{
  "id": "1301.5004",
  "version": "v1",
  "published": "2013-01-21T21:06:26.000Z",
  "updated": "2013-01-21T21:06:26.000Z",
  "title": "Planar functions and perfect nonlinear monomials over finite fields",
  "authors": [
    "Michael Zieve"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CO",
    "cs.IT",
    "math.IT",
    "math.NT"
  ],
  "abstract": "The study of finite projective planes involves planar functions, namely, functions f : F_q --> F_q such that, for each nonzero a in F_q, the function c --> f(c+a) - f(c) is a bijection on F_q. Planar functions are also used in the construction of DES-like cryptosystems, where they are called perfect nonlinear functions. We determine all planar functions on F_q of the form c --> c^t, under the assumption that q >= (t-1)^4. This implies two conjectures of Hernando, McGuire and Monserrat. Our arguments also yield a new proof of a conjecture of Segre and Bartocci from 1971 about monomial hyperovals in finite Desarguesian projective planes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-21T21:06:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51E20",
      "11T06",
      "11T71",
      "05B05"
    ],
    "keywords": [
      "planar functions",
      "perfect nonlinear monomials",
      "finite fields",
      "finite desarguesian projective planes",
      "perfect nonlinear functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.5004Z"
    }
  }
}