Deterministic root finding over finite fields using Graeffe transforms

Bruno Grenet (1) , Joris van der Hoeven (2) , Grégoire Lecerf (2)

1 ECO - Exact Computing
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

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

Domaines: Calcul formel [cs.SC]

Bruno Grenet, Joris van der Hoeven, Grégoire Lecerf. Deterministic root finding over finite fields using Graeffe transforms. 2015. ⟨hal-01104251⟩