
                c2Value := addmod(calldataload(pEval_z), mulmod(hw, t1, q), q)
                c2Value := addmod(c2Value, mulmod(mulmod(hw, hw, q), t2, q), q)
                gamma := addmod(gamma, mulmod(c2Value, mulmod(num, mload(add(pMem, add(pLiS2Inv, mul(0, 32)))), q), q), q)

                hw := mload(add(pMem, pH2w3_1))
                c2Value := addmod(calldataload(pEval_z), mulmod(hw, t1, q), q)
                c2Value := addmod(c2Value, mulmod(mulmod(hw, hw, q), t2, q), q)
                gamma := addmod(gamma, mulmod(c2Value, mulmod(num, mload(add(pMem, add(pLiS2Inv, mul(1, 32)))), q), q), q)

                hw := mload(add(pMem, pH2w3_2))
                c2Value := addmod(calldataload(pEval_z), mulmod(hw, t1, q), q)
                c2Value := addmod(c2Value, mulmod(mulmod(hw, hw, q), t2, q), q)
                gamma := addmod(gamma, mulmod(c2Value, mulmod(num, mload(add(pMem, add(pLiS2Inv, mul(2, 32)))), q), q), q)

                hw := mload(add(pMem, pH3w3_0))
                c2Value := addmod(calldataload(pEval_zw), mulmod(hw, calldataload(pEval_t1w), q), q)
                c2Value := addmod(c2Value, mulmod(mulmod(hw, hw, q), calldataload(pEval_t2w), q), q)
                gamma := addmod(gamma, mulmod(c2Value, mulmod(num, mload(add(pMem, add(pLiS2Inv, mul(3, 32)))), q), q), q)

                hw := mload(add(pMem, pH3w3_1))
                c2Value := addmod(calldataload(pEval_zw), mulmod(hw, calldataload(pEval_t1w), q), q)
                c2Value := addmod(c2Value, mulmod(mulmod(hw, hw, q), calldataload(pEval_t2w), q), q)
                gamma := addmod(gamma, mulmod(c2Value, mulmod(num, mload(add(pMem, add(pLiS2Inv, mul(4, 32)))), q), q), q)

                hw := mload(add(pMem, pH3w3_2))
                c2Value := addmod(calldataload(pEval_zw), mulmod(hw, calldataload(pEval_t1w), q), q)
                c2Value := addmod(c2Value, mulmod(mulmod(hw, hw, q), calldataload(pEval_t2w), q), q)
                gamma := addmod(gamma, mulmod(c2Value, mulmod(num, mload(add(pMem, add(pLiS2Inv, mul(5, 32)))), q), q), q)

                mstore(add(pMem, pR2), gamma)
            }

            // G1 function to accumulate a G1 value to an address
            function g1_acc(pR, pP) {
                let mIn := mload(0x40)
                mstore(mIn, mload(pR))
                mstore(add(mIn, 32), mload(add(pR, 32)))
                mstore(add(mIn, 64), mload(pP))
                mstore(add(mIn, 96), mload(add(pP, 32)))

                let success := staticcall(sub(gas(), 2000), 6, mIn, 128, pR, 64)

                if iszero(success) {
                    mstore(0, 0)
                    return(0, 0x20)
                }
            }

            // G1 function to multiply a G1 value to value in an address
            function g1_mulAcc(pR, pP, s) {
                let success
                let mIn := mload(0x40)
                mstore(mIn, calldataload(pP))
                mstore(add(mIn, 32), calldataload(add(pP, 32)))
                mstore(add(mIn, 64), s)

                success := staticcall(sub(gas(), 2000), 7, mIn, 96, mIn, 64)

                if iszero(success) {
                    mstore(0, 0)
                    return(0, 0x20)
                }

                mstore(add(mIn, 64), mload(pR))
                mstore(add(mIn, 96), mload(add(pR, 32)))

                success := staticcall(sub(gas(), 2000), 6, mIn, 128, pR, 64)

                if iszero(success) {
                    mstore(0, 0)
                    return(0, 0x20)
                }
            }

            // G1 function to multiply a G1 value(x,y) to value in an address
            function g1_mulAccC(pR, x, y, s) {
                let success
                let mIn := mload(0x40)
                mstore(mIn, x)
                mstore(add(mIn, 32), y)
                mstore(add(mIn, 64), s)

                success := staticcall(sub(gas(), 2000), 7, mIn, 96, mIn, 64)

                if iszero(success) {
                    mstore(0, 0)
                    return(0, 0x20)
                }

                mstore(add(mIn, 64), mload(pR))
                mstore(add(mIn, 96), mload(add(pR, 32)))

                success := staticcall(sub(gas(), 2000), 6, mIn, 128, pR, 64)

                if iszero(success) {
                    mstore(0, 0)
                    return(0, 0x20)
                }
            }

            function computeFEJ(pMem) {
                // Prepare shared numerator between F, E and J to reuse it
                let y := mload(add(pMem, pY))
                let numerator := addmod(y, mod(sub(q, mload(add(pMem, pH0w8_0))), q), q)
                numerator := mulmod(numerator, addmod(y, mod(sub(q, mload(add(pMem, pH0w8_1))), q), q), q)
                numerator := mulmod(numerator, addmod(y, mod(sub(q, mload(add(pMem, pH0w8_2))), q), q), q)
                numerator := mulmod(numerator, addmod(y, mod(sub(q, mload(add(pMem, pH0w8_3))), q), q), q)
                numerator := mulmod(numerator, addmod(y, mod(sub(q, mload(add(pMem, pH0w8_4))), q), q), q)
                numerator := mulmod(numerator, addmod(y, mod(sub(q, mload(add(pMem, pH0w8_5))), q), q), q)
                numerator := mulmod(numerator, addmod(y, mod(sub(q, mload(add(pMem, pH0w8_6))), q), q), q)
                numerator := mulmod(numerator, addmod(y, mod(sub(q, mload(add(pMem, pH0w8_7))), q), q), q)

