A parallel GNFS algorithm with the biorthogonal block lanczos method for integer factorization

Abstract

Currently, RSA is a very popular, widely used and secure public key cryptosystem, but the security of the RSA cryptosystem is based on the difficulty of factoring large integers. The General Number Field Sieve (GNFS) algorithm is the best known method for factoring large integers over 110 digits. Our previous work on the parallel GNFS algorithm, which integrated the Montgomery's block Lanczos algorithm to solve the large and sparse linear systems over GF(2), has one major disadvantage, namely the input has to be symmetric (we have to symmetrize the input for nonsymmetric case and this will shrink the rank). In this paper, we successfully implement the parallel General Number Field Sieve (GNFS) algorithm and integrate with a new algorithm called the biorthogonal block Lanczos algorithm for solving large and sparse linear systems over GF(2). This new algorithm is based on the biothorgonal technique, can find more solutions or dependencies than Montgomery's block Lanczos method with less iterations. The detailed experimental results on a SUN cluster will be presented as well. © Springer-Verlag Berlin Heidelberg 2006.