{
  "abstract": "Let $δ\\in\\mathbb{F}_{2^n}$ satisfy $\\operatorname{Tr}_{\\mathbb{F}_{2^n}/\\mathbb{F}_2}(δ)=1$. We study the permutation behavior of $$ f(x) = \\left(\\frac{1}{x^2+x+δ}\\right)^{2^k}+x $$ over $\\mathbb{F}_{2^n}$. Helleseth and Zinoviev proved that $f(x)$ is a permutation for $k=0,1$, and remarked that numerical evidence suggests that no other cases occur. In this paper, we confirm their assertion by proving that, for $0\\leq k<n$, $f(x)$ is a permutation of $\\mathbb{F}_{2^n}$ if and only if $k=0$ or $k=1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03353",
  "author_total": 1,
  "authors": [
    {
      "name": "Yi Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5sx5a4747eher.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03353",
  "fulltext_url": null,
  "id": "5sx5a4747eher",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03353",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7208758897",
  "pdf_url": "https://arxiv.org/pdf/2609.03353",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5sx5a4747eher.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03353",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03353"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Resolving a conjecture on permutation polynomials over $\\mathbb{F}_{2^n}$",
  "title_text": "Resolving a conjecture on permutation polynomials over 𝔽_2^n",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/5sx5a4747eher",
  "venue": "arXiv",
  "versions": []
}