                // Prepare shared quotient between F and E to reuse it
                let quotient1 := mulmod(mload(add(pMem, pAlpha)), mulmod(numerator, mload(add(pMem, pDenH1)), q), q)
                let quotient2 := mulmod(mulmod(mload(add(pMem, pAlpha)), mload(add(pMem, pAlpha)), q), mulmod(numerator, mload(add(pMem, pDenH2)), q), q)

                // Compute full batched polynomial commitment [F]_1
                mstore(add(pMem, pF), C0x)
                mstore(add(pMem, add(pF, 32)), C0y)
                g1_mulAcc(add(pMem, pF), pC1, quotient1)
                g1_mulAcc(add(pMem, pF), pC2, quotient2)

                // Compute group-encoded batch evaluation [E]_1
                g1_mulAccC(add(pMem, pE), G1x, G1y, addmod(mload(add(pMem, pR0)), addmod(mulmod(quotient1, mload(add(pMem, pR1)),q), mulmod(quotient2, mload(add(pMem, pR2)),q), q), q))

                // Compute the full difference [J]_1
                g1_mulAcc(add(pMem, pJ), pW1, numerator)
            }

            // Validate all evaluations with a pairing checking that e([F]_1 - [E]_1 - [J]_1 + y[W2]_1, [1]_2) == e([W']_1, [x]_2)
            function checkPairing(pMem) -> isOk {
                let mIn := mload(0x40)

                // First pairing value
                // Compute -E
                mstore(add(add(pMem, pE), 32), mod(sub(qf, mload(add(add(pMem, pE), 32))), qf))
                // Compute -J
                mstore(add(add(pMem, pJ), 32), mod(sub(qf, mload(add(add(pMem, pJ), 32))), qf))
                // F = F - E - J + y·W2
                g1_acc(add(pMem, pF), add(pMem, pE))
                g1_acc(add(pMem, pF), add(pMem, pJ))
                g1_mulAcc(add(pMem, pF), pW2, mload(add(pMem, pY)))

                mstore(mIn, mload(add(pMem, pF)))
                mstore(add(mIn, 32), mload(add(add(pMem, pF), 32)))

                // Second pairing value
                mstore(add(mIn, 64), G2x2)
                mstore(add(mIn, 96), G2x1)
                mstore(add(mIn, 128), G2y2)
                mstore(add(mIn, 160), G2y1)

                // Third pairing value
                // Compute -W2
                mstore(add(mIn, 192), calldataload(pW2))
                let s := calldataload(add(pW2, 32))
                s := mod(sub(qf, s), qf)
                mstore(add(mIn, 224), s)

                // Fourth pairing value
                mstore(add(mIn, 256), X2x2)
                mstore(add(mIn, 288), X2x1)
                mstore(add(mIn, 320), X2y2)
                mstore(add(mIn, 352), X2y1)

                let success := staticcall(sub(gas(), 2000), 8, mIn, 384, mIn, 0x20)

                isOk := and(success, mload(mIn))
            }

            let pMem := mload(0x40)
            mstore(0x40, add(pMem, lastMem))

            // Check proof data is well-formed
            checkProofData()

            // Compute the challenges: beta, gamma, xi, alpha and y ∈ F, h1w4/h2w3/h3w3 roots, xiN and zh(xi)
            computeChallenges(pMem, pubSignals)

            // To divide prime fields the Extended Euclidean Algorithm for computing modular inverses is needed.
            // The Montgomery batch inversion algorithm allow us to compute n inverses reducing to a single one inversion.
            // More info: https://vitalik.ca/general/2018/07/21/starks_part_3.html
            // To avoid this single inverse computation on-chain, it has been computed in proving time and send it to the verifier.
            // Therefore, the verifier:
            //      1) Prepare all the denominators to inverse
            //      2) Check the inverse sent by the prover it is what it should be
            //      3) Compute the others inverses using the Montgomery Batched Algorithm using the inverse sent to avoid the inversion operation it does.
            computeInversions(pMem)

            // Compute Lagrange polynomial evaluations Li(xi)
            computeLagrange(pMem)

            // Compute public input polynomial evaluation PI(xi) = \sum_i^l -public_input_i·L_i(xi)
            computePi(pMem, pubSignals)

            // Computes r1(y) and r2(y)
            computeR0(pMem)
            computeR1(pMem)
            computeR2(pMem)

            // Compute full batched polynomial commitment [F]_1, group-encoded batch evaluation [E]_1 and the full difference [J]_1
            computeFEJ(pMem)

            // Validate all evaluations
            let isValid := checkPairing(pMem)

            mstore(0, isValid)
            return(0, 0x20)
        }
    }
}