gen-define-elliptic-curve-cryptography-and-give-one-short-worked-04a2

Define elliptic-curve cryptography and give one short worked example.

# Elliptic-Curve Cryptography (ECC) **Definition:** Elliptic-curve cryptography is a public-key cryptographic system based on the algebraic structure of elliptic curves over finite fields. It relies on the difficulty of the **Elliptic Curve Discrete Logarithm Problem (ECDLP)**: given points P and Q = kP on an elliptic curve, finding the integer k is computationally hard. An elliptic curve has the form: **y² = x³ + ax + b** Points on the curve form a group under a special addition operation, enabling cryptographic protocols with smaller key sizes than RSA. --- ## Worked Example: Simple Key Exchange **Setup:** Use curve y² = x³ + 7 over finite field F₁₇ (mod 17) - Base point: **G = (5, 1)** - (You can verify: 1² = 5³ + 7 = 132 ≡ 13² ≡ 1 (mod 17) ✓) **Alice's keys:** - Private key: a = 3 - Public key: A = 3G = G + G + G = **(10, 6)** (after curve addition) **Bob's keys:** - Private key: b = 4 - Public key: B = 4G = **(13, 7)** **Shared secret:** - Alice computes: a·B = 3·(13, 7) = **(10, 11)** - Bob computes: b·A = 4·(10, 6) = **(10, 11)** Both arrive at the same point **(10, 11)**, which becomes their shared secret key. An eavesdropper knowing only G, A, and B cannot easily compute this shared secret.