Deterministic root finding over finite fields using Graeffe transforms

  • Bruno Grenet
  • Joris van der Hoeven
  • Grégoire Lecerf
Original Paper

Abstract

We design new deterministic algorithms, based on Graeffe transforms, to compute all the roots of a polynomial which splits over a finite field \(\mathbb {F}_q\). Our algorithms were designed to be particularly efficient in the case when the cardinality \(q - 1\) of the multiplicative group of \(\mathbb {F}_q\) is smooth. Such fields are often used in practice because they support fast discrete Fourier transforms. We also present a new nearly optimal algorithm for computing characteristic polynomials of multiplication endomorphisms in finite field extensions. This algorithm allows for the efficient computation of Graeffe transforms of arbitrary orders.

Keywords

Polynomial root finding Finite field Graeffe transform Deterministic algorithm 

Mathematics Subject Classification

13P05 12Y05 68W30 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Bruno Grenet
    • 1
  • Joris van der Hoeven
    • 2
  • Grégoire Lecerf
    • 2
  1. 1.Laboratoire d’informatique, de robotique et de microélectronique de Montpellier LIRMM, UMR 5506 CNRS, CC477Université MontpellierMontpellier Cedex 5France
  2. 2.Laboratoire d’informatique de l’École polytechnique, LIX, UMR 7161 CNRSCampus de l’École polytechniquePalaiseauFrance

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