NFS with four large primes: An explosive experiment

Abstract

The purpose of this paper is to report the unexpected results that we obtained while experimenting with the multi-large prime variation of the general number field sieve integer factoring algorithm (NFS, cf. [8]). For traditional factoring algorithms that make use of at most two large primes, the completion time can quite accurately be predicted by extrapolating an almost quartic and entirely ‘smooth’ function that counts the number of useful combinations among the large primes [1]. For NFS such extrapolations seem to be impossible—the number of useful combinations suddenly ‘explodes’ in an as yet unpredictable way, that we have not yet been able to understand completely. The consequence of this explosion is that NFS is substantially faster than expected, which implies that factoring is somewhat easier than we thought.