Three Infinite Classes of APN Permutations on Zn
Abstract
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 ℤn 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, (3d−1)/2 for integers d≥2, and 2p for primes p>5 with p≡5pmod6. 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 ℤn reported since 2011 that yield infinitely many composite orders outside these standard forms.
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BibTeX
@misc{tang2026three,
title = {{Three Infinite Classes of APN Permutations on $Z_n$}},
author = {Deng Tang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05917},
eprint = {2609.05917},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05917},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Three Infinite Classes of APN Permutations on Z_n
AU - Deng Tang
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05917
UR - https://arxiv.org/abs/2609.05917
AB - 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 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≥2, and 2p for primes p>5 with p≡5pmod6. 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 ℤ_n reported since 2011 that yield infinitely many composite orders outside these standard forms.
ER -