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{ |
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"cells": [ |
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{ |
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"cell_type": "markdown", |
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"id": "d172899c", |
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"metadata": {}, |
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"source": [ |
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"# Mahdavi et al.'s OT-MP-PSI Protocol\n", |
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"## An implementation in SageMath for research purposes\n", |
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"\n", |
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"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", |
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"\n", |
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"### Mathematical Preamble\n" |
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] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": 23, |
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"id": "42670e84", |
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"metadata": {}, |
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"outputs": [], |
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"source": [ |
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"import hashlib\n", |
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"import random\n", |
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"\n", |
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"# Prime order of the finite field\n", |
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"q = 11\n", |
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"p = 2*q + 1 # 23 is prime!\n", |
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"# Threshold value\n", |
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"t = 4\n", |
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"# Number of participants (holders of a dataset)\n", |
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"m = 7\n", |
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"# Secret to reconstruct\n", |
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"S = 0\n", |
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"\n", |
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"Fp = GF(p)\n", |
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"R = PolynomialRing(Fp,\"x\")\n", |
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"\n", |
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"G = Integers(q)\n", |
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"\n", |
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"# Paillier Cryptosystem\n", |
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"p2 = 13\n", |
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"q2 = 17\n", |
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"assert(p2.is_prime())\n", |
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"assert(q2.is_prime())\n", |
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"N = p2 * q2\n", |
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"\n", |
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"alpha = 0\n", |
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"\n", |
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"#FN2 = FiniteField(N^2, 'g')\n", |
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"ZN2 = Zmod(N^2)\n", |
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"FN2 = [a for a in ZN2 if gcd(a,N^2) == 1]\n", |
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"\n", |
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"g = random.choice(FN2)" |
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] |
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}, |
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{ |
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"cell_type": "markdown", |
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"id": "8c2c98e4", |
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"metadata": {}, |
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"source": [ |
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"### Paillier Encryption" |
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] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": 24, |
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"id": "a264ea15", |
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"metadata": {}, |
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"outputs": [], |
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"source": [ |
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"def paillier_encrypt(element):\n", |
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" r = random.randint(0, N-1)\n", |
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" return pow(g, element) * pow(r, N)" |
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] |
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}, |
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{ |
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"cell_type": "markdown", |
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"id": "d76a9195", |
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"metadata": {}, |
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"source": [ |
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"### Initialize Shamir's Secret Sharing" |
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] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": 25, |
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"id": "695e93bf", |
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"metadata": {}, |
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"outputs": [], |
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"source": [ |
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"def sample_coefficients(t):\n", |
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" coefficients = []\n", |
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" coefficients.append(S)\n", |
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" for i in range(0, t-1):\n", |
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" coefficients.append(Fp.random_element())\n", |
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" print(coefficients)\n", |
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" return coefficients" |
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] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": 26, |
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"id": "7c2565c5", |
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"metadata": {}, |
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"outputs": [], |
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"source": [ |
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"# Construct Polynomial\n", |
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"def poly(coefficients):\n", |
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" return R(coefficients)" |
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] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": 27, |
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"id": "b61c1603", |
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"metadata": {}, |
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"outputs": [ |
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{ |
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"name": "stdout", |
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"output_type": "stream", |
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"text": [ |
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"[0, 6, 6, 0]\n" |
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] |
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}, |
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{ |
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"data": { |
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"text/plain": [ |
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"5" |
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] |
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}, |
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"execution_count": 27, |
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"metadata": {}, |
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"output_type": "execute_result" |
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} |
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], |
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"source": [ |
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"f_x = poly(sample_coefficients(t))\n", |
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"f_x(4)" |
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] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": 28, |
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"id": "4c8c5660", |
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"metadata": {}, |
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"outputs": [], |
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"source": [ |
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"def H(element):\n", |
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" h = hashlib.new('sha256')\n", |
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" h.update(bytes(element))\n", |
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" return int(h.hexdigest(), 16) % q" |
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] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": 29, |
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"id": "54e8a4bf", |
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"metadata": {}, |
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"outputs": [ |
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{ |
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"data": { |
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"text/plain": [ |
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"7" |
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] |
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}, |
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"execution_count": 29, |
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"metadata": {}, |
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"output_type": "execute_result" |
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} |
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], |
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"source": [ |
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"H(0b10011101)" |
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] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": 39, |
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"id": "b9bada7f", |
