Analysis of the Lenstra Elliptic Curves Factorization Method

The problem of factoring a composite natural number into the product of prime factors is a hard computational problem. This problem lies in the base of the well-known open-key RSA cyphering algorithm. In the paper we investigate Lenstra's Elliptic Curves Factoring Algorithm, which is the third ( after the Number Field Sieve and the Quadratic Sieve) by speed algorithm in the world classification of factoring algorithms. Moreover, the speed of this algorithm depends mostly on the size of the minor factor, so it can be applied to very large record numbers. Original Lenstra's algorithm of 1987 year consisted of one stage while later it was shown that in many cases it is more effectively to use the two-stage version. But when we are able to use multi-processors systems, we can arrange parallel computations which use different versions of the basic algorithm and are more effective. In the paper, we consider several improvements to the LenstraBasic Algorithm and compare them by speed. In particular, we study the special Montgomery presentation of elliptic curves and show that they give a considerable acceleration of the factoring procedure. We measure the speed of four realizations of Lenstra's algorithm to choose the best one.