The three-large-primes variant of the number field sieve

The Number Field Sieve (NFS) is the asymptotically fastest known factoring algorithm for large integers. This method was proposed by John Pollard [20] in 1988. Since then several variants have been implemented with the objective of improving the siever which is the most time consuming part of this method (but fortunately, also the easiest to parallelise). Pollard's original method allowed one large prime. After that the two-large-primes variant led to substantial improvements [11]. In this paper we investigate whether the three-large-primes variant may lead to any further improvement. We present theoretical expectations and experimental results. We assume the reader to be familiar with the NFS. As a side-result, we improved some formulae for Taylor coefficients of Dickman's p function given by Patterson and Rumsey[3] and Marsaglia, Zaman and Marsaglia[16].