GF2 tests.

gf2-tests.js

'use strict';

const assert = require('assert');

// Code ..

console.log(`remainder => ${ffmul25(26, 5)}`);

{
  const mul = ffmul25(0x1d, 0x02);
  console.log(`0x1d(29) * 0x02(2) mod 0x29(41) = ${byte2hex(mul)}(${mul})`)
  console.log(`--
  (${bit2pol6(0x1d)})\t(0x1d -- 29)
  *
  (${bit2pol6(0x02)})\t(0x02 -- 2)
  mod
  (${bit2pol6(0x29)})\t(0x29 -- 41)
  =
  (${bit2pol6(mul)})\t(0x13 -- 19)
--`);
}

/**
 * Multiplication with generator
 * and log/exp tables.
 * We use simplest generator x + 1 over GF(2^8).
 */
const GENERATOR = 0x03; // x + 1

// generate tables
//const GF256_EXP = generateExps(GENERATOR);
//const GF256_LOG = generateLogs(GF256_EXP);
//console.log(print256(GF256_EXP));
//console.log(print256(GF256_LOG));

const GF256_EXP = [
  0x01, 0x03, 0x05, 0x0f, 0x11, 0x33, 0x55, 0xff,
  0x1a, 0x2e, 0x72, 0x96, 0xa1, 0xf8, 0x13, 0x35,
  0x5f, 0xe1, 0x38, 0x48, 0xd8, 0x73, 0x95, 0xa4,
  0xf7, 0x02, 0x06, 0x0a, 0x1e, 0x22, 0x66, 0xaa,
  0xe5, 0x34, 0x5c, 0xe4, 0x37, 0x59, 0xeb, 0x26,
  0x6a, 0xbe, 0xd9, 0x70, 0x90, 0xab, 0xe6, 0x31,
  0x53, 0xf5, 0x04, 0x0c, 0x14, 0x3c, 0x44, 0xcc,
  0x4f, 0xd1, 0x68, 0xb8, 0xd3, 0x6e, 0xb2, 0xcd,
  0x4c, 0xd4, 0x67, 0xa9, 0xe0, 0x3b, 0x4d, 0xd7,
  0x62, 0xa6, 0xf1, 0x08, 0x18, 0x28, 0x78, 0x88,
  0x83, 0x9e, 0xb9, 0xd0, 0x6b, 0xbd, 0xdc, 0x7f,
  0x81, 0x98, 0xb3, 0xce, 0x49, 0xdb, 0x76, 0x9a,
  0xb5, 0xc4, 0x57, 0xf9, 0x10, 0x30, 0x50, 0xf0,
  0x0b, 0x1d, 0x27, 0x69, 0xbb, 0xd6, 0x61, 0xa3,
  0xfe, 0x19, 0x2b, 0x7d, 0x87, 0x92, 0xad, 0xec,
  0x2f, 0x71, 0x93, 0xae, 0xe9, 0x20, 0x60, 0xa0,
  0xfb, 0x16, 0x3a, 0x4e, 0xd2, 0x6d, 0xb7, 0xc2,
  0x5d, 0xe7, 0x32, 0x56, 0xfa, 0x15, 0x3f, 0x41,
  0xc3, 0x5e, 0xe2, 0x3d, 0x47, 0xc9, 0x40, 0xc0,
  0x5b, 0xed, 0x2c, 0x74, 0x9c, 0xbf, 0xda, 0x75,
  0x9f, 0xba, 0xd5, 0x64, 0xac, 0xef, 0x2a, 0x7e,
  0x82, 0x9d, 0xbc, 0xdf, 0x7a, 0x8e, 0x89, 0x80,
  0x9b, 0xb6, 0xc1, 0x58, 0xe8, 0x23, 0x65, 0xaf,
  0xea, 0x25, 0x6f, 0xb1, 0xc8, 0x43, 0xc5, 0x54,
  0xfc, 0x1f, 0x21, 0x63, 0xa5, 0xf4, 0x07, 0x09,
  0x1b, 0x2d, 0x77, 0x99, 0xb0, 0xcb, 0x46, 0xca,
  0x45, 0xcf, 0x4a, 0xde, 0x79, 0x8b, 0x86, 0x91,
  0xa8, 0xe3, 0x3e, 0x42, 0xc6, 0x51, 0xf3, 0x0e,
  0x12, 0x36, 0x5a, 0xee, 0x29, 0x7b, 0x8d, 0x8c,
  0x8f, 0x8a, 0x85, 0x94, 0xa7, 0xf2, 0x0d, 0x17,
  0x39, 0x4b, 0xdd, 0x7c, 0x84, 0x97, 0xa2, 0xfd,
  0x1c, 0x24, 0x6c, 0xb4, 0xc7, 0x52, 0xf6, 0x01
];

