Trading GRH for algebra: Algorithms for factoring polynomials and related structures

Abstract

In this paper we develop a general technique to eliminate the assumption of the Generalized Riemann Hypothesis (GRH) from various deterministic polynomial factoring algorithms over finite fields. It is the first bona fide progress on that issue for more than 25 years of study of the problem. Our main results are basically of the following form: either we construct a nontrivial factor of a given polynomial or compute a nontrivial automorphism of the factor algebra of the given polynomial. Probably the most notable application of such automorphisms is efficiently finding zero divisors in noncommutative algebras. The proof methods used in this paper exploit virtual roots of unity and lead to efficient actual polynomial factoring algorithms in special cases.