Some integer factorization algorithms using elliptic curves

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.

Corrected version of a paper that appeared in Australian Computer Science Communications 8 (1986), with postscript added 1998. For further details see http://wwwmaths.anu.edu.au/~brent/pub/pub102.html