Solving Homogeneous Linear Equations Over GF(2) via Block Wiedemann Algorithm

Abstract

We propose a method of solving large sparse systems of homogeneous linear equations over G F ( 2 ) , the field with two elements. We modify an algorithm due to Wiedemann. A block version of the algorithm allows us to perform 32 matrix-vector operations for the cost of one. The resulting algorithm is competitive with structured Gaussian elimination in terms of time and has much lower space requirements. It may be useful in the last stage of integer factorization. We address here the problem of solving a large sparse system of homogeneous linear equations over GF(2) , the field with two elements. One important application, which motivates the present work, arises in integer factorization. During the last stage of most integer factorization algorithms, we are presented with a large sparse integer matrix and are asked to find linear combinations of the columns of this matrix which vanish modulo 2. For example [7], the matrix may have 100,000 columns, with an average of 15 nonzero entries per column. For this application we would like to obtain several solutions, because a given solution will lead to a nontrivial factorization with probability 112 ; with n independent solutions, our probability of finding a factorization rises to 1 2- . Structured Gaussian elimination can be used [7], but as problems get larger, it may become infeasible to store the matrices obtained in the intermediate stages of Gaussian elimination. The Wiedemann algorithm [9, 71 has smaller storage requirements (one need only store a few vectors and an encoding of a sparse matrix, not a dense matrix as occurs in Gaussian elimination after fillin), and it may have fewer computational steps (since one takes advantage of the sparseness of the matrix). But its efficiency is hampered by the fact that the algorithm acts on only one bit at a time. In the present paper we work with blocks of vectors at a single time. By treating 32 vectors at a time (on a machine with 32-bit words), we can perform 32 matrix-vector products at once, thus considerably decreasing the cost of indexing. This can be viewed as a block Wiedemann algorithm. The main technical difficulty is in obtaining the correct generalization of the Berlekamp-Massey algorithm to a block version, namely, a multidimensional version of the extended Euclidean algorithm. Received by the editor November 20, 1991 and, in revised form, July 24, 1992. 1991 Mathematics Subject Classijication. Primary 15A33, 11Y05, 11-04, 15-04. @ 1994 Amencan Mathernat~cal Soc~ety 0025-571 8/94 $1 .OO + $.25 per page