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arXiv · 0905.1642

Fast construction of irreducible polynomials over finite fields

Jean-Marc Couveignes·Reynald Lercier
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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.

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Jean-Marc Couveignes, Reynald Lercier. 2009-05-11. Fast construction of irreducible polynomials over finite fields. https://arxiv.org/abs/0905.1642

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