{
  "abstract": "Complete permutation polynomials play an important role in cryptography, particularly in the design of cryptographic primitives such as the Lai--Massey scheme and S-boxes. We generalize a result of Sun, Li, Guo, and Qu (2021) by characterizing the complete permutation behavior of the mapping $Ψ(X)=M(X+ψ(AX))$ over $\\mathbb{F}_q^n$, where $\\mathbb{F}_q$ is a finite field of $q$ elements with $q$ being a prime power, $M\\in GL(n, \\mathbb{F}_q)$, $GL(n, \\mathbb{F}_q)$ is the general linear group of order $n$ over $\\mathbb{F}_q$, $A_{m \\times n}$ is a full-rank matrix over $\\mathbb{F}_q$, and $ψ=(ψ_1,ψ_2,\\ldots,ψ_n)$ with each component function $ψ_i:\\mathbb{F}_q^m\\to\\mathbb{F}_q$. Furthermore, we establish criteria for the permutation and complete permutation properties of the mapping $F(X)=T(X+B^tf(AX))$ over $\\mathbb{F}_{q}^n$, $f: \\mathbb{F}_{q}^{m} \\rightarrow \\mathbb{F}_{q}^{n-m}$, $T \\in GL(n, \\mathbb{F}_q)$, $A_{m \\times n}$ and $B_{(n-m)\\times n}$ are full-rank matrices over $\\mathbb{F}_q$, $B^t$ represents the transpose of the matrix $B$, and $0<m<n$ are integers. These results also generalize an earlier result of Gravel and Panario (2023), who showed that any arbitrary function $f$ from $\\mathbb{F}_q^m$ to $\\mathbb{F}_q^{\\,n-m}$ can be extended to a bijection over $\\mathbb{F}_{q}^n$ through the mapping $F(X)=T(X+B^tf(AX))$, under the condition $AB^t=0$. Here we do not impose the restriction that $AB^t=0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05564",
  "author_total": 5,
  "authors": [
    {
      "name": "Sartaj Ul Hasan"
    },
    {
      "name": "Ramandeep Kaur"
    },
    {
      "name": "Hridesh Kumar"
    },
    {
      "name": "Daniel Panario"
    },
    {
      "name": "Qiang Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4pz7pzx8tj879.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05564",
  "fulltext_url": null,
  "id": "4pz7pzx8tj879",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05564",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CR"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7212010796",
  "pdf_url": "https://arxiv.org/pdf/2609.05564",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4pz7pzx8tj879.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05564",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05564"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Constructions of complete permutations over $\\mathbb{F}_q^n$",
  "title_text": "Constructions of complete permutations over 𝔽_q^n",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/4pz7pzx8tj879",
  "venue": "arXiv",
  "versions": []
}