The Improvement of the Commonly Used Linear Polynomial Selection Methods

Abstract

The general number field sieve (GNFS) is asymptotically the fastest algorithm known for factoring large integers. One of the most important steps of GNFS is to select a good polynomial pair. Whether we can select a good polynomial pair, directly affects the efficiency of the sieving step. Now there are three linear methods widely used which base-m method, Murphy method and Klein Jung method. Base-m method is to construct polynomial based on the m-expansion of the integer, Murphy method mainly focuses on the root property of the polynomial and Klein Jung method restricts the first three coefficients size of the polynomial in a certain range. In this paper, we compare the size property and root property of the polynomials which select from the three methods. A good leading coefficient ad of f1 is important. The good means that ad has some small prime divisors. Klein Jung method does not give a concrete method to choose a good ad in its first step. We choose these better ad first and store in a set Ad, then take the Ad as input. Through the pretreatment we can get better polynomial and speed up the efficiency of Klein Jung method at the same time.