Factoring Polynomials over Finite Fields and Stable Colorings of Tournaments

Abstract

In this paper we develop new algorithms for factoring polynomials over finite fields by exploring an interesting connection between the algebraic factoring problem and the combinatorial problem of stable coloring of tournaments. We present an algorithm which can be viewed as a recursive refinement scheme through which most cases of polynomials are completely factored in deterministic polynomial time within the first level of refinement, most of the remaining cases are factored completely before the end of the second level refinement, and so on. The algorithm has average polynomial time complexity and (n $^{\rm log{\it n}}$ logp)^O(1) worst case complexity. Under a purely combinatorial conjecture concerning tournaments, the algorithm has worst case complexity (n $^{\rm loglog{\it n}}$ logp)^O(1). Our approach is also useful in reducing the amount of randomness needed to factor a polynomial completely in expected polynomial time. We present a random polynomial time algorithm for factoring polynomials over finite fields which requires only log p random bits. All these results assume the Extended Riemann Hypothesis.