Using zeta functions to factor polynomials over finite fields

TL;DR

This paper explores a novel approach to polynomial factorization over finite fields by leveraging zeta functions, building on ideas from elliptic curve point counting algorithms and their generalizations.

Contribution

It provides an exposition of Kayal’s idea and details a generalization involving Pila’s algorithm for abelian varieties, advancing deterministic polynomial-time factorization methods.

Findings

Proposes a new approach using zeta functions for polynomial factorization.

Extends Kayal’s idea with a generalization involving Pila's algorithm.

Lays groundwork for deterministic polynomial-time algorithms.

Abstract

In 2005, Kayal suggested that Schoof's algorithm for counting points on elliptic curves over finite fields might yield an approach to factor polynomials over finite fields in deterministic polynomial time. We present an exposition of his idea and then explain details of a generalization involving Pila's algorithm for abelian varieties.