Elliptic Curves for Primality Proving

Synonyms

ECPP

Related Concepts

Elliptic Curves; Elliptic Curve Method for Factoring; Integer Factoring; Prime Number; Primality Test

Definition

Elliptic Curve Primality Proving (ECPP for short) is a method to prove primality of an integer n that uses elliptic curves modulo n.

Background

Proving the primality of an integer N (Primality Proving Algorithm) is easy if N−1 can be factored: N is prime if and only if the multiplicative group of invertible elements ${(\mathbb{Z}/N\mathbb{Z})}^{{_\ast}}$ is cyclic of order N;−;1 (Modular Arithmetic). To prove that an integer g is a generator of ${(\mathbb{Z}/N\mathbb{Z})}^{{_\ast}}$ and hence that the group is cyclic, it suffices to check that g ^N;−;1;≡;1{\rm mod}\,\,N and ${g}^{(N-1)/q}\not\equiv 1{\rm mod}\,\,N$ for all prime factors q of N;−;1. (It is quite easy to find a generator, or to prove that none exists, given the prime factors.)

The above method is the converse of Fermat’s Little Theorem. However, it is rare that N;−;1 is...