elliptic-curve cryptography
Elliptic-curve cryptography (ECC) builds public-key crypto on the points of an elliptic curve — a smooth equation like y^2 = x^3 + ax + b over a finite field. The magic ingredient is a well-defined way to 'add' two curve points and get a third point that is also on the curve.
Point addition has a geometric picture: draw a line through two points, find where it meets the curve a third time, and reflect over the x-axis. Adding a fixed base point G to itself k times gives kG — scalar multiplication, the one-way function at the heart of ECC. Computing P = kG from k and G is fast (double-and-add); recovering k from P and G is the elliptic-curve discrete logarithm problem (ECDLP), believed hard. The best generic attack, Pollard's rho, costs about 2^(n/2) for a curve whose group order is n bits.
That square-root hardness is why ECC keys are so small: a 256-bit curve gives roughly 128-bit security, matching 3072-bit RSA. The private key is a scalar k, and the public key is the point kG. Almost every blockchain signs with ECC — Bitcoin and Ethereum on secp256k1, many newer chains on Ed25519 over Curve25519 — because compact keys and signatures save scarce on-chain bytes.
ECC security rests on the discrete-logarithm problem, not on integer factoring. A large fault-tolerant quantum computer running Shor's algorithm would break both — which is exactly what motivates post-quantum and hash-based signatures.