ECDLP

The Elliptic Curve Discrete Logarithm Problem (ECDLP) asks to determine an integer k given a point P and Q = kP on an elliptic curve E defined over a finite field. The group operation is point addition on E, and P generates a cyclic subgroup of order n. The difficulty of ECDLP underpins elliptic curve cryptography (ECC).

In ECC-based protocols, the public key Q is obtained as Q = dP from a private key d. Recovering d from Q and P would break schemes such as ECDH (elliptic curve Diffie–Hellman) for key exchange and ECDSA or EdDSA for digital signatures. The security of these systems rests on the hardness of ECDLP.

The best generic attacks on ECDLP have time complexity proportional to the square root of n, as in Pollard’s rho algorithm. This means key sizes can be substantially smaller than those required for RSA to achieve a similar security level. There are specialized attacks on curves with small embedding degrees (for example MOV and Frey–Rueppel) that can reduce ECDLP to a finite-field problem, so curves are chosen to resist such attacks by ensuring a large embedding degree and a prime-order subgroup.

Common parameter choices include curves over prime fields or binary fields with a base point G that has prime order n and a small cofactor h. Widely used standards include NIST curves P-256, P-384, P-521 and secp256k1. In general, a 256-bit EC key provides about 128-bit security; 384-bit offers about 192-bit security; 521-bit about 256-bit.

Security considerations emphasize using vetted curves, constant-time implementations, and protections against side-channel and invalid-curve attacks. ECC offers comparable security with much smaller keys, contributing to efficiency in platforms with limited resources.