R1CS-QAP.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "f885a6f3",
   "metadata": {},
   "source": [
    "Evaluate: $ x^{3} + x + 5 $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "c4fc2b48",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(3, 9, 30, 35)\n",
      "(3, 3, 1, 1)\n",
      "(9, 27, 30, 35)\n",
      "True\n"
     ]
    }
   ],
   "source": [
    "x = 3\n",
    "v1 = x*x\n",
    "v2 = v1*x\n",
    "v3 = v2 + x\n",
    "out = v3 + 5\n",
    "\n",
    "F73 = GF(73)\n",
    "\n",
    "w = vector(F73,[1, out, x, v1, v2, v3])\n",
    "\n",
    "A = Matrix(F73, [\n",
    "    [0, 0, 1, 0, 0, 0],\n",
    "    [0, 0, 0, 1, 0, 0],\n",
    "    [0, 0, 1, 0, 1, 0],\n",
    "    [5, 0, 0, 0, 0, 1]\n",
    "])\n",
    "\n",
    "B = Matrix(F73, [\n",
    "    [0, 0, 1, 0, 0, 0],\n",
    "    [0, 0, 1, 0, 0, 0],\n",
    "    [1, 0, 0, 0, 0, 0],\n",
    "    [1, 0, 0, 0, 0, 0],\n",
    "])\n",
    "\n",
    "C = Matrix(F73, [\n",
    "    [0, 0, 0, 1, 0, 0],\n",
    "    [0, 0, 0, 0, 1, 0],\n",
    "    [0, 0, 0, 0, 0, 1],\n",
    "    [0, 1, 0, 0, 0, 0]\n",
    "])\n",
    "\n",
    "Aw = A*w\n",
    "Bw = B*w\n",
    "Cw = C*w\n",
    "\n",
    "print(Aw)\n",
    "print(Bw)\n",
    "print(Cw)\n",
    "\n",
    "# Compute the Hadamard product\n",
    "AwBw = vector([Aw[i]*Bw[i] for i in range(len(Aw))])\n",
    "\n",
    "print(Cw == AwBw)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "id": "129cb758",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A\n",
      "[68 70 68 13]\n",
      "[ 0  0  0  0]\n",
      "[ 8 13  5 48]\n",
      "[67 46 69 37]\n",
      "[ 4 66 40 36]\n",
      "[72 14 72 61]\n",
      "\n",
      "B\n",
      "[ 3  7 39 24]\n",
      "[ 0  0  0  0]\n",
      "[71 66 34 49]\n",
      "[ 0  0  0  0]\n",
      "[ 0  0  0  0]\n",
      "[ 0  0  0  0]\n",
      "\n",
      "C\n",
      "[ 0  0  0  0]\n",
      "[72 14 72 61]\n",
      "[ 0  0  0  0]\n",
      "[ 4 20 38 12]\n",
      "[67 46 69 37]\n",
      "[ 4 66 40 36]\n",
      "\n"
     ]
    }
   ],
   "source": [
    "M = [A, B, C]\n",
    "PolyMtx = []\n",
    "\n",
    "for m in M:\n",
    "    PolyList = []\n",
    "    for i in range(m.ncols()):\n",
    "        points = []\n",
    "        for j in range(m.nrows()):\n",
    "            points.append([j+1, m[j, i]])\n",
    "            \n",
    "        Poly = R73.lagrange_polynomial(points).coefficients()\n",
    "        \n",
    "        if (len(Poly) < m.nrows()):\n",
    "            diff = m.nrows() - len(Poly)\n",
    "            for c in range(diff):\n",
    "                Poly.append(0)\n",
    "                \n",
    "        PolyList.append(Poly)\n",
    "        \n",
    "    PolyMtx.append(Matrix(F73, PolyList))\n",
    "        \n",
    "print(f'''A\n",
    "{PolyMtx[0]}\n",
    "''')\n",
    "\n",
    "print(f'''B\n",
    "{PolyMtx[1]}\n",
    "''')\n",
    "\n",
    "print(f'''C\n",
    "{PolyMtx[2]}\n",
    "''')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "70224c7a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ax =  7*x^3 + 2*x^2 + 24*x + 43\n",
      "Bx =  25*x^3 + 68*x^2 + 59*x + 70\n",
      "Cx =  15*x^3 + 12*x^2 + 23*x + 32\n"
     ]
    }
   ],
   "source": [
    "Ax = R73(list(w*PolyMtx[0]))\n",
    "Bx = R73(list(w*PolyMtx[1]))\n",
    "Cx = R73(list(w*PolyMtx[2]))\n",
    "\n",
    "print(\"Ax = \", Ax)\n",
    "print(\"Bx = \", Bx)\n",
    "print(\"Cx = \", Cx)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "id": "b5163ac0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "T =  29*x^6 + 15*x^5 + 54*x^4 + 15*x^3 + 15*x^2 + 33*x + 58\n",
      "Polynomial roots\n",
      "T(1) = 0\n",
      "T(2) = 0\n",
      "T(3) = 0\n",
      "T(4) = 0\n",
      "Z = 24\n",
      "Quotient of Z/T = 59*x^6 + 28*x^5 + 57*x^4 + 28*x^3 + 28*x^2 + 47*x + 45\n",
      "Remainder of Z/T = 0\n"
     ]
    }
   ],
   "source": [
    "T = Ax*Bx - Cx\n",
    "\n",
    "print(\"T = \", T)\n",
    "print(\"Polynomial roots\")\n",
    "print(\"T(1) =\", T(1))\n",
    "print(\"T(2) =\", T(2))\n",
    "print(\"T(3) =\", T(3))\n",
    "print(\"T(4) =\", T(4))\n",
    "\n",
    "x = 5\n",
    "Z = R73((x-1)*(x-2)*(x-3)*(x-4))\n",
    "\n",
    "print(\"Z =\", Z)\n",
    "\n",
    "H = T.quo_rem(Z)\n",
    "print(\"Quotient of Z/T = \", end=\"\")\n",
    "print(H[0])\n",
    "print(\"Remainder of Z/T = \", end=\"\")\n",
    "print(H[1])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a11e86e1",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 10.0",
   "language": "sage",
   "name": "sagemath-10.0"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}