{
  "abstract": "We say an $(n,n)$-function $F \\colon \\mathbb{F}_2^n \\to \\mathbb{F}_2^n$ is a crooked function if for any nonzero $a \\in \\mathbb{F}_2^n$, the image of $D_aF(x)=F(x)+F(x+a)$ is an affine hyperplane. The only known examples of crooked functions are all quadratic almost perfect nonlinear (APN), or equivalently, for every known crooked function, $D_aF$ is affine for all $a \\in \\mathbb{F}_2^n$. The ortho-derivative $π_F \\colon\\mathbb{F}_2^n \\to \\mathbb{F}_2^n$ of a crooked function $F$ is the function such that $π_F(0)=0$, and for any nonzero $a$, the set $\\{0,π_F(a)\\}^\\perp$ is the underlying vector space of $\\mathrm{Im}(D_aF)$. We prove that for $n \\geq 4$ and a crooked function $F$, if $k$ is a non-negative integer such that $F$ has $2^k-1$ quadratic component functions, $π_F$ has at least $2^n-2^{n-k}$ component functions of algebraic degree $n-2$. In particular, we resolve Gorodilova's conjecture that every component function of $π_F$ has algebraic degree $n-2$ when $F$ is quadratic APN. As a second main result, for $n \\geq 4$, we associate to a crooked function $F$ a quadratic function $\\varepsilon_F \\colon \\mathbb{F}_2^n \\to \\mathbb{F}_2^n$ that satisfies a strong geometric-combinatorial condition regarding the sums of $F$ over $2$-dimensional linear subspaces. As a corollary to both of our main results, we prove that for any even $n \\geq 4$, any quadratic APN $(n,n)$-function has at least $n$ semi-bent components. Furthermore, we obtain a congruence result on a problem on $m$-sequences introduced by Johansen, Helleseth, and Kholosha, and we determine the exact algebraic degrees of some Boolean functions associated to the bent and near-bent components of particular classes of plateaued vectorial functions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.23888",
  "author_total": 2,
  "authors": [
    {
      "name": "Claude Carlet"
    },
    {
      "name": "Darrion Thornburgh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7561g919bt10h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.23888",
  "fulltext_url": null,
  "id": "7561g919bt10h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.23888",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7204260943",
  "pdf_url": "https://arxiv.org/pdf/2608.23888",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7561g919bt10h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2608.23888",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.23888"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Resolving a conjecture on quadratic APN functions and a new quadratic $(n,n)$-function associated to crooked functions",
  "title_text": "Resolving a conjecture on quadratic APN functions and a new quadratic (n,n)-function associated to crooked functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/7561g919bt10h",
  "venue": "arXiv",
  "versions": []
}