Cryptography Summary

Number Theory · Axiom Academy

SUMMARY Cryptography Applications Let's review the key concepts from Unit 8: Cryptography and its mathematical foundations in number theory. Caesar Cipher: Shifts each letter by a fixed amount (mod 26), easily breakable by frequency analysis Substitution Ciphers: Replace each letter with another letter according to a fixed permutation Affine Cipher: Uses linear transformation E(x) = (ax + b) mod 26, requires gcd(a, 26) = 1 Limitation: All classical ciphers rely on shared secret keys and are vulnerable to modern attacks Key Innovation: Separates encryption key (public) from decryption key (private), solving key distribution problem Mathematical Foundation: Based on computationally hard problems from number theory One-Way Functions: Easy to compute but extremely difficult to reverse without special information Security Basis: Relies on computational complexity, not just key secrecy Example Recap: RSA Key Generation Step 1 - Choose Primes: Select two large distinct primes p = 61 and q = 53, compute n = 61 × 53 = 3233 Step 2 - Compute Totient: Calculate φ(n) = (61-1)(53-1) = 60 × 52 = 3120 Step 3 - Select Public Exponent: Choose e = 17 (must satisfy gcd(17, 3120) = 1) Step 4 - Compute Private Key: Find d where 17d ≡ 1 (mod 3120) using Extended Euclidean Algorithm, getting d = 2753 Result: Public key (3233, 17) can be shared openly; private key 2753 remains secret Purpose: Allows two parties to establish a shared secret over an insecure channel

This is the written version of the interactive lesson above. See the full Number Theory course.