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Deterministic root finding over finite fields using Graeffe transforms

Bruno Grenet 1 Joris van der Hoeven 2 Grégoire Lecerf 2
1 ECO - Exact Computing
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : We design new deterministic algorithms, based on Graeffe transforms, to compute all the roots of a polynomial which splits over a finite field Fq. Our algorithms were designed to be particularly efficient in the case when the cardinality q−1 of the multiplicative group of Fq 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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Submitted on : Tuesday, June 7, 2016 - 1:36:11 PM
Last modification on : Thursday, March 5, 2020 - 6:29:39 PM

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Bruno Grenet, Joris van der Hoeven, Grégoire Lecerf. Deterministic root finding over finite fields using Graeffe transforms. Applicable Algebra in Engineering, Communication and Computing, Springer Verlag, 2016, 27 (3), pp.237-257. ⟨10.1007/s00200-015-0280-5⟩. ⟨lirmm-01328010⟩



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