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An analytic approach to smooth polynomials over finite fields

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Algorithmic Number Theory (ANTS 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1423))

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Abstract

We consider the largest degrees that occur in the decomposition of polynomials over finite fields into irreducible factors. We expand the range of applicability of the Dickman function as an approximation for the number of smooth polynomials, which provides precise estimates for the discrete logarithm problem. In addition, we characterize the distribution of the two largest degrees of irreducible factors, a problem relevant to polynomial factorization. As opposed to most earlier treatments, our methods are based on a combination of exact descriptions by generating functions and a specific complex asymptotic method.

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Joe P. Buhler

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© 1998 Springer-Verlag Berlin Heidelberg

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Panario, D., Gourdon, X., Flajolet, P. (1998). An analytic approach to smooth polynomials over finite fields. In: Buhler, J.P. (eds) Algorithmic Number Theory. ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054865

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  • DOI: https://doi.org/10.1007/BFb0054865

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64657-0

  • Online ISBN: 978-3-540-69113-6

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