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 O(n3/2logq+nlog2q) 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