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   "source": [
    "# Mahdavi et al.'s OT-MP-PSI Protocol\n",
    "## An implementation in SageMath for research purposes\n",
    "\n",
    "We implement the OT-MP-PSI protocol of Mahdavi et al. It is based on Shamir's Secret Sharing and the homomorphic Paillier encryption scheme.\n",
    "\n",
    "### Mathematical Preamble\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "42670e84",
   "metadata": {},
   "outputs": [],
   "source": [
    "import hashlib\n",
    "import random\n",
    "\n",
    "# Prime order of the finite field\n",
    "q = 11\n",
    "p = 2*q + 1 # 23 is prime!\n",
    "# Threshold value\n",
    "t = 4\n",
    "# Number of participants (holders of a dataset)\n",
    "m = 7\n",
    "# Secret to reconstruct\n",
    "S = 0\n",
    "\n",
    "Fp = GF(p)\n",
    "R = PolynomialRing(Fp,\"x\")\n",
    "\n",
    "G = Integers(q)\n",
    "\n",
    "# Paillier Cryptosystem\n",
    "p2 = 13\n",
    "q2 = 17\n",
    "assert(p2.is_prime())\n",
    "assert(q2.is_prime())\n",
    "N  = p2 * q2\n",
    "\n",
    "alpha = 0\n",
    "\n",
    "#FN2 = FiniteField(N^2, 'g')\n",
    "ZN2 = Zmod(N^2)\n",
    "FN2 = [a for a in ZN2 if gcd(a,N^2) == 1]\n",
    "\n",
    "g = random.choice(FN2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8c2c98e4",
   "metadata": {},
   "source": [
    "### Paillier Encryption"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "a264ea15",
   "metadata": {},
   "outputs": [],
   "source": [
    "def paillier_encrypt(element):\n",
    "    r = random.randint(0, N-1)\n",
    "    return pow(g, element) * pow(r, N)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d76a9195",
   "metadata": {},
   "source": [
    "### Initialize Shamir's Secret Sharing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "695e93bf",
   "metadata": {},
   "outputs": [],
   "source": [
    "def sample_coefficients(t):\n",
    "    coefficients = []\n",
    "    coefficients.append(S)\n",
    "    for i in range(0, t-1):\n",
    "        coefficients.append(Fp.random_element())\n",
    "    print(coefficients)\n",
    "    return coefficients"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "7c2565c5",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Construct Polynomial\n",
    "def poly(coefficients):\n",
    "    return R(coefficients)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "b61c1603",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, 6, 6, 0]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 27,
     "metadata": {},
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   ],
   "source": [
    "f_x = poly(sample_coefficients(t))\n",
    "f_x(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "4c8c5660",
   "metadata": {},
   "outputs": [],
   "source": [
    "def H(element):\n",
    "    h = hashlib.new('sha256')\n",
    "    h.update(bytes(element))\n",
    "    return int(h.hexdigest(), 16) % q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "54e8a4bf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7"
      ]
     },
     "execution_count": 29,
     "metadata": {},
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   ],
   "source": [
    "H(0b10011101)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "b9bada7f",
   "metadata": {},
   "outputs": [],
   "source": [
    "def keyholder_init():\n",
    "    coefficients = sample_coefficients(t)\n",
    "    \n",
    "def participant_init(element):\n",
    "    global alpha \n",
    "    alpha = random.randint(1,p)\n",
    "    print(alpha)\n",
    "    return (pow(H(element), alpha), pow(g, alpha))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5d43f845",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "fd75b745",
   "metadata": {},
   "outputs": [],
   "source": [
    "def keyholder_processing(part_init_res):\n",
    "    hash_element, g_power_alpha = part_init_res\n",
    "    print(hash_element)\n",
    "    random_numbers = []\n",
    "    for i in range(0, t-1):\n",
    "        random_numbers.append(Fp.random_element())\n",
    "    \n",
    "    return_values = []\n",
    "    print(random_numbers)\n",
    "    for j in range(0, t-1):\n",
    "        return_values.append(pow(g_power_alpha, random_numbers[j]) * pow(hash_element, coefficients[j+1]))\n",
    "        \n",
    "    return return_values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "8395e9bd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, 8, 14, 12]\n",
      "[0, 8, 14, 12]\n",
      "3\n",
      "3\n",
      "(64, 16102)\n",
      "64\n",
      "[21, 21, 3]\n",
      "3\n",
      "64\n",
      "[6, 9, 6]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[5806, 32072, 38726]"
      ]
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     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coefficients = sample_coefficients(t)\n",
    "print(coefficients)\n",
    "res = participant_init(10)\n",
    "print(alpha)\n",
    "print(res)\n",
    "keyholder_processing(res)\n",
    "\n",
    "def participant_2(keyholder_values):\n",
    "    intermediate_results = []\n",
    "    for i in keyholder_values:\n",
    "        intermediate_results.append(pow(i, alpha))\n",
    "        \n",
    "    # Paillier Encryption\n",
    "    encrypted_values = []\n",
    "    for i in intermediate_results:\n",
    "        encrypted_values.append(paillier_encrypt(i))\n",
    "    \n",
    "    return encrypted_values\n",
    "    \n",
    "    \n",
    "participant_2(keyholder_processing(participant_init(10)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6cc33c78",
   "metadata": {},
   "outputs": [],
   "source": [
    "def keyholder_2(encrypted_values):\n",
    "    intermediate_values = []\n",
    "    for i in encrypted_values:\n",
    "        intermediate_values.append(i * )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "dcb3185e",
   "metadata": {},
   "outputs": [],
   "source": []
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