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PreprintCC0arXiv

The APN Property of Two Families of functions over F2n from the Switching Construction

  • Chenmiao Shi
  • Longjiang Qu
  • Kangquan Li
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Abstract

Let m≥5 be odd with 3nmid m. Helleseth, Ichanska, and Kaleyski \cite{HIK} conjectured that two norm-perturbed Budaghyan--Helleseth--Kaleyski functions F2,F3 over F22m are APN. In this paper, we prove this conjecture. First, by using the basis (1,β) of F22m over F2m, we show that F2 and F3 are EA-equivalent to two functions 𝒢1 and 𝒢0, respectively. Their derivatives can be written as two trace equations for a matrix P∈GL(2,F2m). We then prove that these two equations have no solution. The proof uses only identities over finite fields, the characteristic polynomial of a 2×2 matrix, and elementary properties of Frobenius automorphisms. The repeated-root case and all parity cases are considered separately. It follows that the derivative of each function has kernel of size two in every nonzero direction, and hence both F2 and F3 are APN for every admissible m.

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Cite

BibTeXRIS
BibTeX
@misc{shi2026property,
  title = {{The APN Property of Two Families of functions over $\F_2^n$ from the Switching Construction}},
  author = {Chenmiao Shi and Longjiang Qu and Kangquan Li},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05014},
  eprint = {2611.05014},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05014},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - The APN Property of Two Families of functions over F_2^n from the Switching Construction
AU  - Chenmiao Shi
AU  - Longjiang Qu
AU  - Kangquan Li
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.05014
UR  - https://arxiv.org/abs/2611.05014
AB  - Let m≥5 be odd with 3nmid m. Helleseth, Ichanska, and Kaleyski \cite{HIK} conjectured that two norm-perturbed Budaghyan--Helleseth--Kaleyski functions F_2,F_3 over F_2^2m are APN. In this paper, we prove this conjecture. First, by using the basis (1,β) of F_2^2m over F_2^m, we show that F_2 and F_3 are EA-equivalent to two functions 𝒢_1 and 𝒢_0, respectively. Their derivatives can be written as two trace equations for a matrix P∈GL(2,F_2^m). We then prove that these two equations have no solution. The proof uses only identities over finite fields, the characteristic polynomial of a 2×2 matrix, and elementary properties of Frobenius automorphisms. The repeated-root case and all parity cases are considered separately. It follows that the derivative of each function has kernel of size two in every nonzero direction, and hence both F_2 and F_3 are APN for every admissible m.
ER  -