{
  "id": "2010.03801",
  "version": "v1",
  "published": "2020-10-08T07:06:08.000Z",
  "updated": "2020-10-08T07:06:08.000Z",
  "title": "On functions with the maximal number of bent components",
  "authors": [
    "Nurdagül Anbar",
    "Tekgül Kalaycı",
    "Wilfried Meidl",
    "László Mérai"
  ],
  "categories": [
    "math.NT",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "A function $F:\\mathbb{F}_2^n\\rightarrow \\mathbb{F}_2^n$, $n=2m$, can have at most $2^n-2^m$ bent component functions. Trivial examples are obtained as $F(x) = (f_1(x),\\ldots,f_m(x),a_1(x),\\ldots, a_m(x))$, where $\\tilde{F}(x)=(f_1(x),\\ldots,f_m(x))$ is a vectorial bent function from $\\mathbb{F}_2^n$ to $\\mathbb{F}_2^m$, and $a_i$, $1\\le i\\le m$, are affine Boolean functions. A class of nontrivial examples is given in univariate form with the functions $F(x) = x^{2^r}{\\rm Tr^n_m}(\\Lambda(x))$, where $\\Lambda$ is a linearized permutation of $\\mathbb{F}_{2^m}$. In the first part of this article it is shown that plateaued functions with $2^n-2^m$ bent components can have nonlinearity at most $2^{n-1}-2^{\\lfloor\\frac{n+m}{2}\\rfloor}$, a bound which is attained by the example $x^{2^r}{\\rm Tr^n_m}(x)$, $1\\le r<m$ (Pott et al. 2018). This partially solves Question 5 in Pott et al. 2018. We then analyse the functions of the form $x^{2^r}{\\rm Tr^n_m}(\\Lambda(x))$. We show that for odd $m$, only $x^{2^r}{\\rm Tr^n_m}(x)$, $1\\le r<m$, has maximal nonlinearity, whereas there are more of them for even $m$, of which we present one more infinite class explicitly. In detail, we investigate Walsh spectrum, differential spectrum and their relations for the functions $x^{2^r}{\\rm Tr^n_m}(\\Lambda(x))$. Our results indicate that this class contains many nontrivial EA-equivalence classes of functions with the maximal number of bent components, if $m$ is even, several with maximal possible nonlinearity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-08T07:06:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "maximal number",
      "vectorial bent function",
      "affine boolean functions",
      "nonlinearity",
      "bent component functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}