In This Lesson 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 Why Number Theory? RSA Algorithm Diffie-Hellman Key Exchange Modern Topics 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 Why Number Theory? Public-key cryptography relies on trapdoor functions — easy to compute but hard to invert. The difficulty of factoring large numbers into primes and computing discrete logarithms provides these trapdoors. What was once "pure" mathematics is now essential to internet security.
RSA Algorithm 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 The RSA cryptosystem (Rivest-Shamir-Adleman):
Key generation: Choose large primes p, q. Compute n = pq, φ(n) = (p−1)(q−1) Public key: Choose e coprime to φ(n). Public key = (n, e) Private key: Find d where ed ≡ 1 (mod φ(n)) — the modular inverse Encrypt: c = mᵉ mod 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 Decrypt: m = cᵈ mod n Example: Toy RSA p = 61, q = 53. n = 3233, φ(n) = 3120
e = 17 (coprime to 3120). d = 2753 (since 17 × 2753 = 46801 ≡ 1 mod 3120)
Encrypt m = 65: c = 65¹⁷ mod 3233 = 2790
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 Decrypt: 2790²⁷⁵³ mod 3233 = 65 ✓
Security relies on: factoring n = pq is hard when p, q are ~1024 bits each. Euler's theorem guarantees decryption works: m^(ed) ≡ m^(1 + kφ(n)) ≡ m (mod n).
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 Diffie-Hellman Key Exchange
Public: large prime p, generator g
Alice sends: A = gᵃ mod p (a is secret)
Bob sends: B = gᵇ mod p (b is secret)
Shared secret: s = Bᵃ = Aᵇ = gᵃᵇ mod p
Security relies on the Discrete Logarithm Problem : given g, p, and gᵃ mod p, finding a is computationally difficult. This uses the same modular exponentiation techniques.
Modern Topics Elliptic Curve Cryptography (ECC): Uses algebraic curves over finite fields — same security with smaller keys Hash functions: One-way mappings used in digital signatures Post-quantum cryptography: Lattice-based methods using high-dimensional linear algebra Zero-knowledge proofs: Prove knowledge without revealing information
Quantum computers could break RSA and Diffie-Hellman using Shor's algorithm (efficient prime factorization). The cybersecurity community is actively transitioning to quantum-resistant algorithms based on lattices, codes, and hash functions — all rooted in mathematics from
linear algebra and
number theory .