{
  "id": "0905.1642",
  "version": "v3",
  "published": "2009-05-11T15:44:25.000Z",
  "updated": "2011-11-19T17:46:09.000Z",
  "title": "Fast construction of irreducible polynomials over finite fields",
  "authors": [
    "Jean-Marc Couveignes",
    "Reynald Lercier"
  ],
  "comment": "To appear in the Israel Journal of Mathematics",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-11-19T17:46:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G20",
      "11Y16",
      "14H45",
      "68Q25"
    ],
    "keywords": [
      "finite field",
      "fast construction",
      "elementary operations",
      "real positive function",
      "random irreducible polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0905.1642C"
    }
  }
}