const GF256_LOG = [
  -0x1, 0x00, 0x19, 0x01, 0x32, 0x02, 0x1a, 0xc6,
  0x4b, 0xc7, 0x1b, 0x68, 0x33, 0xee, 0xdf, 0x03,
  0x64, 0x04, 0xe0, 0x0e, 0x34, 0x8d, 0x81, 0xef,
  0x4c, 0x71, 0x08, 0xc8, 0xf8, 0x69, 0x1c, 0xc1,
  0x7d, 0xc2, 0x1d, 0xb5, 0xf9, 0xb9, 0x27, 0x6a,
  0x4d, 0xe4, 0xa6, 0x72, 0x9a, 0xc9, 0x09, 0x78,
  0x65, 0x2f, 0x8a, 0x05, 0x21, 0x0f, 0xe1, 0x24,
  0x12, 0xf0, 0x82, 0x45, 0x35, 0x93, 0xda, 0x8e,
  0x96, 0x8f, 0xdb, 0xbd, 0x36, 0xd0, 0xce, 0x94,
  0x13, 0x5c, 0xd2, 0xf1, 0x40, 0x46, 0x83, 0x38,
  0x66, 0xdd, 0xfd, 0x30, 0xbf, 0x06, 0x8b, 0x62,
  0xb3, 0x25, 0xe2, 0x98, 0x22, 0x88, 0x91, 0x10,
  0x7e, 0x6e, 0x48, 0xc3, 0xa3, 0xb6, 0x1e, 0x42,
  0x3a, 0x6b, 0x28, 0x54, 0xfa, 0x85, 0x3d, 0xba,
  0x2b, 0x79, 0x0a, 0x15, 0x9b, 0x9f, 0x5e, 0xca,
  0x4e, 0xd4, 0xac, 0xe5, 0xf3, 0x73, 0xa7, 0x57,
  0xaf, 0x58, 0xa8, 0x50, 0xf4, 0xea, 0xd6, 0x74,
  0x4f, 0xae, 0xe9, 0xd5, 0xe7, 0xe6, 0xad, 0xe8,
  0x2c, 0xd7, 0x75, 0x7a, 0xeb, 0x16, 0x0b, 0xf5,
  0x59, 0xcb, 0x5f, 0xb0, 0x9c, 0xa9, 0x51, 0xa0,
  0x7f, 0x0c, 0xf6, 0x6f, 0x17, 0xc4, 0x49, 0xec,
  0xd8, 0x43, 0x1f, 0x2d, 0xa4, 0x76, 0x7b, 0xb7,
  0xcc, 0xbb, 0x3e, 0x5a, 0xfb, 0x60, 0xb1, 0x86,
  0x3b, 0x52, 0xa1, 0x6c, 0xaa, 0x55, 0x29, 0x9d,
  0x97, 0xb2, 0x87, 0x90, 0x61, 0xbe, 0xdc, 0xfc,
  0xbc, 0x95, 0xcf, 0xcd, 0x37, 0x3f, 0x5b, 0xd1,
  0x53, 0x39, 0x84, 0x3c, 0x41, 0xa2, 0x6d, 0x47,
  0x14, 0x2a, 0x9e, 0x5d, 0x56, 0xf2, 0xd3, 0xab,
  0x44, 0x11, 0x92, 0xd9, 0x23, 0x20, 0x2e, 0x89,
  0xb4, 0x7c, 0xb8, 0x26, 0x77, 0x99, 0xe3, 0xa5,
  0x67, 0x4a, 0xed, 0xde, 0xc5, 0x31, 0xfe, 0x18,
  0x0d, 0x63, 0x8c, 0x80, 0xc0, 0xf7, 0x70, 0x07
];

console.log(`0xb6 * 0x53 mod 0x1b = ${byte2hex(ffmul28(0xb6, 0x53, 0x1b))}`)
console.log(`0xb6 * 0x53 mod 0x1b = ${byte2hex(ffmulFast(0xb6, 0x53))}`);

// functions ..

