Polynomial Factorization over Finite Fields By Computing Euler-Poincare Characteristics of Drinfeld Modules

TL;DR

This paper introduces two randomized algorithms for factoring polynomials over finite fields using Drinfeld modules, significantly improving efficiency by leveraging Euler-Poincare characteristics and establishing new bounds on polynomial factorization patterns.

Contribution

The paper presents novel algorithms utilizing Drinfeld modules for polynomial factorization, reducing worst-case complexity to average-case and analyzing degree distributions in factorization patterns.

Findings

Algorithms achieve nearly linear time complexity in polynomial degree.

New bounds on degree distributions in polynomial factorization patterns.

Reduction of worst-case to average-case complexity for polynomial factorization.

Abstract

We propose and rigorously analyze two randomized algorithms to factor univariate polynomials over finite fields using rank $2$ Drinfeld modules. The first algorithm estimates the degree of an irreducible factor of a polynomial from Euler-Poincare characteristics of random Drinfeld modules. Knowledge of a factor degree allows one to rapidly extract all factors of that degree. As a consequence, the problem of factoring polynomials over finite fields in time nearly linear in the degree is reduced to finding Euler-Poincare characteristics of random Drinfeld modules with high probability. Notably, the worst case complexity of polynomial factorization over finite fields is reduced to the average case complexity of a problem concerning Drinfeld modules. The second algorithm is a random Drinfeld module analogue of Berlekamp's algorithm. During the course of its analysis, we prove a new bound on degree distributions in factorization patterns of polynomials over finite fields in certain short intervals.