Elliptic Curve Scalar Multiplication: The Double-and-Add Algorithm Explained

Master the most critical operation in elliptic curve cryptography—scalar multiplication! This presentation explains how to efficiently compute kP (a point added to itself k times) using the double-and-add algorithm, which transforms an impossible O(k) operation into a practical O(log k) one. Learn why the naive approach fails for cryptographic key sizes, understand point doubling formulas derived from calculus, and discover how binary representation makes the algorithm work. Includes complete worked examples computing 2P and 3P with all algebraic steps shown explicitly, plus complexity analysis demonstrating the exponential speedup that makes 256-bit ECC possible. Covers real-world applications in Bitcoin's ECDSA signatures and TLS's ECDH key exchange, explaining why computing kP is fast while finding k from kP is computationally infeasible. Perfect for anyone implementing elliptic curve cryptography, studying the mathematics behind blockchain security, or understanding why ECC provides equivalent security to RSA with much smaller key sizes. All calculations shown algebraically with real numbers—no programming required!