Deterministic Polynomial Factoring and Association Schemes

TL;DR

This paper presents a deterministic polynomial-time algorithm for factoring prime degree polynomials over finite fields, leveraging association schemes and algebraic-combinatorial structures, under certain number-theoretic conditions and the GRH.

Contribution

It introduces a new deterministic algorithm for polynomial factoring that improves previous results by exploiting association schemes and smooth divisors, especially for prime degrees.

Findings

Deterministic polynomial-time factoring for prime degree polynomials under GRH.

Identification of special association scheme structures in the factoring process.

Algorithm outperforms previous methods for infinitely many prime degrees under certain conditions.

Abstract

The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the generalized Riemann hypothesis (GRH). In this work we improve the state of the art by focusing on prime degree polynomials; let n be the degree. If (n-1) has a `large' r-smooth divisor s, then we find a nontrivial factor of f(x) in deterministic poly(n^r,log q) time; assuming GRH and that s > sqrt{n/(2^r)}. Thus, for r = O(1) our algorithm is polynomial time. Further, for r > loglog n there are infinitely many prime degrees n for which our algorithm is applicable and better than the best known; assuming GRH. Our methods build on the algebraic-combinatorial framework of m-schemes initiated by Ivanyos, Karpinski and Saxena (ISSAC 2009). We show that the m-scheme on n points, implicitly appearing in our factoring algorithm, has an exceptional structure; leading us to the improved time complexity. Our structure theorem proves the existence of small intersection numbers in any association scheme that has many relations, and roughly equal valencies and indistinguishing numbers.