{
  "abstract": "Let $q=p^n>3$ be an odd prime power and let $g:\\F_q\\to\\F_q$ be an even perfect nonlinear (PN) function with $g(0)=0$. We construct almost perfect nonlinear (APN) functions $G_{λ,M}(x)=g(x)+M(x)+λ(x^{q-1}-1)$, where $λ\\in\\F_q^*$, $-λ\\notin g(\\F_q)$, and $M$ is an $\\F_p$-linearized permutation polynomial. These one-point modifications of $g+M$ have boomerang uniformity one, the smallest possible value for APN functions in odd characteristic. We give complete differential and boomerang spectrum formulas and obtain closed-form evaluations for selected constructions based on Coulter--Matthews monomials and Dembowski--Ostrom polynomials. For $q>7$, comparison of differential spectra establishes CCZ-inequivalence to all APN monomial families listed in the paper and every APN binomial in the original Ness--Helleseth family or its extension by Zeng et al. We characterize CCZ-equivalence within our construction, showing that any two linearly inequivalent even PN functions over the same field yield CCZ-inequivalent APN functions. We obtain three pairwise CCZ-inequivalent APN functions with boomerang uniformity one over $\\F_{p^{15\\ell}}$ for every $\\ell\\ge1$, six over $\\F_{p^{120r}}$ for every $r\\ge1$, and unbounded numbers of such classes in characteristic three. We further prove that CCZ-equivalence and EA-equivalence coincide for functions at Hamming distance at most $m$ from a PN function, provided that $m(m+1)<q-q/p$. We also show that all our APN functions have the maximal possible algebraic degree $n(p-1)$. Finally, we exhibit both CCZ-equivalent and CCZ-inequivalent APN pairs with boomerang uniformity one at Hamming distance one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.10766",
  "author_total": 2,
  "authors": [
    {
      "name": "Namhun Koo"
    },
    {
      "name": "Soonhak Kwon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ntp5dngf89cp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.10766",
  "fulltext_url": null,
  "id": "8ntp5dngf89cp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.10766",
  "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.CR"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.10766",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ntp5dngf89cp.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:33Z",
      "native_id": "2611.10766",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.10766"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "New Classes of CCZ-Inequivalent APN Functions with Boomerang Uniformity One in Odd Characteristic: Differential and Boomerang Spectra",
  "title_text": "New Classes of CCZ-Inequivalent APN Functions with Boomerang Uniformity One in Odd Characteristic: Differential and Boomerang Spectra",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:33Z",
  "url": "https://research.draconitic.com/paper/8ntp5dngf89cp",
  "venue": "arXiv",
  "versions": []
}