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"metadata": {}, |
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"outputs": [], |
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"source": [ |
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"def keyholder_init():\n", |
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" coefficients = sample_coefficients(t)\n", |
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" \n", |
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"def participant_init(element):\n", |
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" global alpha \n", |
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" alpha = random.randint(1,p)\n", |
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" print(alpha)\n", |
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" return (pow(H(element), alpha), pow(g, alpha))" |
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] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": null, |
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"id": "5d43f845", |
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"metadata": {}, |
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"outputs": [], |
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"source": [] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": 40, |
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"id": "fd75b745", |
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"metadata": {}, |
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"outputs": [], |
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"source": [ |
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"def keyholder_processing(part_init_res):\n", |
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" hash_element, g_power_alpha = part_init_res\n", |
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" print(hash_element)\n", |
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" random_numbers = []\n", |
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" for i in range(0, t-1):\n", |
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" random_numbers.append(Fp.random_element())\n", |
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" \n", |
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" return_values = []\n", |
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" print(random_numbers)\n", |
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" for j in range(0, t-1):\n", |
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" return_values.append(pow(g_power_alpha, random_numbers[j]) * pow(hash_element, coefficients[j+1]))\n", |
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" \n", |
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" return return_values" |
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] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": 41, |
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"id": "8395e9bd", |
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"metadata": {}, |
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"outputs": [ |
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{ |
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"name": "stdout", |
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"output_type": "stream", |
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"text": [ |
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"[0, 8, 14, 12]\n", |
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"[0, 8, 14, 12]\n", |
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"3\n", |
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"3\n", |
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"(64, 16102)\n", |
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"64\n", |
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"[21, 21, 3]\n", |
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"3\n", |
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"64\n", |
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"[6, 9, 6]\n" |
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] |
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}, |
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{ |
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"data": { |
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"text/plain": [ |
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"[5806, 32072, 38726]" |
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] |
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}, |
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"execution_count": 41, |
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"metadata": {}, |
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"output_type": "execute_result" |
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} |
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], |
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"source": [ |
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"coefficients = sample_coefficients(t)\n", |
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"print(coefficients)\n", |
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"res = participant_init(10)\n", |
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"print(alpha)\n", |
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"print(res)\n", |
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"keyholder_processing(res)\n", |
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"\n", |
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"def participant_2(keyholder_values):\n", |
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" intermediate_results = []\n", |
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" for i in keyholder_values:\n", |
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" intermediate_results.append(pow(i, alpha))\n", |
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" \n", |
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" # Paillier Encryption\n", |
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" encrypted_values = []\n", |
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" for i in intermediate_results:\n", |
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" encrypted_values.append(paillier_encrypt(i))\n", |
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" \n", |
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" return encrypted_values\n", |
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" \n", |
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" \n", |
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"participant_2(keyholder_processing(participant_init(10)))" |
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] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": null, |
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"id": "6cc33c78", |
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"metadata": {}, |
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"outputs": [], |
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"source": [ |
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"def keyholder_2(encrypted_values):\n", |
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" intermediate_values = []\n", |
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" for i in encrypted_values:\n", |
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" intermediate_values.append(i * )" |
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] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": null, |
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"id": "dcb3185e", |
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"metadata": {}, |
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"outputs": [], |
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"source": [] |
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} |
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], |
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"metadata": { |
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"kernelspec": { |
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"display_name": "SageMath 9.5", |
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"language": "sage", |
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"name": "sagemath" |
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}, |
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"language_info": { |
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"codemirror_mode": { |
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"name": "ipython", |
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"version": 3 |
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}, |
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"file_extension": ".py", |
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"mimetype": "text/x-python", |
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"name": "python", |
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"nbconvert_exporter": "python", |
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"pygments_lexer": "ipython3", |
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"version": "3.10.12" |
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} |
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}, |
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"nbformat": 4, |
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"nbformat_minor": 5 |
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} |
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{ |
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"cells": [ |
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{ |
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"cell_type": "code", |
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"execution_count": 2, |
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"id": "8a105f7c", |
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"metadata": {}, |
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"outputs": [ |
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{ |
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"name": "stdout", |
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"output_type": "stream", |
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"text": [ |
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"[0, 15, 12, 18]\n", |
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"18*x^3 + 12*x^2 + 15*x\n", |
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"share 1: 7\n", |
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"share 2: 13\n", |
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"share 3: 12\n", |
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"share 4: 17\n", |
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"share 5: 3\n", |
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"share 6: 2\n", |
