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.