notebooks/16_building_blocks_for_cryptography.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "8a3f72de",
   "metadata": {},
   "source": [
    "# Integer Factorization\n",
    "\n",
    "## Prime Numbers\n",
    "\n",
    "Checking whether a number is prime:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "d74a3af3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "The smallest divisor>1 of 2 is 2 (hence, 2 is prime)\n",
      "The smallest divisor>1 of 3 is 3 (hence, 3 is prime)\n",
      "The smallest divisor>1 of 4 is 2\n",
      "The smallest divisor>1 of 5 is 5 (hence, 5 is prime)\n",
      "The smallest divisor>1 of 6 is 2\n",
      "The smallest divisor>1 of 7 is 7 (hence, 7 is prime)\n",
      "The smallest divisor>1 of 8 is 2\n",
      "The smallest divisor>1 of 9 is 3\n",
      "The smallest divisor>1 of 10 is 2\n",
      "The smallest divisor>1 of 11 is 11 (hence, 11 is prime)\n",
      "The smallest divisor>1 of 12 is 2\n",
      "The smallest divisor>1 of 13 is 13 (hence, 13 is prime)\n",
      "The smallest divisor>1 of 14 is 2\n",
      "The smallest divisor>1 of 15 is 3\n",
      "The smallest divisor>1 of 16 is 2\n",
      "The smallest divisor>1 of 17 is 17 (hence, 17 is prime)\n",
      "The smallest divisor>1 of 18 is 2\n",
      "The smallest divisor>1 of 19 is 19 (hence, 19 is prime)\n",
      "The smallest divisor>1 of 20 is 2\n",
      "The smallest divisor>1 of 21 is 3\n",
      "The smallest divisor>1 of 22 is 2\n",
      "The smallest divisor>1 of 23 is 23 (hence, 23 is prime)\n",
      "The smallest divisor>1 of 24 is 2"
     ]
    }
   ],
   "source": [
    "# Finds the smallest divisor>1 of the given integer m>1\n",
    "def min_divisor(m):\n",
    "    for d in range(2, m + 1):\n",
    "        if m % d == 0:\n",
    "            return d\n",
    "        # optimization:\n",
    "        if d ** 2 > m:\n",
    "            return m\n",
    "\n",
    "\n",
    "for i in range(2, 25):\n",
    "    divisor = min_divisor(i)\n",
    "    print(f'\\nThe smallest divisor>1 of {i} is {divisor}', end='')\n",
    "    if divisor == i:\n",
    "        print(f' (hence, {i} is prime)', end='')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6944fd77",
   "metadata": {},
   "source": [
    "List of the first $n$ primes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "cbf01af7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The first ten primes:\n",
      "[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]\n"
     ]
    }
   ],
   "source": [
    "def is_prime(m):\n",
    "    return m == min_divisor(m)\n",
    "\n",
    "\n",
    "def primes_list(n):\n",
    "    lst = []\n",
    "    boundary = 2\n",
    "    # primes < boundary are in lst\n",
    "    while len(lst) < n:\n",
    "        if is_prime(boundary):\n",
    "            lst.append(boundary)\n",
    "        boundary += 1\n",
    "\n",
    "    return lst\n",
    "\n",
    "\n",
    "print('The first ten primes:')\n",
    "print(primes_list(10))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7d8a203d",
   "metadata": {},
   "source": [
    "## Factoring"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "04bbdf2a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "7 = 7\n",
      "20 = 2*2*5\n",
      "60 = 2*2*3*5\n",
      "1001 = 7*11*13\n",
      "4294967297 = 641*6700417\n",
      "18446744073709551617 = 274177*67280421310721\n"
     ]
    }
   ],
   "source": [
    "def factoring(m):\n",
    "    if is_prime(m):\n",
    "        return [m]\n",
    "    else:\n",
    "        divisor = min_divisor(m)\n",
    "        return [divisor] + factoring(m // divisor)\n",
    "\n",
    "\n",
    "for i in (7, 20, 60, 1001, 2 ** 32 + 1, 2 ** 64 + 1):\n",
    "    print(i, '=', '*'.join(map(str, factoring(i))))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8188e0c8",
   "metadata": {},
   "source": [
    "Factoring Fermat numbers:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "833f34d9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{3: 1}\n",
      "{5: 1}\n",
      "{17: 1}\n",
      "{257: 1}\n",
      "{65537: 1}\n",
      "{641: 1, 6700417: 1}\n",
      "{274177: 1, 67280421310721: 1}\n"
     ]
    }
   ],
   "source": [
    "from sympy import factorint\n",
    "\n",
    "for n in range(7):\n",
    "    print(factorint(2 ** (2 ** n) + 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0d715b2a",
   "metadata": {},
   "source": [
    "# Chinese Remainder Theorem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "90ca1757",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0: 0 mod 2=0, 0 mod 3=0\n",
      "1: 1 mod 2=1, 1 mod 3=1\n",
      "2: 2 mod 2=0, 2 mod 3=2\n",
      "3: 3 mod 2=1, 3 mod 3=0\n",
      "4: 4 mod 2=0, 4 mod 3=1\n",
      "5: 5 mod 2=1, 5 mod 3=2\n",
      "6: 6 mod 2=0, 6 mod 3=0\n",
      "7: 7 mod 2=1, 7 mod 3=1\n",
      "8: 8 mod 2=0, 8 mod 3=2\n",
      "9: 9 mod 2=1, 9 mod 3=0\n",
      "10: 10 mod 2=0, 10 mod 3=1\n",
      "11: 11 mod 2=1, 11 mod 3=2\n",
