{
  "id": "1105.5852",
  "version": "v1",
  "published": "2011-05-30T03:22:20.000Z",
  "updated": "2011-05-30T03:22:20.000Z",
  "title": "On Taking r-th Roots without r-th Nonresidues over Finite Fields and Its Applications",
  "authors": [
    "Tsz-Wo Sze"
  ],
  "comment": "28 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We first show a deterministic algorithm for taking $r$-th roots over $\\F_q$ without being given any $r$-th nonresidue, where $\\F_q$ is a finite field with $q$ elements and $r$ is a small prime such that $r^2$ divides of $q-1$. As applications, we illustrate deterministic algorithms over $\\F_q$ for constructing $r$-th nonresidues, constructing primitive elements, solving polynomial equations and computing elliptic curve \"$n$-th roots\", and a deterministic primality test for the generalized Proth numbers. All algorithms are proved without assuming any unproven hypothesis. They are efficient only if all the factors of $q-1$ are small and some primitive roots of unity can be constructed efficiently over $\\F_q$. In some cases, they are the fastest among the known deterministic algorithms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-05-30T03:22:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "12Y05",
      "11Y11",
      "14H52"
    ],
    "keywords": [
      "finite field",
      "r-th nonresidues",
      "r-th roots",
      "applications",
      "illustrate deterministic algorithms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.5852S"
    }
  }
}