What Do Random Polynomials over Finite Fields Look Like?

Abstract

In this paper, we survey old and new results about random univariate polynomials over a finite field \(\mathbb{F}_q\). We are interested in three aspects: (1) the decomposition of a random polynomial in terms of its irreducible factors, (2) the usage of random polynomials in algorithms, and (3) the average-case analysis of algorithms that use polynomials over finite fields.KeywordsFinite FieldDiscrete LogarithmDiscrete Logarithm ProblemIrreducible FactorSaddle Point MethodThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.