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 $d^{1+o(1)} \times (\log q)^{5+o(1)}$ elementary operations. The $o(1)$ in $d^{1+o(1)}$ is a function of $d$ that tends to zero when $d$ tends to infinity. And the $o(1)$ in $(\log q)^{5+o(1)}$ is a function of $q$ that tends to zero when $q$ tends to infinity. In particular, the complexity is quasi-linear in the degree $d$.
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Jean-Marc Couveignes, Reynald Lercier. Fast construction of irreducible polynomials over finite fields. Israel Journal of Mathematics, The Hebrew University Magnes Press, 2013, 194 (1), pp.77-105. ⟨10.1007/s11856-012-0070-8⟩. ⟨hal-00456456⟩

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