Tests and Constructions of Irreducible Polynomials over Finite Fields

Abstract

In this paper we focus on tests and constructions of irre- ducible polynomials over finite fields. We revisit Rabin's 1980 algo-rithm providing a variant of it that improves Rabin's cost estimate bya logn factor. We give a precise analysis of the probability that a ran-dom polynomial of degree n contains no irreducible factors of degree lessthan O logn . This probability is naturally related to Ben-Or's 1981 algorithm for testing irreducibility of polynomials over finite fields. Wealso compute the probability of a polynomial being irreducible when ithas no irreducible factors of low degree. This probability is useful in theanalysis of various algorithms for factoring polynomials over finite fields.We present an experimental comparison of these irreducibility methodswhen testing random polynomials.