637 lines
17 KiB
Plaintext
637 lines
17 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# GCD / Euclidean algorithm\n",
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"Explaination: https://brilliant.org/wiki/extended-euclidean-algorithm/"
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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": 1,
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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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"5"
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]
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},
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"execution_count": 1,
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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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"# Returns the greatest common divisor of a and b.\n",
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"def gcd(a, b):\n",
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" if b > a:\n",
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" return gcd(b, a)\n",
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" if a % b == 0:\n",
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" return b\n",
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" return gcd(b, a % b)\n",
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"\n",
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"gcd(35,30)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Back substitution / Extended euclidean algorithm\n",
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"Explaination: https://brilliant.org/wiki/extended-euclidean-algorithm/"
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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": 2,
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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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"ax + by = gcd(a,b) ==> 35*1 + 10*-3 = gcd(35,10)\n",
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"\n",
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"gcd(35,10) = 5\n",
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"x = 1, y = -3\n"
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]
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}
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],
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"source": [
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"def gcdExtended(a, b): \n",
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" # Base Case \n",
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" if a == 0 : \n",
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" return b,0,1\n",
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"\n",
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" gcd,x1,y1 = gcdExtended(b%a, a)\n",
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"\n",
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" # Update x and y using results of recursive call \n",
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" x = y1 - (b//a) * x1 \n",
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" y = x1 \n",
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"\n",
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" return gcd,x,y \n",
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"\n",
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"a, b = 35,10\n",
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"g, x, y = gcdExtended(a, b)\n",
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"print(f\"ax + by = gcd(a,b) ==> {a}*{x} + {b}*{y} = gcd({a},{b})\\n\")\n",
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"\n",
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"print(f\"gcd({a},{b}) = {g}\")\n",
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"print(f\"x = {x}, y = {y}\")"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Modular inverse, A^-1\n",
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"Explanation: https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/modular-inverses"
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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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"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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"3 * 5 ≡ 1 (mod 7)\n",
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"Modular inverse for 3 (mod 7) is 5\n",
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"a^-1 = 5\n"
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]
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}
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],
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"source": [
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"# Naive method of finding modular inverse for A (mod C) \n",
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"def inverse(a, c):\n",
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" for b in range(c):\n",
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" if (a*b)%c == 1: return b\n",
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" print(\"NO INVERSE FOUND\")\n",
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"\n",
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"a,c = 3,7\n",
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"b = inverse(a,c)\n",
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"\n",
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"print(f\"{a} * {b} ≡ 1 (mod {c})\")\n",
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"print(f\"Modular inverse for {a} (mod {c}) is {b}\")\n",
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"print(f\"a^-1 = {b}\")"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Chineese remainder theorem\n",
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"Source: https://rosettacode.org/wiki/Chinese_remainder_theorem#Python"
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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": 4,
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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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"23\n"
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]
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}
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],
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"source": [
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"from functools import reduce\n",
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"def crt(n, a):\n",
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" sum = 0\n",
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" prod = reduce(lambda a, b: a*b, n)\n",
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" for n_i, a_i in zip(n, a):\n",
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" p = prod // n_i\n",
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" sum += a_i * mul_inv(p, n_i) * p\n",
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" return sum % prod\n",
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"\n",
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"def mul_inv(a, b):\n",
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" b0 = b\n",
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" x0, x1 = 0, 1\n",
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" if b == 1: return 1\n",
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" while a > 1:\n",
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" q = a // b\n",
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" a, b = b, a%b\n",
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" x0, x1 = x1 - q * x0, x0\n",
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" if x1 < 0: x1 += b0\n",
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" return x1\n",
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"\n",
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"n = [3, 5, 7]\n",
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"a = [2, 3, 2]\n",
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"print(crt(n, a))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Euler function φ and Z*n\n",
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"From slides 2020-4135-l07 - Number Theory for Public Key Cryptography\n",
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"\n",
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"> $\\phi(16) = \\phi(2^{4}) = 16*\\left(1-\\frac{1}{2}\\right)$"
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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": 5,
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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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"φ(16) = 8 since the 8 items in Z*16 are each relatively prime to 16.\n",
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"Z*16 = [1, 3, 5, 7, 9, 11, 13, 15]\n"
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]
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}
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],
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"source": [
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"# Relatively prime means that gcd equals 1\n",
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"def is_relative_prime(a,b):\n",
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" if b == 0 or a == 0: return False\n",
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" return gcd(a,b) == 1\n",
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"\n",
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"# The set of positive integers less than n and relatively prime \n",
