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Tests and Constructions of Irreducible Polynomials over Finite Fields

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Foundations of Computational Mathematics

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Abstract

In this paper we focus on tests and constructions of irreducible polynomials over finite fields. We revisit Rabin’s (1980) algorithm providing a variant of it that improves Rabin’s cost estimate by a log n factor. We give a precise analysis of the probability that a random polynomial of degree n contains no irreducible factors of degree less than O(log n). This probability is naturally related to Ben-Or’s (1981) algorithm for testing irreducibility of polynomials over finite fields. We also compute the probability of a polynomial being irreducible when it has no irreducible factors of low degree. This probability is useful in the analysis of various algorithms for factoring polynomials over finite fields. We present an experimental comparison of these irreducibility methods when testing random polynomials.

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Authors
  1. Shuhong Gao
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  2. Daniel Panario
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Editor information

Editors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, Tat Cheet Avenue, 83 Tat Chee Avenue, Kowloon, Hong Kong

    Felipe Cucker

  2. Departament d’Economia, Universitat Pompeu Fabra, Ramon Trias Fargas 25-27, 08005, Barcelona, Spain

    Felipe Cucker

  3. I.B.M. T.J. Watson Center, 10598, Yorktown Heights, New York, USA

    Michael Shub

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© 1997 Springer-Verlag Berlin Heidelberg

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Gao, S., Panario, D. (1997). Tests and Constructions of Irreducible Polynomials over Finite Fields. In: Cucker, F., Shub, M. (eds) Foundations of Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60539-0_27

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  • DOI: https://doi.org/10.1007/978-3-642-60539-0_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61647-4

  • Online ISBN: 978-3-642-60539-0

  • eBook Packages: Springer Book Archive

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