Deterministic root finding over finite fields using Graeffe transforms
Résumé
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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