{
  "abstract": "Symmetric primitives over Galois rings need nonlinear components whose differentials with respect to ring addition are controlled. Let $R=\\mathrm{GR}(4,m)$ and $q=2^m$, so that $(R,+)\\cong(\\mathbb{Z}/4\\mathbb{Z})^m$ has order $N=q^2$. For every $m\\ge1$ we construct an explicit permutation $G_c$ of $R$ all of whose differentials with a nonzero input difference have probability at most $36/N$, whereas every permutation polynomial of $R$ has such a differential of probability at least $2/\\sqrt{N}$ (Ronjom--Sandrib). The bound holds for the fixed permutation itself, not on average over keys. Equivalently, $Δ(G_c)\\le36$ against $Δ\\ge2\\sqrt{N}$ for permutation polynomials; since every permutation has $Δ\\ge2$, the least differential uniformity of a permutation of $(\\mathbb{Z}/4\\mathbb{Z})^m$ lies between $2$ and $36$ for every $m$. In Teichmuller digits $(a,b)\\in\\mathbb{F}_q^2$ and for $c\\in\\mathbb{F}_q$ of absolute trace $1$, $G_c(a,b)=(L,(L^2+L+c)a)$ with $L=(a^2+a+c)b$. The map is $F_c\\circ J\\circ F_c$, where $F_c(a,b)=(a,(a^2+a+c)b)$ is induced by a polynomial over $R$ and $J$ exchanges the digits: a single digit exchange between two polynomial layers yields a bound independent of $m$. The proof reduces the difference equations, in each of three types of directions, to one-variable polynomials whose degrees do not depend on $m$. For general maps $F\\circ J\\circ F$ with $F(a,b)=(p(a),h(a)+d(a)b)$, where $p,d,h$ have degrees bounded independently of $m$, $p'\\ne0$ and $d$ has no zeros, a Jacobian identity in characteristic $2$ bounds the differentials with nonzero low input and output differences independently of $m$; the remaining directions need a separate argument, which we give for an explicit family ($Δ\\le440$, resp. $510$, for odd $m$).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02572",
  "author_total": 1,
  "authors": [
    {
      "name": "Hiroki Fukui"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7c4s7avm09rm8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02572",
  "fulltext_url": null,
  "id": "7c4s7avm09rm8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02572",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02572",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7c4s7avm09rm8.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02572",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02572"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Explicit permutations of $(\\mathbb{Z}/4\\mathbb{Z})^m$ with uniformly bounded differential uniformity",
  "title_text": "Explicit permutations of (ℤ/4ℤ)^m with uniformly bounded differential uniformity",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/7c4s7avm09rm8",
  "venue": "arXiv",
  "versions": []
}