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

TL;DR

This paper explores algebraic problems related to polynomial factorization over finite fields, showing that improving the exponent below 3/2 in these problems would lead to faster factorization algorithms.

Contribution

It establishes a web of reductions among algebraic problems, linking improvements in any one to faster polynomial factorization over finite fields.

Findings

Reductions connect algebraic problems to polynomial factorization.

Improving exponent below 3/2 in these problems implies faster algorithms.

Current best algorithms have exponent 3/2, and breaking this barrier is open.

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 $\widetilde{O}(n^{3/2}\log q + n \log^2 q)$ 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 $3/2$.