Polynomials over finite fields free from large and small degree irreducible factors

Authors:
Theodoulos Garefalakis;Daniel Panario
Affiliations:
Department of Mathematics, Royal Hollaway, University of London, Egham, Surrey TW20 0EX, United Kingdom;School of Mathematics and Statistics, Carleton University, Ottawa, ON, Canada K1S 5B6
Venue:
Journal of Algorithms - Analysis of algorithms
Year:
2002

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Abstract

We study the number Nq(n, m1, m2) of polynomials of degree n over a finite field Fq with all irreducible factors of degree bigger than m2 and less than or equal to m1. Applying the saddle point method, we obtain estimates for Nq(n, m1, m2) in the range m1 = o(n), which have the flavor of de Bruijn, Canfield et al., and Friedlander for the corresponding problem for integers. Our results have applications in computational number theory and cryptography, and include as a particular case the smooth polynomials studied by Odlyzko and others.