      "12: 12 mod 2=0, 12 mod 3=0\n",
      "13: 13 mod 2=1, 13 mod 3=1\n",
      "14: 14 mod 2=0, 14 mod 3=2\n"
     ]
    }
   ],
   "source": [
    "for i in range(15):\n",
    "    print(f'{i}: {i} mod 2={i % 2}, {i} mod 3={i % 3}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "531f51c7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " 0: (0, 0) (0, 0) (0, 0)\n",
      " 1: (1, 1) (1, 1) (1, 1)\n",
      " 2: (0, 2) (2, 2) (2, 2)\n",
      " 3: (1, 3) (0, 3) (3, 3)\n",
      " 4: (0, 0) (1, 0) (4, 0)\n",
      " 5: (1, 1) (2, 1) (5, 1)\n",
      " 6: (0, 2) (0, 2) (0, 2)\n",
      " 7: (1, 3) (1, 3) (1, 3)\n",
      " 8: (0, 0) (2, 0) (2, 0)\n",
      " 9: (1, 1) (0, 1) (3, 1)\n",
      "10: (0, 2) (1, 2) (4, 2)\n",
      "11: (1, 3) (2, 3) (5, 3)\n",
      "12: (0, 0) (0, 0) (0, 0)\n",
      "13: (1, 1) (1, 1) (1, 1)\n",
      "14: (0, 2) (2, 2) (2, 2)\n"
     ]
    }
   ],
   "source": [
    "for i in range(15):\n",
    "    print(f'{i:2d}: ({i % 2}, {i % 4}) '\n",
    "          f'({i % 3}, {i % 4}) '\n",
    "          f'({i % 6}, {i % 4})')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "f2f4083b",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "a, b = 10, 5\n",
    "n = a * b\n",
    "\n",
    "plt.plot([i % a for i in range(n)], [i % b for i in range(n)],\n",
    "         color='green', linestyle='dashed', linewidth=3,\n",
    "         marker='o', markerfacecolor='blue', markersize=12)\n",
    "  \n",
    "plt.axis('square')\n",
    "plt.xlim(-1, a)\n",
    "plt.ylim(-1, b)\n",
    "plt.xlabel(f'm mod {a}')\n",
    "plt.ylabel(f'm mod {b}')\n",
    "\n",
    "plt.savefig('crt-10-5.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2b96fc58",
   "metadata": {},
   "source": [
    "Algorithmic version of Chinese Reminder Theorem:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "7bd8f9fe",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "If x=3 mod 5 and x=7 mod 12, then x=43\n",
      "If x=3 mod 10 and x=8 mod 13, then x=73\n",
      "If x=3 mod 10 and x=1 mod 14, then x=113\n"
     ]
    }
   ],
   "source": [
    "from sympy import gcdex\n",
    "\n",
    "\n",
    "def chinese_remainder_theorem(n1, r1, n2, r2):\n",
    "    x, y, d = gcdex(n1, n2)\n",
    "    assert n1 * x + n2 * y == d  # == gcd(n1, n2)\n",
    "    y = -y\n",
    "    assert n1 * x - n2 * y == d\n",
    "    assert (r2 - r1) % d == 0\n",
    "    x *= (r2 - r1) // d\n",
    "    y *= (r2 - r1) // d\n",
    "    assert n1 * x - n2 * y == r2 - r1\n",
    "    return (n1 * x + r1) % (n1 * n2)\n",
    "\n",
    "\n",
    "for n1, r1, n2, r2 in (\n",
    "    (5, 3, 12, 7),\n",
    "    (10, 3, 13, 8),\n",
    "    (10, 3, 14, 1)\n",
    "):\n",
    "    result = chinese_remainder_theorem(n1, r1, n2, r2)\n",
    "    print(f'If x={r1} mod {n1} and x={r2} mod {n2}, then x={result}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0db887b6",
   "metadata": {},
   "source": [
    "# Modular Exponentiation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "43e9e4f2",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "38\n"
     ]
    }
   ],
   "source": [
    "print((314 ** 271) % 123)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "0b1c3235",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "38\n",
      "32073907\n"
     ]
    }
   ],
   "source": [
    "print(pow(314, 271, 123))\n",
    "print(pow(314159265358, 2718281828, 123456789))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "6770d102",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "32073907\n"
     ]
    }
   ],
   "source": [
    "def fast_modular_exponentiation(b, e, m):\n",
    "    assert m > 0 and e >= 0\n",
    "    if e == 0:\n",
    "        return 1\n",
    "    if e == 1:\n",
    "        return b\n",
    "    if e % 2 == 0:\n",
    "        return fast_modular_exponentiation((b * b) % m, e // 2, m)\n",
    "    else:\n",
    "        return (fast_modular_exponentiation(b, e - 1, m) * b) % m\n",
    "\n",
    "\n",
    "print(fast_modular_exponentiation(314159265358, 2718281828, 123456789))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "49137147",
   "metadata": {},
   "source": [
    "## Fermat's Little Theorem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "76f46ada",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import networkx as nx\n",
    "import matplotlib.pyplot as plt\n",
    "from math import cos, sin, pi\n",
    "\n",
    "n, factor, rad = 10, 3, 10\n",
    "\n",
    "graph = nx.DiGraph()\n",
    "graph.add_edges_from([(i, i * factor % n) for i in range(n)])\n",
    "\n",
    "positions = [(rad * cos(i * 2 * pi / n), (rad * sin(i * 2 * pi / n)))\n",
    "             for i in range(n)]\n",
    "\n",
    "nx.draw(graph, pos=positions, with_labels=True,\n",
    "        font_color='white', node_size=400, font_size=12)\n",
    "\n",
    "plt.gca().set_aspect('equal')\n",
    "plt.savefig('fermat.png')"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.13"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}