Polygon zkEVM中的常量多项式

1. 引言

具体见：

https://github.com/0xPolygonHermez/zkevm-proverjs/blob/main/pil/l

2. Global.pil中的常量多项式

Polygon zkEVM全局多项式Global.pil中包含3个constant多项式：

1）L1 constant多项式
2）BYTE constant多项式
3）BYTE2 constant多项式

namespace Global(%N);
pol constant L1;    // 1, 0, 0, 0, 0,
pol constant BYTE;
pol constant BYTE2;

这些全局constant多项式的基本赋值情况为：

module.exports.buildConstants = async function (pols) {

    const F = new F1Field("0xFFFFFFFF00000001");

    const N = pols.BYTE.length;
    buidBYTE(pols.BYTE, F, N);
    buidBYTE2(pols.BYTE2, F, N);
    buildL1(pols.L1, F, N);

};

function buidBYTE2(pol, F, N) {
    const m = 1 A
        CNT_KECCAK_F: ASSERT
        CNT_MEM_ALIGN :ASSERT
        CNT_POSEIDON_G :ASSERT
        CNT_PADDING_PG :ASSERT

end:
       0 => A,B,C,D,E,CTX, SP, PC, GAS, MAXMEM, SR

finalWait:
        ${beforeLast()}  : JMPN(finalWait)

                         : JMP(start)
opINVALID:

经const rom = await zkasm.compile(path.join(__dirname, "zkasm", zkasmFile)); zkasmcom 编译后的结果为：

