Some integer factorization algorithms using elliptic curves

TL;DR

This paper proposes enhancements to Lenstra's elliptic curve-based integer factorization algorithm, introducing a second phase to significantly improve its speed and efficiency, with practical and theoretical implications.

Contribution

It introduces a second phase to Lenstra's elliptic curve factorization algorithm, significantly speeding up the process under certain assumptions.

Findings

Speedup of order log(p) under plausible assumptions

Practical improvements in factorization speed

Connections with Pollard's p-1 algorithm

Abstract

Lenstra's integer factorization algorithm is asymptotically one of the fastest known algorithms, and is ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase. Under some plausible assumptions, the speedup is of order log(p), where p is the factor which is found. In practice the speedup is significant. We mention some refinements which give greater speedup, an alternative way of implementing a second phase, and the connection with Pollard's "p-1" factorization algorithm.