Subquadratic Computation of Vector Generating Polynomials and Improvement of the Block Wiedemann Algorithm

This paper describes a new algorithm for computing linear generators (vector generating polynomials) for matrix sequences, running in subquadratic time. This algorithm applies in particular to the sequential stage of Coppersmith’s block Wiedemann algorithm. Experiments showed that our method can be substituted in place of the quadratic one proposed by Coppersmith, yielding important speedups even for realistic matrix sizes. The base fields we were interested in were finite fields of large characteristic. As an example, we have been able to compute a linear generator for a sequence of 4 × 4 matrices of length 242 304 defined over F 2607 − 1 in less than 2 days on one 667 MHz alpha ev67 CPU.