{
  "abstract": "We study Carlet's cyclic-additive conjecture for the Kasami almost perfect nonlinear (APN) function $F(x)=x^{4^k-2^k+1}$ on $GF(2^n)$, $\\gcd(k,n)=1$: for the $2^{n-1}$-element set $Δ=\\{F(b)+F(b+1)+1: b\\in GF(2^n)\\}$ and all distinct nonzero $v_1,v_2\\in GF(2^n)$, \\[ \\bigl|\\{(x,y,z)\\inΔ^3 : v_1x+v_2y+(v_1+v_2)z=0\\}\\bigr| \\;=\\; 2^{2n-3}. \\] This exact triple-count condition was first formulated by Carlet in his 2018 cyclic-additive difference-set framework; the Kasami instance was subsequently posed as an open problem at the NSUCRYPTO 2019 cryptographic olympiad, whose individual proposer was not publicly disclosed. We prove the conjecture for $k\\bmod n\\in\\{1,2,n-2,n-1\\}$, in particular a complete proof for $k=2$ ($d=13$) via a quadratic-form theory and an exact root-count reduction, and we verify it exhaustively by computer for every admissible $(n,k)$ with $n\\le13$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.18584",
  "author_total": 2,
  "authors": [
    {
      "name": "Gábor P. Nagy"
    },
    {
      "name": "Attila Vajda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f121b0v42ww2p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.18584",
  "fulltext_url": null,
  "id": "f121b0v42ww2p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.18584",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7203928703",
  "pdf_url": "https://arxiv.org/pdf/2608.18584",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f121b0v42ww2p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2608.18584",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.18584"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On a conjecture on the Kasami APN function: reductions, structure theorems, a proof for $k\\bmod n\\in\\{1,2,n{-}2,n{-}1\\}$, and exhaustive verification for $n\\le 13$",
  "title_text": "On a conjecture on the Kasami APN function: reductions, structure theorems, a proof for kbmod n∈{1,2,n−2,n−1}, and exhaustive verification for n≤ 13",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/f121b0v42ww2p",
  "venue": "arXiv",
  "versions": []
}