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

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A Pub Date : 2024-12-19 DOI:10.1016/j.jcta.2024.106001

Gábor P. Nagy

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引用次数: 0

Abstract

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.

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西顿集，瘦集，和向量布尔函数的非线性

向量值函数的向量非线性是它到仿射函数集合的距离。2017年，Liu、Mesnager和Chen推测了向量线性的一般上界。最近，Carlet建立了微分均匀性的下界。本文改进了Carlet下界。我们的方法是基于这样一个事实，即向量布尔函数的水平集是瘦集。特别是，APN函数的水平集是Sidon集，因此，Liu-Mesnager-Chen猜想预测，在F2n中，对于所有n，应该存在大小至少为2n/2+1的Sidon集。本文概述了F2n中已知的大Sidon集，并检验了由有限仿射平面的双曲线和椭圆导出的大Sidon集的完备性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Combinatorial Theory Series A 数学-数学

CiteScore

2.90

自引率

9.10%

发文量

审稿时长

12 months

期刊介绍： The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.