On generalizing cryptographic results to Sidon sets in $\mathbb{F}_2^n$
Darrion Thornburgh
TL;DR
This work establishes a deep link between Sidon sets in $\mathbb{F}_2^n$ and cryptographic object classifications (APN/AB) via Fourier-analytic and graph-theoretic tools. It generalizes differential/linear-attack notions to set-based counterparts $\delta_S$ and $\gamma_S$, showing that $k$-covers correspond to Sidon sets with minimal linearity, and that Cayley graphs of $\gamma_S$ are strongly regular precisely in this regime when separability holds. A central result is a classification: if $\mathrm{Cay}(\gamma_S)$ has two eigenvalues, then $n\in\{1,2,4\}$ with restricted $(n,s)$; if three or more eigenvalues occur, $S$ is a $k$-cover iff $\mathrm{Cay}(\gamma_S)$ is SRG and $S$ is separable, leading to explicit SRG parameters. The paper also constructs notable objects, including the unique rank-3 SRG with $(2048,276,44,36)$ from a $1$-cover example, and improves the best-known lower bounds on the largest Sidon set in $\mathbb{F}_2^{4t+1}$ by leveraging Kloosterman-sum bounds. These results illuminate the cryptographic significance of Sidon-structured sets and their rich combinatorial-graph-theoretic manifestations.
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 $t \geq 4$.