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Fast construction of irreducible polynomials over finite fields

Israel Journal of Mathematics volume 194, pages 77–105 (2013)Cite this article

Abstract

We present a randomized algorithm that on inputting a finite field K with q elements and a positive integer d outputs a degree d irreducible polynomial in K[x]. The running time is d 1+ɛ(d)×(log q)5+ɛ(q) elementary operations. The function ɛ in this expression is a real positive function belonging to the class o(1), especially, the complexity is quasi-linear in the degree d. Once given such an irreducible polynomial of degree d, we can compute random irreducible polynomials of degree d at the expense of d 1+ɛ(d) × (log q)1+ɛ(q) elementary operations only.

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Authors and Affiliations

  1. INRIA Bordeaux Sud-Ouest & Université de Toulouse, Université de Toulouse le Mirail, Département de Mathématiques et Informatique, 5 allées Antonio Machado, F-31058, Toulouse cédex 9, France

    Jean-Marc Couveignes

  2. La Roche Marguerite, DGA, F-35174, Bruz Cedex, France

    Reynald Lercier

  3. Institut de Recherche Mathématique de Rennes, Université de Rennes 1, Campus de Beaulieu, F-35042, Rennes, France

    Reynald Lercier

Authors
  1. Jean-Marc Couveignes
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  2. Reynald Lercier
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Correspondence to Jean-Marc Couveignes.

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Couveignes, JM., Lercier, R. Fast construction of irreducible polynomials over finite fields. Isr. J. Math. 194, 77–105 (2013). https://doi.org/10.1007/s11856-012-0070-8

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  • DOI: https://doi.org/10.1007/s11856-012-0070-8

Keywords

  • Elliptic Curve
  • Elliptic Curf
  • Prime Power
  • Minimal Polynomial
  • Irreducible Polynomial