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"[(1, 7), (2, 13), (3, 12), (4, 17), (5, 3), (6, 2)]\n", |
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"0\n", |
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"[0, 15, 12, 18]\n", |
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"0\n", |
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"0\n" |
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] |
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} |
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], |
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"source": [ |
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"import hashlib\n", |
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"import random\n", |
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"\n", |
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"# Prime order of the finite field\n", |
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"q = 19\n", |
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"\n", |
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"Fq = GF(q)\n", |
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"R = PolynomialRing(Fq,\"x\")\n", |
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"\n", |
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"t = 4\n", |
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"m = 7\n", |
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"S = 0\n", |
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"\n", |
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"coefficients = []\n", |
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"coefficients.append(S)\n", |
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"for i in range(1, t):\n", |
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" coefficients.append(Fq.random_element())\n", |
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"\n", |
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"print(coefficients)\n", |
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"\n", |
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"f_x = R(coefficients)\n", |
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"print(f_x)\n", |
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"shares = []\n", |
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"for i in range(1, m):\n", |
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" print(f\"share {i}: {f_x(i)}\")\n", |
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" shares.append((i, f_x(i)))\n", |
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"\n", |
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"def secret_recovery(shares): \n", |
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" return sum([i[1] * prod([(0-j[0] * pow(i[0] - j[0], -1, q)) for j in shares[:t] if i[0] != j[0]]) for i in shares[:t]])\n", |
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"\n", |
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"S_man = 0\n", |
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"for i in shares[:t]:\n", |
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" im_prod = i[1]\n", |
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" for j in shares[:t]:\n", |
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" if j[0] != i[0]:\n", |
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" im_prod *= ((-j[0]) * pow(i[0] - j[0], -1, q)) % q\n", |
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" S_man += im_prod\n", |
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" \n", |
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"\n", |
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"print(shares)\n", |
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"S_sage = R.lagrange_polynomial(shares[:t])\n", |
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"print(S_sage[0])\n", |
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"print(coefficients)\n", |
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"print(S_man % q)\n", |
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"print(secret_recovery(shares[:t]))" |
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] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": 3, |
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"id": "e5c47c9d", |
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"metadata": {}, |
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"outputs": [ |
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{ |
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"name": "stdout", |
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"output_type": "stream", |
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"text": [ |
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"Information Word a: \t[0, 15, 10, 13, 13, 17, 5, 2, 8, 12, 7]\n", |
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"Code Word D: \t\t[7, 9, 5, 9, 0, 2, 9, 1, 7, 16, 17, 12, 18, 6, 12, 11, 8, 3]\n", |
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"a_0: \t\t\t0\n", |
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"Manipulated Codeword: \t[6, 9, 5, 9, 0, 2, 9, 1, 7, 16, 16, 12, 18, 6, 12, 11, 8, 3]\n", |
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"a_0: \t\t\t2\n" |
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] |
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} |
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], |
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"source": [ |
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"non_zero_elem_of_Fq = [x for x in Fq if x != 0]\n", |
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"k = 11\n", |
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"\n", |
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"information_word = []\n", |
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"information_word.append(S)\n", |
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"for _ in range(0, k-1):\n", |
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" information_word.append(Fq.random_element())\n", |
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"\n", |
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"print(f\"Information Word a: \\t{information_word}\")\n", |
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"\n", |
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"code_word = []\n", |
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"for i in range(0, q-1):\n", |
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" code_word.append(sum([information_word[j] * pow(non_zero_elem_of_Fq[i], j) for j in range(0, k)]))\n", |
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"\n", |
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"print(f\"Code Word D: \\t\\t{code_word}\")\n", |
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"\n", |
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"secret = -sum([code_word[d] for d in range(0, q-1)])\n", |
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"print(f\"a_0: \\t\\t\\t{secret}\")\n", |
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"\n", |
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"assert(secret == information_word[0] == S)\n", |
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"\n", |
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"manipulated_codeword = []\n", |
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"for i in range(0, len(code_word)):\n", |
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" if i % 10 == 0:\n", |
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" manipulated_codeword.append(code_word[i]-1 % q)\n", |
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" else:\n", |
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" manipulated_codeword.append(code_word[i])\n", |
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"\n", |
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"print(f\"Manipulated Codeword: \\t{manipulated_codeword}\")\n", |
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"\n", |
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"secret_man = -sum([manipulated_codeword[d] for d in range(0, q-1)])\n", |
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"print(f\"a_0: \\t\\t\\t{secret_man}\")\n" |
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] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": null, |
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"id": "cff95199", |
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"metadata": {}, |
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"outputs": [], |
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"source": [] |
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}, |
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{ |
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"cell_type": "code", |
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"execution_count": null, |
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"id": "152ae6a0", |
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"metadata": {}, |
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"outputs": [], |
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"source": [] |
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} |
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], |
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"metadata": { |
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"kernelspec": { |
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"display_name": "SageMath 10.6", |
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"language": "sage", |
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"name": "sagemath" |
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}, |
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"language_info": { |
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"codemirror_mode": { |
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"name": "ipython", |
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"version": 3 |
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}, |
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"file_extension": ".py", |
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"mimetype": "text/x-python", |
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"name": "python", |
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"nbconvert_exporter": "python", |
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"pygments_lexer": "ipython3", |
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"version": "3.12.3" |
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} |
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}, |
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"nbformat": 4, |
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"nbformat_minor": 5 |
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} |
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