Enumerating all the Irreducible Polynomials over Finite Field

TL;DR

This paper presents efficient algorithms for enumerating all irreducible polynomials over finite fields and their roots, utilizing Lyndon words enumeration techniques to achieve near-linear time complexity.

Contribution

It introduces a novel approach combining Lyndon words enumeration with polynomial irreducibility testing, improving enumeration efficiency over previous methods.

Findings

Enumeration time is quasilinear in output size

Algorithm achieves linear delay in listing Lyndon words

Efficient enumeration of irreducible polynomials and roots

Abstract

In this paper we give a detailed analysis of deterministic and randomized algorithms that enumerate any number of irreducible polynomials of degree $n$ over a finite field and their roots in the extension field in quasilinear where $N=n^2$ is the size of the output.} time cost per element. Our algorithm is based on an improved algorithm for enumerating all the Lyndon words of length $n$ in linear delay time and the known reduction of Lyndon words to irreducible polynomials.