On different notions related to APN mappings
Abstract
An APN mapping F:𝔽2n→ 𝔽2n 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 𝔽2n 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.
Related
- On the stability of proximal operators in Wasserstein spaces under different notions of convexity
- Separating Notions of Graph Width: the Adaptive, Normal, Linear, Entropic, and Submodular Width
- Rigidity and weak notions of spectral scalar curvature
- Notions of equivalence among constant Yang--Baxter Operators: To ∞-Equivalence and Beyond
- Some New Notions of Mathematical Integral Inequalities: Theory and Applications
BibTeX
@misc{rodriguezaldama2026different,
title = {{On different notions related to APN mappings}},
author = {René Rodríguez-Aldama and Ajla Šehović and Enes Pasalic and Sadmir Kudin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.22394},
eprint = {2609.22394},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22394},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On different notions related to APN mappings
AU - René Rodríguez-Aldama
AU - Ajla Šehović
AU - Enes Pasalic
AU - Sadmir Kudin
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2609.22394
UR - https://arxiv.org/abs/2609.22394
AB - An APN mapping F:𝔽_2^n→ 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 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.
ER -