Improving the Berlekamp algorithm for binomials x n - a

Abstract

In this paper, we describe an improvement of the Berlekamp algorithm, a method for factoring univariate polynomials over finite fields, for binomials x n - a over finite fields double-struck F q. More precisely, we give a deterministic algorithm for solving the equation h(x) q ≡ h(x) (mod x n - a) directly without applying the sweeping-out method to the corresponding coefficient matrix. We show that the factorization of binomials using the proposed method is performed in O~(n log q) operations in double-struck F q if we apply a probabilistic version of the Berlekamp algorithm after the first step in which we propose an improvement. Our method is asymptotically faster than known methods in certain areas of q, n and as fast as them in other areas. © 2012 Springer-Verlag.