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 \(\mathbb{F}_{q}\). 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.
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Harasawa, R., Sueyoshi, Y., Kudo, A. (2012). Improving the Berlekamp Algorithm for Binomials x n − a . In: Özbudak, F., Rodríguez-Henríquez, F. (eds) Arithmetic of Finite Fields. WAIFI 2012. Lecture Notes in Computer Science, vol 7369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31662-3_16
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DOI: https://doi.org/10.1007/978-3-642-31662-3_16
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