Sidon sets, thin sets, and the nonlinearity of vectorial Boolean functions

The vectorial nonlinearity of a vector-valued function is its distance from the set of affine functions. In 2017, Liu, Mesnager, and Chen conjectured a general upper bound for the vectorial linearity. Recently, Carlet established a lower bound in terms of differential uniformity. In this paper, we i...

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Bibliographic Details
Published in:Journal of combinatorial theory. Series A Vol. 212; p. 106001
Main Author: Nagy, Gábor P.
Format: Journal Article
Language:English
Published: Elsevier Inc 01.05.2025
Subjects:
APN functions
Error correcting codes
Hamming distance
Sidon sets
Vectorial Boolean functions
APN functions
Hamming distance
Sidon sets
Vectorial Boolean functions
Error correcting codes
ISSN:0097-3165
Online Access:Get full text
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Summary:The vectorial nonlinearity of a vector-valued function is its distance from the set of affine functions. In 2017, Liu, Mesnager, and Chen conjectured a general upper bound for the vectorial linearity. Recently, Carlet established a lower bound in terms of differential uniformity. In this paper, we improve Carlet's lower bound. Our approach is based on the fact that the level sets of a vectorial Boolean function are thin sets. In particular, level sets of APN functions are Sidon sets, hence the Liu-Mesnager-Chen conjecture predicts that in F2n, there should be Sidon sets of size at least 2n/2+1 for all n. This paper provides an overview of the known large Sidon sets in F2n, and examines the completeness of the large Sidon sets derived from hyperbolas and ellipses of the finite affine plane.
ISSN:0097-3165
DOI:10.1016/j.jcta.2024.106001