xor.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {},
   "outputs": [],
   "source": [
    "def bit_indices(n):\n",
    "    i = 0\n",
    "    res = []\n",
    "    while n > 0:\n",
    "        if (n & 1) != 0:\n",
    "            res.append(i)\n",
    "        n //= 2\n",
    "        i += 1\n",
    "    return res\n",
    "def parity(n): return len(bit_indices(n)) % 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {},
   "outputs": [],
   "source": [
    "R = IntegerModRing(8)\n",
    "n = 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {},
   "outputs": [],
   "source": [
    "alphas = [chr(ord('a') + i) for i in range(n)]\n",
    "var_names = [''.join(alphas[i] for i in bit_indices(S)) for S in range(1, 2**n - 1)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = Matrix(R, [[parity(S & T) for T in range(1, 2**n - 1)] for S in range(0, 2**n)])\n",
    "b = vector(R, [parity(S) for S in range(0, 2**n)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {},
   "outputs": [],
   "source": [
    "sol = A.solve_right(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-3*a - 3*b - 3*ab - 3*c + ac + bc + abc + 3*d + 3*ad - bd + 3*abd - cd - acd + 3*bcd"
      ]
     },
     "execution_count": 171,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_with_vars = R[[var(name) for name in var_names]]\n",
    "expr = sum(R_with_vars(a) * x for a, x in zip(sol, R_with_vars.gens()))\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0^0^0^0: ref(0) =?= f(0): True\n",
      "0^0^0^1: ref(1) =?= f(1): True\n",
      "0^0^1^0: ref(1) =?= f(1): True\n",
      "0^0^1^1: ref(0) =?= f(0): True\n",
      "0^1^0^0: ref(1) =?= f(1): True\n",
      "0^1^0^1: ref(0) =?= f(0): True\n",
      "0^1^1^0: ref(0) =?= f(0): True\n",
      "0^1^1^1: ref(1) =?= f(1): True\n",
      "1^0^0^0: ref(1) =?= f(1): True\n",
      "1^0^0^1: ref(0) =?= f(0): True\n",
      "1^0^1^0: ref(0) =?= f(0): True\n",
      "1^0^1^1: ref(1) =?= f(1): True\n",
      "1^1^0^0: ref(0) =?= f(0): True\n",
      "1^1^0^1: ref(1) =?= f(1): True\n",
      "1^1^1^0: ref(1) =?= f(1): True\n",
      "1^1^1^1: ref(0) =?= f(0): True\n"
     ]
    }
   ],
   "source": [
    "def evalxor3(f):\n",
    "    for (i, j, k, l) in [(i, j, k, l) for i in [0, 1] for j in [0, 1] for k in [0, 1] for l in [0, 1]]:\n",
    "      ref = i^^j^^k^^l\n",
    "      eval = f.substitute(a=i, b=j, c=k, d=l, ab=i^^j, ac=i^^k, ad=i^^l, bc=j^^k, bd=j^^l, cd=k^^l, abc=i^^j^^k, abd=i^^j^^l,\n",
    "                          acd=i^^k^^l, bcd=j^^k^^l)\n",
    "      print(\"%s^%s^%s^%s: ref(%s) =?= f(%s): %s\" % \n",
    "        (i, j, k, l, ref, eval, ref == eval))\n",
    "evalxor3(expr)"
   ]
  }
 ],
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