On generalizing cryptographic results to Sidon sets in $\mathbb{F}_2^n$

TL;DR

This paper explores properties of Sidon sets in binary vector spaces, generalizing cryptographic results to classify minimal linearity sets, connect them to Cayley graphs, and improve bounds on Sidon set sizes.

Contribution

It generalizes cryptographic results to Sidon sets, classifies minimal linearity sets as k-covers, and constructs specific strongly regular graphs.

Findings

Classified Sidon sets with minimal linearity as k-covers.

Connected k-covers to Cayley graphs of Boolean functions.

Improved lower bounds on the size of Sidon sets in certain dimensions.

Abstract

A Sidon set $S$ in $\mathbb{F}_2^n$ is a set such that $x+y=z+w$ has no solutions $x,y,z,w \in S$ with $x,y,z,w$ all distinct. In this paper, we prove various results on Sidon sets by using or generalizing known cryptographic results. In particular, we generalize known results on the Walsh transform of almost perfect nonlinear (APN) functions to Sidon sets. One such result is that we classify Sidon sets with minimal linearity as those that are $k$-covers. That is, Sidon sets with minimal linearity are those Sidon sets $S \subseteq \mathbb{F}_2^n$ such that there exists $k > 0$ such that for any $p \in \mathbb{F}_2^n \setminus S$, there are exactly $k$ subsets $\{x,y,z\} \subseteq S$ such that $x+y+z = p$. From this, we also classify $k$-covers by means of the Cayley graph of a particular Boolean function, and we construct the unique rank $3$ strongly regular graph with parameters $(2048, 276, 44, 36)$ as the Cayley graph of a Boolean function. Finally, by computing the linearity of a particular family of Sidon sets, we increase the best-known lower bound of the largest Sidon set in $\mathbb{F}_2^{4t+1}$ by $1$ for all $t \geq 4$.