Improving the berlekamp algorithm for binomials xn-a

Authors:
Ryuichi Harasawa;Yutaka Sueyoshi;Aichi Kudo
Affiliations:
Graduate School of Engineering, Nagasaki University, Nagasaki-shi, Nagasaki, Japan;Graduate School of Engineering, Nagasaki University, Nagasaki-shi, Nagasaki, Japan;Graduate School of Engineering, Nagasaki University, Nagasaki-shi, Nagasaki, Japan
Venue:
WAIFI'12 Proceedings of the 4th international conference on Arithmetic of Finite Fields
Year:
2012

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Abstract

In this paper, we describe an improvement of the Berlekamp algorithm, a method for factoring univariate polynomials over finite fields, for binomials xn−a over finite fields $\mathbb{F}_{q}$. More precisely, we give a deterministic algorithm for solving the equation $h(x)^{q} \equiv h(x) \ ({\rm mod}\ x^{n} -a)$ 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 $O \, \tilde{}\, (n \log q)$ operations in $\mathbb{F}_{q}$ 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.