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Improving the Berlekamp Algorithm for Binomials x n − a

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Book cover Arithmetic of Finite Fields (WAIFI 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7369))

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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 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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Authors and Affiliations

  1. Graduate School of Engineering, Nagasaki University, 1-14 Bunkyomachi, Nagasaki-shi, Nagasaki, 852-8521, Japan

    Ryuichi Harasawa, Yutaka Sueyoshi & Aichi Kudo

Authors
  1. Ryuichi Harasawa
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  2. Yutaka Sueyoshi
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  3. Aichi Kudo
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Editor information

Editors and Affiliations

  1. Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey

    Ferruh Özbudak

  2. Computer Science Section, CINVESTAV, Mexico DF, Mexico

    Francisco Rodríguez-Henríquez

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© 2012 Springer-Verlag Berlin Heidelberg

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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

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31661-6

  • Online ISBN: 978-3-642-31662-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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