/**
 * GF(32) Addition and Multiplication
 */

/**
 * Addition for coeffients
 * @param {Number} a - 5 bits
 * @param {Number} b - 5 bits
 */

function add2(a, b) {
  assert(typeof a === 'number', 'a must be a number.');
  assert(typeof b === 'number', 'b must be a number.');

  // take only five bits
  a = a & 0x1f;
  b = b & 0x1f;

  return a ^ b;
}

/**
 * This function will compute multiplication of two
 * polynomials withing GF(2^5)
 * @param {Number} a - 5 bits will be used
 * @param {Number} b - only 5 bits will be used and 
 */
function mul2(a, b) {
  // take only five bits
  a = a & 0x1f;
  b = b & 0x1f;

  let r = 0;

  // if we have a(x) * x^0
  if (a & 0x01) r ^= b << 0; // r' = r(x) + b(x) * x^0

  // if we have a(x) * x^1
  if (a & 0x02) r ^= b << 1; // r' = r(x) + b(x) * x^1

  // ...
  if (a & 0x04) r ^= b << 2;
  if (a & 0x08) r ^= b << 3;
  if (a & 0x10) r ^= b << 4;

  return r;
}

function ffmul2(a, b, m, d) {
  a = a & 0xff;
  b = b & 0xff;
  m = m & 0xff;
  d = d & 0xff;

  let r = 0;
  let t;

  while (a != 0) {
    // do we have coeffient?
    if ((a & 1) != 0)
      r ^= b; // then add those coeffient.

    t = b & d; // highest degree?
    b <<= 1;

    // if we have higher degree then subtract m
    if (t != 0)
      b ^= m;

    a >>>= 1;
  }

  return r;
}

/**
 * This function will multiply two polynomials
 * over GF(2) with maximum degree of 5
 * @param {Number} a
 * @param {Number} b
 * @param {Number} [m=x^5 + x^3 + 1] - modulo
 * @returns {Number}
 */

function ffmul25(a, b, m = 0x29) {
  return ffmul2(a & 0x1f, b & 0x1f, m, 0x10);
}

/**
 * Same as ffmul25, but with max degree of 8
 * @param {Number} a
 * @param {Number} b
 * @param {Number} m
 * @returns {Number}
 */
function ffmul28(a, b, m) {
  return ffmul2(a, b, m, 0x80);
}

/**
 * ffmul210, max degree ??
 * @param {Number} a
 * @param {Number} b
 * @param {Number} [m=???] - modulo
 * @returns {Number}
 */

function ffmul210(a, b, m) {
  return ffmul2(a & 0x3ff, b & 0x3ff, m, 0x400);
}


/**
 * Polynomial multiplication over GF(2^8)
 * using precomputed tables
 * @param {Number} a
 * @param {Number} b
 * @returns {Number}
 */

function ffmulFast(a, b) {
  if (a === 0 || b === 0)
    return 0;

  let t = GF256_LOG[a] + GF256_LOG[b];

  if (t > 0xff)
    t -= 255;

  return GF256_EXP[t];
}


function generateExps(generator) {
  const exps = new Array(256);

  let x = 0x01;

  exps[0] = x;
  for (let i = 1; i < 256; i++) {
    const y = ffmul28(x, GENERATOR, 0x1b);
    exps[i] = y;
    x = y;
  }

  return exps;
}

function generateLogs(exps) {
  const logs = new Array(256);

  logs[1] = 0;
  for (let i = 1; i < 255; i++) {
    logs[exps[i] & 0xff] = i;
  }

  return logs;
}

/**
 * Helpers
 */

// format binary
function fb(a, bits = 5) {
  return ('0'.repeat(bits) + a.toString(2)).slice(-bits);
}

// for print256
function byte2hex(a) {
  if (!Number.isInteger(a)) return '-0x1';

  return '0x' + ('00' + a.toString(16)).slice(-2);
}

// print 6 bit representation as polynomial
function bit2pol6(n) {
  let pol = '';

  if (n & 0x20) pol += '+ x^5 ';
  if (n & 0x10) pol += '+ x^4 ';
  if (n & 0x08) pol += '+ x^3 ';
  if (n & 0x04) pol += '+ x^2 ';
  if (n & 0x02) pol += '+ x^1 ';
  if (n & 0x01) pol += '+ 1 ';

  return pol.substring(2, pol.length - 1);
}

// hex print tables for GF(256)
function print256(table) {
  const str = [];

  str.push('[\n');

  while (table.length > 0) {
    str.push('  ');

    for (let i = 0; i < 8; i++)
      str.push(byte2hex(table.shift()), ', ');

    str.push('\n');
  }

  str.splice(-2, 1);
  str.push(']');


  return str.join('');
}