{
  "abstract": "We construct infinite families of reduced almost perfect nonlinear (APN) permutations of $\\mathbb{F}_{2^m}$ that admit coefficientwise permutation lifts to $\\mathrm{GR}(2^k,m)$ for every $k>1$. These families disprove Conjecture~2 of Rønjom and Sandrib (Cryptogr. Commun., 2026), even for reduced polynomial representatives. For every odd $m\\ge5$ with $3\\nmid m$, we obtain explicit seven-term polynomials over $\\mathbb{F}_2$ with formal derivative identically one. We derive them from a complete criterion for the reduced representative of $A(x)^{2^s+1}$ to have no critical point in $\\mathbb{F}_{2^m}$, where $A$ is an invertible linearized polynomial and $\\gcd(s,m)=1$. We then apply the cube case to construct a second family of degree $2^{m-1}+2$, whose nonconstant derivatives avoid zero by an explicit trace identity. A fixed parameter in $\\mathbb{F}_{32}$ gives this family in every dimension $m=5r$ with $r$ odd, including a degree-$18$ example over $\\mathbb{F}_{32}$. Both constructions remain in Gold linear equivalence classes, so liftability of reduced representatives is not invariant under invertible $\\mathbb{F}_2$-linear input changes. We obtain the lifting conclusions from the standard Galois-ring permutation criterion. We also verify a degree-$24$ example supplied by Zhang and Haobo, prove a trace-adjoint criterion for reduced quadratic functions, and interpret the derivative condition through ramification at rational points.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30808",
  "author_total": 2,
  "authors": [
    {
      "name": "Daniele Bartoli"
    },
    {
      "name": "Pantelimon Stanica"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fdd5r5ydsscr4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30808",
  "fulltext_url": null,
  "id": "fdd5r5ydsscr4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30808",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2608.30808",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fdd5r5ydsscr4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2608.30808",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30808"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Reduced APN permutations and a disproof of the Galois-ring lifting conjecture",
  "title_text": "Reduced APN permutations and a disproof of the Galois-ring lifting conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/fdd5r5ydsscr4",
  "venue": "arXiv",
  "versions": []
}