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Three Infinite Classes of APN Permutations on $Z_n$ — arXiv:2609.05917

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.05917
License (as archived): CC0-1.0
Archived: 2026-10-16T14:48:29.223Z
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Three Infinite Classes of APN Permutations on $Z_n$ — arXiv:2609.05917 — arXiv (en), https://arxiv.org/abs/2609.05917 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_84da274d2ffa (Metraipolis archive dc_84da274d2ffa, sha256 9669d503b1e418c581c46b4d3c1d76dfe9e069b3c5291e197a8a7b3febc0475a)

BibTeX

@misc{dc84da274d2ffa,
  title = {Three Infinite Classes of APN Permutations on $Z_n$ — arXiv:2609.05917},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.05917},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_84da274d2ffa; sha256 9669d503b1e418c581c46b4d3c1d76dfe9e069b3c5291e197a8a7b3febc0475a},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.05917. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

For any permutation of a nontrivial finite abelian group, the differential uniformity is at least two; permutations attaining this bound are called almost perfect nonlinear (APN). We construct three infinite classes of APN permutations on the cyclic group $\mathbb{Z}_n$ using Singer cycles, binomials inducing projective permutations, and completed reciprocals combined with parity and quadratic characters. The respective domain orders are $q+1$ for prime powers $q>2$, $(3^d-1)/2$ for integers $d\ge2$, and $2p$ for primes $p>5$ with $p\equiv5\pmod6$. Each class contains an infinite subclass of composite orders outside the standard forms $r-1$, $r-2$, $r-3$, and $r-4$, where $r$ is a prime power. These forms arise in the Welch--Costas, Panario--Sakzad--Stevens--Wang, and Golomb constructions. To the best of our knowledge, these are the first infinite APN constructions on $\mathbb{Z}_n$ reported since 2011 that yield infinitely many composite orders outside these standard forms.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-09 in math.CO).
Categorie arXiv: Combinatorics (math.CO) ; Information Theory (cs.IT).
Pagina del lavoro: https://arxiv.org/abs/2609.05917 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T14:32:29.767Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-09?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).