Fast construction of irreducible polynomials 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 $d^{1+ε(d)} \times (\log q)^{5+ε(q)}$ elementary operations. The function $ε$ 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 $d^{1+ε(d)} \times (\log q)^{1+ε(q)}$ elementary operations only.