Factorization of polynomials over finite fields in subexponential time under GRH

Abstract

We show assuming the Generalized Riemann Hypothethis that the factorization of a one-variable polynomial of degree n over an explicitly given finite field of cardinality q can be done in deterministic time (n log n log q)O(1). Since we need the hypothesis only to take roots in finite fields in polynomial time, the result can also be formulated in the following way: a polynomial equation over a finite field can be solved “by radicals” in subexponential time.