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Deterministic root finding over finite fields using Graeffe transforms

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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.

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Authors and Affiliations

  1. Laboratoire d’informatique, de robotique et de microélectronique de Montpellier LIRMM, UMR 5506 CNRS, CC477, Université Montpellier, 161, rue Ada, 34095, Montpellier Cedex 5, France

    Bruno Grenet

  2. Laboratoire d’informatique de l’École polytechnique, LIX, UMR 7161 CNRS, Campus de l’École polytechnique, 1, rue Honoré d’Estienne d’Orves, Bâtiment Alan Turing, CS35003, 91120, Palaiseau, France

    Joris van der Hoeven & Grégoire Lecerf

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  1. Bruno Grenet
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  2. Joris van der Hoeven
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  3. Grégoire Lecerf
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Correspondence to Grégoire Lecerf.

Additional information

Bruno Grenet was partially supported by a LIX-Qualcomm\(^{\circledR }\)—Carnot postdoctoral fellowship.

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Grenet, B., van der Hoeven, J. & Lecerf, G. Deterministic root finding over finite fields using Graeffe transforms. AAECC 27, 237–257 (2016). https://doi.org/10.1007/s00200-015-0280-5

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  • DOI: https://doi.org/10.1007/s00200-015-0280-5

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