Elliptic Curve Cryptography

Elliptic curve cryptography (ECC) is a public-key cryptosystem based on the algebraic structure of elliptic curves over finite fields. Unlike RSA, which depends on the difficulty of integer factorization, ECC relies on the elliptic curve discrete logarithm problem (ECDLP): given two points P and Q on an elliptic curve, find the integer k such that Q = kP. No efficient general algorithm for this problem is known, and the best known attacks are exponentially slower than the best factoring algorithms — which means ECC can achieve equivalent security with much smaller key sizes.

The security of ECC rests on the interaction between algebraic geometry and computational complexity. The group of points on an elliptic curve over a finite field has a rich structure that is well-understood algebraically but poorly understood computationally. This gap — between what mathematics can describe and what algorithms can exploit — is precisely the window that cryptography requires.

ECC is now the default in most modern secure protocols, including TLS, Bitcoin, and Signal. Its efficiency makes it essential for constrained devices, but its relative novelty means that the underlying hardness assumptions have been subjected to less cryptanalytic scrutiny than RSA. The field's confidence in ECC is a sociological judgment about the community's failure to find breaks, not a mathematical proof of security.