Fast construction of irreducible polynomials over finite fields

TL;DR

This paper introduces a randomized, quasi-linear time algorithm for efficiently constructing irreducible polynomials over finite fields, significantly improving computational methods in algebraic applications.

Contribution

The paper presents a novel randomized algorithm that constructs degree d irreducible polynomials over finite fields with near-linear complexity in d, advancing computational algebra techniques.

Findings

Algorithm operates in quasi-linear time in degree d

Efficiently generates multiple irreducible polynomials of degree d

Reduces computational complexity for polynomial construction over finite fields

Abstract

We present a randomized algorithm that on input a finite field $K$ with $q$ elements and a positive integer $d$ outputs a degree $d$ irreducible polynomial in $K[x]$. The running time is $d^{1+\epsilon(d)} \times (\log q)^{5+\epsilon(q)}$ elementary operations. The function $\epsilon$ in this expression is a real positive function belonging to the class $o(1)$, especially, the complexity is quasi-linear in the degree $d$. Once given such an irreducible polynomial of degree $d$, we can compute random irreducible polynomials of degree $d$ at the expense of $d^{1+\epsilon(d)} \times (\log q)^{1+\epsilon(q)}$ elementary operations only.