{
  "abstract": "An APN mapping $F:\\mathbb{F}_{2^n}\\to \\mathbb{F}_{2^n}$ is a polynomial characterized by the non-vanishing property on 2-flats. In this work, we analyze notions that are closely related to this property. To understand which $k$-flats of $\\mathbb{F}_{2^n}$ remain flats under $F$, we study the $k$-breaking. The function $x^{-1}$ has been studied in the past in this context---we extend this study to general mappings and characterize the 2-breaking of APN functions. Recently, two generalizations of the APN property have been introduced: $k$-strongly non-normality and $k$-th-order sum-freedom. Sum-freedom generalizes the non-vanishing property of APN functions to higher dimensional flats. We provide in-depth observations of the relations between the breaking property, strongly non-normality and sum-freedom. We introduce a fourth concept called $k$-strongly breaking, which implies the breaking property. We derive several structural results for both notions and give a characterization of a subclass of APN functions in terms of the 2-strongly breaking property. We propose a different perspective of the non-vanishing property via a natural character transformation, which is closely related to the sum-of-square indicator of the components of $F$. We derive a precise value for the total sum of the sum-of-square indicators of $F$. With this approach, we provide a simple answer to Open Problem 4 in IEEE Trans. Inf. Theory 52(9): 4160-4170, 2006. Moreover, it allows us to explore balancedness properties of polynomials, one of which characterizes component-wise APNness, for odd $n$, and provides a natural extension to any dimension. We show that Dillon's APN permutation and the Gold functions satisfy a related property, termed $k$-balanced, which is presented under our framework.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22394",
  "author_total": 4,
  "authors": [
    {
      "name": "René Rodríguez-Aldama"
    },
    {
      "name": "Ajla Šehović"
    },
    {
      "name": "Enes Pasalic"
    },
    {
      "name": "Sadmir Kudin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cq3n65zgj8y2d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22394",
  "fulltext_url": null,
  "id": "cq3n65zgj8y2d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22394",
  "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.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7214035210",
  "pdf_url": "https://arxiv.org/pdf/2609.22394",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cq3n65zgj8y2d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2609.22394",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22394"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On different notions related to APN mappings",
  "title_text": "On different notions related to APN mappings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/cq3n65zgj8y2d",
  "venue": "arXiv",
  "versions": []
}