R1CS-QAP2.ipynb

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   "source": [
    "Evaluate: $ x^{3} + x + 5 $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "c4fc2b48",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "(3, 9)\n",
      "(3, 3)\n",
      "(9, 33)\n",
      "False\n"
     ]
    }
   ],
   "source": [
    "x = 3\n",
    "v1 = x*x\n",
    "out = v1*x + x + 5\n",
    "\n",
    "F73 = GF(73)\n",
    "\n",
    "w = vector(F73,[1, out, x, v1])\n",
    "\n",
    "A = Matrix(F73, [\n",
    "    [0, 0, 1, 0],\n",
    "    [0, 0, 0, 1]\n",
    "])\n",
    "\n",
    "B = Matrix(F73, [\n",
    "    [0, 0, 1, 0],\n",
    "    [0, 0, 1, 0]\n",
    "])\n",
    "\n",
    "C = Matrix(F73, [\n",
    "    [0, 0, 0, 1],\n",
    "    [-5, 1, 1, 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": 5,
   "id": "129cb758",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A\n",
      "[ 0  0]\n",
      "[ 0  0]\n",
      "[ 2 72]\n",
      "[72  1]\n",
      "\n",
      "B\n",
      "[0 0]\n",
      "[0 0]\n",
      "[1 0]\n",
      "[0 0]\n",
      "\n",
      "C\n",
      "[ 5 68]\n",
      "[72  1]\n",
      "[72  1]\n",
      "[ 2 72]\n",
      "\n"
     ]
    }
   ],
   "source": [
    "R73.<x> = PolynomialRing(F73)\n",
    "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": 6,
   "id": "70224c7a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ax =  6*x + 70\n",
      "Bx =  3\n",
      "Cx =  24*x + 58\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": 7,
   "id": "b5163ac0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "T =  67*x + 6\n",
      "Polynomial roots\n",
      "T(1) = 0\n",
      "T(2) = 67\n",
      "T(3) = 61\n",
      "T(4) = 55\n",
      "Z = 24\n",
      "Quotient of Z/T = 18*x + 55\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": []
  }
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