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