Smoothness testing of polynomials over finite fields

Abstract

We present an analysis of Bernstein's batch integer smoothness test when applied to the case of polynomials over a finite field $\mathbb{F}_q.$ We compare the performance of our algorithm with the standard method based on distinct degree factorization from both an analytical and a practical point of view. Our results show that although the batch test is asymptotically better as a function of the degree of the polynomials to test for smoothness, it is unlikely to offer significant practical improvements for cases of practical interest.