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

The fastest known algorithm for factoring univariate polynomials over finite fields is the KedlayaUmans [13] (fast modular composition) implementation of the Kaltofen-Shoup algorithm [12, § 2]. It is randomized and takes Õ(n3/2 log q+n log2 q) time to factor polynomials of degree n over the finite field Fq 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. 1998 ACM Subject Classification F.2.1 Computations in Finite Fields