Schemes for deterministic polynomial factoring

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 mschemes 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 constantsmooth 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. Categories and Subject Descriptors

• Type: Conference
• Year: 2009
• Where: ISSAC
• Authors: Gábor Ivanyos, Marek Karpinski, Nitin Saxena
• Keywords: Deterministic, Deterministic Complexity, ISSAC 2009, Mathematics, Polynomial Factoring Algorithm