Fast and rigorous factorization under the generalized Riemann hypothesis

Abstract

We present an algorithm that finds a non-trivial factor of an odd composite integer n with probability ⩾1/2 - o(1) in expected time bounded by $e^{(1+o(1\left)\right)\sqrt{\log n\log \log n}}$. This result can be rigorously proved under the sole assumption of the generalized Riemann hypothesis. The time bound matches the heuristic time bounds for the continued fraction algorithm, the quadratic sieve algorithm, the Schnorr-Lenstra class group algorithm, and the worst case of the elliptic curve method. The algorithm is based on Seysen's factoring algorithm [14], and the elliptic curve smoothness test from [12].