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"# to n form the reduced residue class Z∗n.\n",
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"def z_star(n):\n",
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" r = []\n",
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" for i in range(1,n):\n",
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" if is_relative_prime(i,n):\n",
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" r.append(i)\n",
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" return r\n",
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"\n",
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"n = 16\n",
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"ar = z_star(n)\n",
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"e = len(ar)\n",
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"print(f\"φ({n}) = {e} since the {e} items in Z*{n} are each relatively prime to {n}.\")\n",
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"print(f\"Z*{n} = {ar}\")"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Euler’s Totient\n",
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"- https://www.dcode.fr/euler-totient\n",
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"- https://www.ibnus.io/eulers-totient-function-using-python/\n",
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"\n",
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"In general:\n",
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"\n",
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"> $\\varphi(n) = n \\prod_{p \\mid n} \\left( 1 - \\frac{1}{p} \\right)$\n",
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"\n",
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"Examples:\n",
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"> $\\varphi(16) = \\varphi(2^{4}) = 16*\\left(1-\\frac{1}{2}\\right) = 8$\n",
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"\n",
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"> $\\varphi(1125) = \\varphi(33555) = 1125\\left(1-\\frac{1}{3}\\right)\\left(1-\\frac{1}{5}\\right) = 600$\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": 6,
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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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"𝜑(1125) = 600\n"
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]
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}
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],
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"source": [
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"# Euler's Totient Function\n",
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"# using Euler's product formula\n",
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"def totient(n) :\n",
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" result = n # Initialize result as n\n",
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"\n",
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" # Consider all prime factors\n",
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" # of n and for every prime\n",
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" # factor p, multiply result with (1 – 1/p)\n",
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" p = 2\n",
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" while(p * p <= n) :\n",
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"\n",
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" # Check if p is a prime factor.\n",
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" if (n % p == 0) :\n",
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"\n",
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" # If yes, then update n and result\n",
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" while (n % p == 0) :\n",
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" n = n // p\n",
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" result = result * (1.0 - (1.0 / (float) (p)))\n",
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" p = p + 1\n",
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"\n",
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" # If n has a prime factor\n",
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" # greater than sqrt(n)\n",
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" # (There can be at-most one\n",
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" # such prime factor)\n",
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" if (n > 1) :\n",
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" result = result * (1.0 - (1.0 / (float)(n)))\n",
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"\n",
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" return (int)(result)\n",
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"\n",
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"n = 1125\n",
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"print(f\"𝜑({n}) = {totient(1125)}\")"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Testing for primality\n",
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"From slides 2020-4135-l07 - Number Theory for Public Key Cryptography"
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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": 7,
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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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"137 True\n"
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]
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}
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],
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"source": [
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"from math import sqrt\n",
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"# Inefficient, but accurate method\n",
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"def is_prime(n):\n",
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" if n % 2 == 0 and n > 2: \n",
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" return False\n",
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" return all(n % i for i in range(3, int(sqrt(n)) + 1, 2))\n",
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"\n",
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"print(137, is_prime(137))"
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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": 8,
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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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"1105 is probable prime\n"
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]
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}
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],
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"source": [
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"# Fermat primality test\n",
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"\n",
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"# Inputs: \n",
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"# n: a value to test for primality;\n",
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"# k: a parameter that determines the number of \n",
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"# times to test for primality\n",
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"from random import randint\n",
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"def fermat_prime(n, k):\n",
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" for _ in range(k):\n",
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" a = randint(2,n-2)\n",
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" # Fermat’s theorem says that if a number p is prime then \n",
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" # a^p−1 mod p = 1 for all a with gcd(a,p) = 1.\n",
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" if not (a**(n-1))%n == 1:\n",
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" return \"composite\"\n",
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" return \"probable prime\"\n",
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"\n",
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"n = 1105 # value to test for primality;\n",
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"k = 1 # number of times to test for primality\n",
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"\n",
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"print(f\"{n} is {fermat_prime(n,k)}\")\n",
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"\n",
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"# First few Carmichael numbers are: 561, 1105, 1729, 2465 \n",
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"# Will output probable prime for every a with gcd(a, n) = 1"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Miller-Rabin\n",
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"From https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test \n",
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"This test is not fooled by Charmichael numbers.\n",
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"\n",
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"To test `n`, you choose different `a` values, and test if \n",
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"\n",
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"> $a^{d*2^i} \\equiv 1 \\pmod{n}$ for $i = (0, 1, 2, ... r)$\n",
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"\n",
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"If this happens, look at the congruence on the calculation just before you got `1`. If that value is not -1 (aka n-1), the value is composite aka not prime."
