Deterministic root finding over finite fields using Graeffe transforms

Abstract

We design new deterministic algorithms, based on Graeffe transforms, to compute all the roots of a polynomial which splits over a finite field $$\mathbb {F}_q$$ F q . Our algorithms were designed to be particularly efficient in the case when the cardinality $$q - 1$$ q - 1 of the multiplicative group of $$\mathbb {F}_q$$ 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.

DOI: 10.1007/s00200-015-0280-5

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Cite this paper

@article{Grenet2015DeterministicRF, title={Deterministic root finding over finite fields using Graeffe transforms}, author={Bruno Grenet and Joris van der Hoeven and Gr{\'e}goire Lecerf}, journal={Applicable Algebra in Engineering, Communication and Computing}, year={2015}, volume={27}, pages={237-257} }