{
 "program": [
  {
   "inSTEP": "1",
   "setA": 1,
   "line": 3,
   "fileName": "arith.zkasm",
   "lineStr": "        STEP => A"
  },
  {
   "CONST": "0",
   "assert": 1,
   "line": 4,
   "fileName": "arith.zkasm",
   "lineStr": "        0 :ASSERT"
  },
  {
   "CONST": "0",
   "setA": 1,
   "line": 6,
   "fileName": "arith.zkasm",
   "lineStr": "        0 => A"
  },
  {
   "inCntArith": "1",
   "assert": 1,
   "line": 7,
   "fileName": "arith.zkasm",
   "lineStr": "        CNT_ARITH :ASSERT"
  },
  {
   "inCntBinary": "1",
   "assert": 1,
   "line": 8,
   "fileName": "arith.zkasm",
   "lineStr": "        CNT_BINARY :ASSERT"
  },
  {
   "inCntKeccakF": "1",
   "assert": 1,
   "line": 9,
   "fileName": "arith.zkasm",
   "lineStr": "        CNT_KECCAK_F: ASSERT"
  },
  {
   "inCntMemAlign": "1",
   "assert": 1,
   "line": 10,
   "fileName": "arith.zkasm",
   "lineStr": "        CNT_MEM_ALIGN :ASSERT"
  },
  {
   "inCntPoseidonG": "1",
   "assert": 1,
   "line": 11,
   "fileName": "arith.zkasm",
   "lineStr": "        CNT_POSEIDON_G :ASSERT"
  },
  {
   "inCntPaddingPG": "1",
   "assert": 1,
   "line": 12,
   "fileName": "arith.zkasm",
   "lineStr": "        CNT_PADDING_PG :ASSERT"
  },
  {
   "CONST": "0",
   "setA": 1,
   "setB": 1,
   "setC": 1,
   "setD": 1,
   "arith": 1,
   "arithEq0": 1,
   "line": 14,
   "fileName": "arith.zkasm",
   "lineStr": "        0 => A,B,C,D    :ARITH"
  },
  {
   "inCntArith": "1",
   "setA": 1,
   "line": 16,
   "fileName": "arith.zkasm",
   "lineStr": "        CNT_ARITH => A"
  },
  {
   "CONST": "1",
   "assert": 1,
   "line": 17,
   "fileName": "arith.zkasm",
   "lineStr": "        1               :ASSERT"
  },
  {
   "inCntArith": "1",
   "setA": 1,
   "line": 19,
   "fileName": "arith.zkasm",
   "lineStr": "        CNT_ARITH => A"
  },
  {
   "CONST": "1",
   "assert": 1,
   "line": 20,
   "fileName": "arith.zkasm",
   "lineStr": "        1               :ASSERT"
  },
  {
  # CONSTL为0x2000000000000000000000000000000000000000000000000000000000000001n，以8个寄存器CONST0~CONST7表示，对应CONST7值为0x20000000=536870912,CONST0=1。
   "CONSTL": "14474011154664524427946373126085988481658748083205070504932198000989141204993",
   "setA": 1,
   "line": 22,
   "fileName": "arith.zkasm",
   "lineStr": "        0x2000000000000000000000000000000000000000000000000000000000000001n => A"
  },
  {
   "CONST": "256",
   "setB": 1,
   "line": 23,
   "fileName": "arith.zkasm",
   "lineStr": "        0x100    => B"
  },
  {
   "CONST": "115",
   "setC": 1,
   "line": 24,
   "fileName": "arith.zkasm",
   "lineStr": "        0x73     => C"
  },
  {
   "CONST": "32",
   "setD": 1,
   "line": 25,
   "fileName": "arith.zkasm",
   "lineStr": "        0x20    => D"
  },
  {
   "CONST": "371",
   "arith": 1,
   "arithEq0": 1,
   "line": 26,
   "fileName": "arith.zkasm",
   "lineStr": "        0x173   :ARITH"
  },
  {
   "CONST": "2",
   "setA": 1,
   "line": 29,
   "fileName": "arith.zkasm",
   "lineStr": "        2 => A"
  },
  {
   "inCntArith": "1",
   "assert": 1,
   "line": 30,
   "fileName": "arith.zkasm",
   "lineStr": "        CNT_ARITH :ASSERT"
  },
  {
   "CONST": "0",
   "setA": 1,
   "line": 32,
   "fileName": "arith.zkasm",
   "lineStr": "        0 => A"
  },
  {
   "inCntKeccakF": "1",
   "assert": 1,
   "line": 33,
   "fileName": "arith.zkasm",
   "lineStr": "        CNT_KECCAK_F: ASSERT"
  },
  {
   "inCntMemAlign": "1",
   "assert": 1,
   "line": 34,
   "fileName": "arith.zkasm",
   "lineStr": "        CNT_MEM_ALIGN :ASSERT"
  },
  {
   "inCntPoseidonG": "1",
   "assert": 1,
   "line": 35,
   "fileName": "arith.zkasm",
   "lineStr": "        CNT_POSEIDON_G :ASSERT"
  },
  {
   "inCntPaddingPG": "1",
   "assert": 1,
   "line": 36,
   "fileName": "arith.zkasm",
   "lineStr": "        CNT_PADDING_PG :ASSERT"
  },
  {
   "CONST": "0",
   "setA": 1,
   "setB": 1,
   "setC": 1,
   "setD": 1,
   "setE": 1,
   "setCTX": 1,
   "setSP": 1,
   "setPC": 1,
   "setGAS": 1,
   "setMAXMEM": 1,
   "setSR": 1,
   "line": 39,
   "fileName": "arith.zkasm",
   "lineStr": "       0 => A,B,C,D,E,CTX, SP, PC, GAS, MAXMEM, SR"
  },
  {
   "freeInTag": {
    "op": "functionCall",
    "funcName": "beforeLast",
    "params": []
   },
   "inFREE": "1",
   "JMPC": 0,
   "JMPN": 1,
   "offset": 27,
   "line": 42,
   "offsetLabel": "finalWait",
   "fileName": "arith.zkasm",
   "lineStr": "        ${beforeLast()}  : JMPN(finalWait)"
  },
  {
   "JMP": 1,
   "JMPC": 0,
   "JMPN": 0,
   "offset": 0,
   "line": 44,
   "offsetLabel": "start",
   "fileName": "arith.zkasm",
   "lineStr": "                         : JMP(start)"
  }
 ],
 "labels": {
  "start": 0,
  "end": 26,
  "finalWait": 27,
  "opINVALID": 29
 }
}

