Add RSA
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3b934c43c2
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168
TTM4135.ipynb
168
TTM4135.ipynb
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@ -1,5 +1,12 @@
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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": null,
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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": "markdown",
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"metadata": {},
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@ -223,9 +230,7 @@
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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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"From slides 2020-4135-l07 - Number Theory for Public Key Cryptography\n"
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]
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},
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{
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@ -593,14 +598,13 @@
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"# Square and multiply\n",
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"from math import log, floor\n",
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"\n",
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"def square_and_multiply(y,e,n, verbose=True):\n",
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"def square_and_multiply(y,e,n, verbose=False):\n",
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" # prep\n",
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" e_bin = bin(e)[:1:-1] # e0, e1, e2, e3, ... , ek\n",
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" k = len(e_bin)\n",
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" z = 1\n",
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" yi = y\n",
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" indices = []\n",
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" # -------------\n",
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"\n",
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" for i in range(k):\n",
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" ei = int(e_bin[i])\n",
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@ -611,12 +615,10 @@
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" if ei:\n",
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" indices.append(i)\n",
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"\n",
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" # post calc\n",
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" mults = str(e_bin).count(\"1\") - 1\n",
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" squarings = floor( log(e, 2) )\n",
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" # -------------\n",
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"\n",
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" if verbose:\n",
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" mults = str(e_bin).count(\"1\") - 1\n",
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" squarings = floor( log(e, 2) )\n",
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"\n",
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" print(f\"{' * '.join([f'{y}^({2**i})' for i in indices])}\")\n",
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" print(f\"squarings: {squarings}\")\n",
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" print(f\"mults: {mults}\")\n",
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@ -630,7 +632,7 @@
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"e = 102900819\n",
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"n = 604604729\n",
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"\n",
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"z = square_and_multiply(y,e,n, True)\n",
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"z = square_and_multiply(y,e,n, verbose=True)\n",
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"\n",
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"print(f\"{y}^{e} mod({n}) = {z}\")"
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]
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@ -702,16 +704,144 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Block ciphers\n",
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"# RSA\n",
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"\n",
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"Notation:\n",
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"### Key generation\n",
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"\n",
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"- `P`: Plaintext block (length n bits) \n",
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"- `C`: Ciphertext block (length n bits) \n",
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"- `K` : Key (length k bits)\n",
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"- `C = E(P, K)`: Encryption function \n",
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"- `P = D(C, K)`: Decryption function\n",
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"- `IV`: Initialisation vector"
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"1. Let $p$ and $q$ be distinct prime numbers, randomly chosen from the set of all prime numbers of a certain size.\n",
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"2. Compute $n = pq$.\n",
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"3. Select $e$ randomly with $gcd(e, \\varphi(n)) = 1$.\n",
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"4. Compute $d = e−1 \\mod \\varphi(n)$.\n",
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"5. The public key is the pair $n$ and $e$.\n",
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"6. The private key consists of the values $p$, $q$ and $d$.\n",
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"\n",
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"### Encryption\n",
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"\n",
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"The public key for encryption is $KE = (n, e)$\n",
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"1. Input is any value $M$ where $0 < M < n$. \n",
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"2. Compute $C = E(M,KE) = Me mod n$.\n",
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"\n",
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"### Decryption\n",
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"\n",
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"The private key for decryption is KD = d (values p and q are not used here).\n",
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"1. Compute D(C,KD) = Cd mod n = M."
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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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"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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"p=43, q=59, n=2537, 𝜑(n)=2436, e=5, d=1949\n",
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"\n",
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"Public key: n=2537, e=5\n",
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"Private key: p=43, q=59, d=1949\n",
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"\n",
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"{'n': 2537, 'e': 5, 'p': 43, 'q': 59, 'd': 1949}\n"
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]
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}
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],
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"source": [
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"import random\n",
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"\n",
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"def rsa_keygen(p=None, q=None, e=None, prime_range=1000, verbose=False):\n",
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" # Generate a set of primes. In practice, this should be\n",
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" # a large number of large primes.\n",
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" primes = [i for i in range(0,prime_range) if is_prime(i)]\n",
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"\n",
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" if not p: p = random.choice(primes)\n",
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" if not q: q = random.choice(primes)\n",
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" n = p*q\n",
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"\n",
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" # Euler function 𝜑(n) \n",
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" phi_n = totient(n)\n",
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"\n",
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" # Select e randomly with gcd(e, φ(n)) = 1.\n",
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" if not e: e = random.randint(1, phi_n)\n",
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" while not gcd(e, phi_n) == 1:\n",
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" e = random.randint(1,phi_n)\n",
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"\n",
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" # Compute d = e−1 mod φ(n).\n",
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" d = inverse(e, phi_n)\n",
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" \n",
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" if verbose:\n",
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" print(f\"p={p}, q={q}, n={n}, 𝜑(n)={phi_n}, e={e}, d={d}\\n\")\n",
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" print(f\"Public key: n={n}, e={e}\")\n",
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" print(f\"Private key: p={p}, q={q}, d={d}\\n\")\n",
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"\n",
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" return {\"n\":n, \"e\":e, \"p\":p, \"q\":q, \"d\":d}\n",
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"\n",
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"# Example from slides\n",
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"rsa_keys = rsa_keygen(p=43, q=59, e=5, verbose=True)\n",
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"\n",
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"# Random keys\n",
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"#rsa_keys = rsa_keygen(verbose=True)\n",
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"print(rsa_keys)"
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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": 22,
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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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"C = M^e (mod n) = 50^102900819 (mod 604604729) = 2488\n",
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"\n",
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"Ciphertext: 2488\n"
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]
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}
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],
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"source": [
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"def rsa_encrypt(M, rsa_keys):\n",
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" # e and n are the public key\n",
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" e = rsa_keys['e']\n",
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" n = rsa_keys['n']\n",
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"\n",
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" # C = M^e (mod n)\n",
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" return square_and_multiply(M,e,n)\n",
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"\n",
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"M = 50\n",
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"C = rsa_encrypt(M, rsa_keys)\n",
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"print(f\"C = M^e (mod n) = {M}^{e} (mod {n}) = {C}\\n\")\n",
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"print(f\"Ciphertext: {C}\")"
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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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"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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"M = C^d (mod n) = 2488^1 (mod 604604729) = 50\n",
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"\n",
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"Message: 50\n"
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]
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}
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],
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"source": [
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"def rsa_decrypt(C, rsa_keys):\n",
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" # private key\n",
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" p = rsa_keys['p']\n",
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" q = rsa_keys['q']\n",
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" d = rsa_keys['d']\n",
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" n = p*q\n",
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"\n",
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" # M = C^d (mod n)\n",
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" return square_and_multiply(C,d,n)\n",
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"\n",
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"M = rsa_decrypt(C, rsa_keys)\n",
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"print(f\"M = C^d (mod n) = {C}^{d} (mod {n}) = {M}\\n\")\n",
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"print(f\"Message: {M}\")"
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]
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}
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],
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