Schemes for deterministic polynomial factoring

Abstract

In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects, we call m-schemes, that are generalizations of permutation groups. We design a new generalization of the known conditional deterministic subexponential time polynomial factoring algorithm to get an underlying m-scheme. We then demonstrate how progress in understanding m-schemes relate to improvements in the deterministic complexity of factoring polynomials, assuming the Generalized Riemann Hypothesis (GRH). In particular, we give the first deterministic polynomial time algorithm (assuming GRH) to find a nontrivial factor of a polynomial of prime degree n where (n-1) is a constant-smooth number. We use a structural theorem about association schemes on a prime number of points, which Hanaki and Uno (2006) proved by representation theory methods.