对应的各常量多项式的赋值为：

indexCONST0CONST1CONST2CONST3CONST4CONST5CONST6CONST7offsetincStackincCodeisStackisCodeisMemindindPRuseCTXmOpmWRsWRsRDaritharithEq0arithEq1arithEq2arithEq3memAlignmemAlignWRmemAlignWR8hashKhashKLenhashDigesthashPhashPLenhashPDigestbinbinOpcodeassertlineinAinBinCinROTL_CinDinEinSRinFREEinCTXinSPinPCinGASinMAXMEMinHASHPOSinSTEPinPRsetAsetBsetCsetDsetEsetSRsetCTXsetSPsetPCsetGASsetMAXMEMsetHASHPOSJMPJMPNJMPCsetPR00000000000000000000000000000000000000000000000000000010100000000000000010000000000000000000000000000000000000110000000000000000000000000000000020000000000000000000000000000000000000020000000000000000100000000000000030000000000000000000000000000000000000130000000000000000000000000000000040000000000000000000000000000000000000140000000000000000000000000000000050000000000000000000000000000000000000150000000000000000000000000000000060000000000000000000000000000000000000160000000000000000000000000000000070000000000000000000000000000000000000170000000000000000000000000000000080000000000000000000000000000000000000180000000000000000000000000000000090000000000000000000001100000000000000090000000000000000111100000000000010000000000000000000000000000000000000001000000000000000001000000000000000111000000000000000000000000000000000000111000000000000000000000000000000001200000000000000000000000000000000000000120000000000000000100000000000000013100000000000000000000000000000000000011300000000000000000000000000000000141000000536870912000000000000000000000000000000140000000000000000100000000000000015256000000000000000000000000000000000000015000000000000000001000000000000001611500000000000000000000000000000000000001600000000000000000010000000000000173200000000000000000000000000000000000001700000000000000000001000000000000183710000000000000000000011000000000000000180000000000000000000000000000000019200000000000000000000000000000000000001900000000000000001000000000000000200000000000000000000000000000000000000120000000000000000000000000000000002100000000000000000000000000000000000000210000000000000000100000000000000022000000000000000000000000000000000000012200000000000000000000000000000000230000000000000000000000000000000000000123000000000000000000000000000000002400000000000000000000000000000000000001240000000000000000000000000000000025000000000000000000000000000000000000012500000000000000000000000000000000260000000000000000000000000000000000000026000000000000000011111111111000002700000000270000000000000000000000000000027000000000000000000000000000001002800000000000000000000000000000000000000280000000000000000000000000000100029000000000000000000000000000000000000002900000000000000000000000000000000 ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ ⋮ \vdots ⋮ 2 21 − 1 2^{21}-1 221−100000000000000000000000000000000000000 2 21 − 1 2^{21}-1 221−100000000000000000000000000000000 5. byte4.pil中的常量多项式

byte4.pil主要用于构建任意32-bit（4字节）数字，其具有常量多项式SET——奇数行为1，偶数行为0。

/*
This state machine is able to builds any number of 4 bytes (32 bits)


Example for building numbers: 0x00030007, 0x12345678, 0x00050009 and 0
        SET     freeIN  out        out'
w^0     1       3       0          3
w^1     0       7       3          0x00030007
w^2     1       0x1234  0x00030007 0x1234
w^3     0       0x5678  0x1234     0x12345678
w^4     1       5       0x12345678 5
w^5     0       9       5          0x50009
w^6     1       0       0x50009    0
w^7     0       0       0          0

*/

include "global.pil";

namespace Byte4(%N);
    /// Constant Polynomials
    pol constant SET;    // 1, 0, 1, 0, 1, 0 ......

    /// State Polynomials
    pol commit freeIN;
    pol commit out;

    freeIN in Global.BYTE2;  // 0, 1, 2,       , 65535

    out' = SET*freeIN +
           (1-SET)*(out * 2**16 + freeIN);

6. padding_kk.pil中的常量多项式

padding_kk.pil中的常量多项式有：

	/* Read Data output
        crLatch * crValid {addr, crOffset - crLen -1, crLen, crV0C, crV1C, crV2C, crV3C, crV4C, crV5C, crV6C, crV7C}
    */

    /* Read Len output
        lastHashLatch {addr, len}
    */

    /* Read Len digest
        lastHashLatch { addr, hash0, hash1, hash2, hash3, hash4, hash5, hash6, hash7 }
    */

namespace PaddingKK(%N);

    // Polynomials that are used to compute a hash chunk
    pol constant r8Id;
	
	pol constant lastBlock;
    pol constant lastBlockLatch;
    pol constant r8valid;
	
	pol constant sOutId;
	
	pol constant forceLastHash;
	
	pol constant k_crOffset, k_crF0, k_crF1, k_crF2, k_crF3, k_crF4, k_crF5, k_crF6, k_crF7;
	
	pol constant crValid;

具体的赋值逻辑为：

const BYTESPERBLOCK = 136;
const BlockSize = 158418;

module.exports.buildConstants = async function (pols) {
    const poseidon = await buildPoseidon();
    const F = poseidon.F;

    const N = pols.lastBlock.length;

    const nBlocks = 9*Math.floor((N-1)/BlockSize);

    let p =0;

    pols.k_crF = [];
    for (let i=0; i

Polygon zkEVM中的常量多项式

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