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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": 9,
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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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"168 = 21*2^3 = 168\n",
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"65536 = 1*2^16 = 65536\n",
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"\n",
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"16127 is probably prime.\n",
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"561 is composite.\n",
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"2465 is composite.\n",
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"65537 is probably prime.\n"
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]
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}
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],
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"source": [
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"# Miller-Rabin primality test\n",
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"# This test is not fooled by Charmichael numbers.\n",
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"\n",
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"from math import log2, floor\n",
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"from random import randint\n",
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"\n",
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"def decompose_even_number(n):\n",
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" '''Find the values r and d such that n == d * 2**r.'''\n",
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" r = 0\n",
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" d = n\n",
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" for i in range(1, floor(log2(n)) + 1):\n",
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" if (d/2).is_integer():\n",
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" d = d/2\n",
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" r += 1\n",
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" else:\n",
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" break\n",
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" return (r, int(d))\n",
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"\n",
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"verbose = False # Print during execution\n",
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"\n",
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"def miller_rabin_prime(n):\n",
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" ''':param n: should be odd, and n>3. No primes can be even, for obvious reasons.'''\n",
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" accuracy = 3 # How many different `a` to test\n",
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" r, d = decompose_even_number(n-1) # n is odd, thus n-1 is even\n",
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" verbose and print(f\"Testing {n} a maximum of {accuracy} times...\")\n",
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" \n",
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" for _ in range(accuracy):\n",
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" try:\n",
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" a = randint(2, n-2) # The \"witness\". TODO: don't reuse a witness.\n",
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" verbose and print(f\"\\tTesting {n} with a={a}\")\n",
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"\n",
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" x = (a**d) % n\n",
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" if x == 1 or x == n-1:\n",
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" verbose and print(f\"\\t({a}**{d})%{n} was 1 or -1 already!\")\n",
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" continue # This witness was immediately 1 or -1 (aka n-1), so it can't help us any more.\n",
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" for squaring in range(r-1):\n",
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" x = (x**2) % n\n",
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" verbose and print(f\"\\tx={x}\")\n",
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" if x == n-1:\n",
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" raise Exception(\"Try next witness\") # This witness can't help any more.\n",
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" return \"composite\" # The witness didnt uphold the criteria for a prime, so n is composite!\n",
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" except Exception as tryNext:\n",
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" continue\n",
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" return \"probably prime\"\n",
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" \n",
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"s, d = decompose_even_number(168)\n",
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"print(f\"168 = {d}*2^{s} = {d*(2**s)}\")\n",
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"s, d = decompose_even_number(65536)\n",
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"print(f\"65536 = {d}*2^{s} = {d*(2**s)}\")\n",
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"print()\n",
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"\n",
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"for value in [16127, 561, 2465, 65537]:\n",
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" print(f\"{value} is {miller_rabin_prime(value)}.\")\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Integer factorisation\n",
