Tests and constructions of irreducible polynomials over finite fields

In this paper we focus on tests and constructions of irreducible polynomials over finite fields. We revisit Rabin’s (1980) algorithm providing a variant of it that improves Rabin’s cost estimate by a log n factor. We give a precise analysis of the probability that a random polynomial of degree n contains no irreducible factors of degree less than O(log n). This probability is naturally related to Ben-Or’s (1981) algorithm for testing irreducibility of polynomials over finite fields. We also compute the probability of a polynomial being irreducible when it has no irreducible factors of low degree. This probability is useful in the analysis of various algorithms for factoring polynomials over finite fields. We present an experimental comparison of these irreducibility methods when testing random polynomials.

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