Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields

Abstract

The fastest known algorithm for factoring univariate polynomials over finite fields is the Kedlaya-Umans (fast modular composition) implementation of the Kaltofen-Shoup algorithm. It is randomized and takes $\tilde{O}(n^{3/2}\log q+n{\log }^{2}q)$ time to factor polynomials of degree $n$ over the finite field ${\mathbb{F}}_q$ with $q$ elements. A significant open problem is if the $3/2$ exponent can be improved. We study a collection of algebraic problems and establish a web of reductions between them. A consequence is that an algorithm for any one of these problems with exponent better than $3/2$ would yield an algorithm for polynomial factorization with exponent better than $3/2$.