{
  "id": "2104.13234",
  "version": "v1",
  "published": "2021-04-27T14:40:00.000Z",
  "updated": "2021-04-27T14:40:00.000Z",
  "title": "Permutation polynomials from a linearized decomposition",
  "authors": [
    "Lucas Reis",
    "Qiang Wang"
  ],
  "comment": "11 pages, comments are welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we discuss the permutational property of polynomials of the form $f(L(x))+k(L(x))\\cdot M(x)\\in \\mathbb F_{q^n}[x]$ over the finite field $\\mathbb F_{q^n}$, where $L, M\\in \\mathbb F_q[x]$ are $q$-linearized polynomials. The restriction $L, M\\in \\mathbb F_q[x]$ implies a nice correspondence between the pair $(L, M)$ and the pair $(g, h)$ of conventional $q$-associates over $\\mathbb F_q$ of degree at most $n-1$. In particular, by using the AGW criterion, permutational properties of our class of polynomials translates to some arithmetic properties of polynomials over $\\mathbb F_q$, like coprimality. This relates the problem of constructing PPs of $\\mathbb F_{q^n}$ to the problem of factorizing $x^n-1$ in $\\mathbb F_q[x]$. We then specialize to the case where $L(x)$ is the trace polynomial from $\\mathbb F_{q^n}$ over $\\mathbb F_q$, providing results on the construction of permutation and complete permutation polynomials, and their inverses. We further demonstrate that the latter can be extended to more general linearized polynomials of degree $q^{n-1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-27T14:40:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linearized decomposition",
      "permutational property",
      "complete permutation polynomials",
      "general linearized polynomials",
      "finite field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}