An analytic approach to smooth polynomials over finite fields | Zendy

Daniel Panario | Zendy; Xavier Gourdon | Zendy; Philippe Flajolet | Zendy

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An analytic approach to smooth polynomials over finite fields

Author(s) -

Daniel Panario,

Xavier Gourdon,

Philippe Flajolet

Publication year - 1998

Publication title -

lecture notes in computer science

Language(s) - English

Resource type - Book series

SCImago Journal Rank - 0.249

H-Index - 400

eISSN - 1611-3349

pISSN - 0302-9743

ISBN - 3-540-64657-4

DOI - 10.1007/bfb0054865

Subject(s) - logarithm , factorization , polynomial , irreducible polynomial , mathematics , range (aeronautics) , factorization of polynomials , finite field , discrete logarithm , decomposition , function (biology) , algebra over a field , discrete mathematics , computer science , pure mathematics , algorithm , mathematical analysis , matrix polynomial , ecology , materials science , evolutionary biology , composite material , biology , encryption , public key cryptography , operating system

A b s t r a c t. We consider the largest degrees that occur in the decomposi-tion 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 dis-tribution of the two largest degrees of irreducible factors, a problem relevant to polynomial factorization. As opposed to most earlier treat-ments, our methods are based on a combination of exact descriptions by generating functions and a specific complex asymptotic method.

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