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"Given an integer, find its prime factors"
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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": 10,
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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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"The factors of 641361 are [3, 7, 7, 4363], verified\n",
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"Manual test: True\n"
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]
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}
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],
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"source": [
|
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"# Brute force solution\n",
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"def prime_factors(n):\n",
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" i = 2\n",
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" factors = []\n",
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" while i * i <= n:\n",
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" if n % i:\n",
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" i += 1\n",
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" else:\n",
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" n //= i\n",
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" factors.append(i)\n",
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" if n > 1:\n",
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" factors.append(n)\n",
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" return factors\n",
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"\n",
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"# Check if the numbers in array a is the factors of n\n",
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"def factor_test(a, n):\n",
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" k = 1\n",
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" for i in factors:\n",
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" k=k*i\n",
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" return k==n\n",
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"\n",
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"n = 641361\n",
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"factors = prime_factors(n)\n",
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||
"print(f\"The factors of {n} are {factors}, {'verified' if factor_test(factors, n) else 'ERROR'}\")\n",
|
||
"\n",
|
||
"print(\"Manual test:\", factor_test([3, 7, 7, 4363], 641361))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"# Discrete logarithm problem\n",
|
||
"Given a prime `p` and an integer `y` with `0 < y < p`, find `x` such that \n",
|
||
"\n",
|
||
"> $y = g^{x}\\mod{p}$\n",
|
||
"\n",
|
||
"No polynomial algorithims exists, but \"Baby-step giant-step\" is a reasonable one. \n",
|
||
"\n",
|
||
"Sources:\n",
|
||
"- https://stackoverflow.com/a/1832648/5976426\n",
|
||
"- https://en.wikipedia.org/wiki/Baby-step_giant-step\n",
|
||
"- https://gist.github.com/0xTowel/b4e7233fc86d8bb49698e4f1318a5a73"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 11,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"x = 2570\n",
|
||
"y = g^x mod p ==> 34 = 7324^2570 mod 4363\n",
|
||
"Test: True\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"from math import ceil, sqrt\n",
|
||
"\n",
|
||
"def bsgs(g, y, p):\n",
|
||
" '''\n",
|
||
" Solve for x in y = g^x mod p given a prime p.\n",
|
||
" If p is not prime, you shouldn't use BSGS anyway.\n",
|
||
" '''\n",
|
||
" N = ceil(sqrt(p - 1)) # phi(p) is p-1 if p is prime\n",
|
||
"\n",
|
||
" # Store hashmap of g^{1...m} (mod p). Baby step.\n",
|
||
" tbl = {pow(g, i, p): i for i in range(N)}\n",
|
||
"\n",
|
||
" # Precompute via Fermat's Little Theorem\n",
|
||
" c = pow(g, N * (p - 2), p)\n",
|
||
"\n",
|
||
" # Search for an equivalence in the table. Giant step.\n",
|
||
" for j in range(N):\n",
|
||
" h = (y * pow(c, j, p)) % p\n",
|
||
" if h in tbl:\n",
|
||
" return j * N + tbl[h]\n",
|
||
"\n",
|
||
" # Solution not found\n",
|
||
" return None\n",
|
||
"\n",
|
||
"def disc_log_test(y, g, x, p):\n",
|
||
" return y == (g**x) % p\n",
|
||
"\n",
|
||
"#y = 355407489\n",
|
||
"#g = 7894352216\n",
|
||
"#p = 604604729 # Must be prime\n",
|
||
"\n",
|
||
"# y = g^x mod p\n",
|
||
"y = 34\n",
|
||
"g = 7324\n",
|
||
"p = 4363 # Must be prime\n",
|
||
"\n",
|
||
"x = bsgs(g, y, p)\n",
|
||
"\n",
|
||
"print(f\"x = {x}\")\n",
|
||
"\n",
|
||
"print(f\"y = g^x mod p ==> {y} = {g}^{x} mod {p}\")\n",
|
||
"\n",
|
||
"# Don't run the test with big numbers!\n",
|
||
"if x<10000: print(f\"Test: {disc_log_test(y,g,x,p)}\")"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"# Block ciphers\n",
|
||
"\n",
|
||
"Notation:\n",
|
||
"\n",
|
||
"- `P`: Plaintext block (length n bits) \n",
|
||
"- `C`: Ciphertext block (length n bits) \n",
|
||
"- `K` : Key (length k bits)\n",
|
||
"- `C = E(P, K)`: Encryption function \n",
|
||
"- `P = D(C, K)`: Decryption function"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 12,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]\n",
|
||
"54\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"# Discarded code\n",
|
||
"\n",
|
||
"def z(n):\n",
|
||
" return [i for i in range(n)]\n",
|
||
"print(z(10))\n",
|
||
"\n",
|
||
"n = 81\n",
|
||
"zn = z(n)\n",
|
||
"rel_primes = list(filter(lambda x: is_relative_prime(x,n), zn))\n",
|
||
"print(len(rel_primes))"
|
||
]
|
||
}
|
||
],
|
||
"metadata": {
|
||
"kernelspec": {
|
||
"display_name": "Python 3",
|
||
"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": "3.7.3"
|
||
}
|
||
},
|
||
"nbformat": 4,
|
||
"nbformat_minor": 4
|
||
}
|