One part of the PhD I'm working on is optimizing ECM, P1 and some other factoring algorithms (maybe P+1, Pollard rho is most likely useless) for NFS with more than two large primes on one side.
Peter's new idea for the P+/1 stage 2 looks very attractive for the job as the asymptotic complexity drops from O(d (log d)^2) to O(d log d), d the degree of the polynomial we evaluate, and perhaps more importantly the implied constant drops by rather a lot. I.e. or a c200, B2=10^9 the old code took 4.0 seconds, the new code takes 1.0 second. I'm hopeful that a properly optimized implementation operating on, say, 96 or 128 bit moduli would be quite useful for refactoring. However, at the moment, even the GMPbased implementation in GMPECM isn't 100% complete so the smallmodulus version will take a while yet.
Alex
