Uniform exclude distributions of Sidon sets

TL;DR

This paper constructs Sidon sets in binary vector spaces using APN plateaued functions, achieving uniform exclude distributions on natural partitions, and characterizes their exclude multiplicity values for Gold and Kasami functions.

Contribution

It introduces a method to create Sidon sets with uniform exclude distributions using APN plateaued functions and characterizes their exclude multiplicity distributions for specific functions.

Findings

Constructed Sidon sets with uniform exclude distributions

Determined exclude multiplicity values for Gold and Kasami functions

Established connections between APN functions and Sidon set properties

Abstract

A Sidon set $S$ in $\mathbb{F}_2^n$ is a set such that the pairwise sums of distinct points are all distinct. The exclude points of a Sidon set $S$ are the sums of three distinct points in $S$, and the exclude multiplicity of a point in $\mathbb{F}_2^n \setminus S$ is the number of such triples in $S$ it is equal to. We call the function $d_S \colon \mathbb{F}_2^n \setminus S \to \mathbb{Z}_{\geq 0}$ taking points in $\mathbb{F}_2^n \setminus S$ to their exclude multiplicity the exclude distribution of $S$. We say that $d_S$ is uniform on $\mathcal{P}$ if $\mathcal{P}$ is an equally-sized partition $\mathcal{P}$ of $\mathbb{F}_2^n \setminus S$ such that $d_S$ takes the same values an equal number of times on every element of $\mathcal{P}$. In this paper, we use APN plateaued functions with all component functions unbalanced to construct Sidon sets $S$ in $(\mathbb{F}_2^n)^2$ whose exclude distributions are uniform on natural partitions of $(\mathbb{F}_2^n)^2 \setminus S$ into $2^n$ elements. We use this result and a result of Carlet to determine exactly what values the exclude distributions of the graphs of the Gold and Kasami functions take and how often they take these values.