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 \(t \geq 4\).

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.

Let \(\mathbb{F}_p^n\) be the \(n\)-dimensional vector space over \(\mathbb{F}_p\). The graph \(\mathcal{G}_F = \{ (x, F(x)) : x \in \mathbb{F}_p^n \}\) of a vectorial function \(F \colon \mathbb{F}_p^n \to \mathbb{F}_p^m\) can have interesting combinatorial properties depending on varying cryptographic conditions on \(F\). A vectorial Boolean function \(F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n\) is almost perfect nonlinear (APN) if there are at most \(2\) solutions to the equation \(F(x+a) + F(x) = b\) for all \(a,b \in \mathbb{F}_2^n\) where \(a \neq 0\). In this paper, we classify APN functions and important subclasses of APN functions in graph theoretical terms using the Kneser graph of all translations of \(\mathcal G_F\). We also study the properties of \(\mathcal G_F\) as a Sidon set. In particular, we introduce the notion of uniform exclude distributions, and we study APN functions whose graphs have uniform exclude distributions.

A cornerstone result of Erdős, Ginzburg, and Ziv (EGZ) states that any sequence of \(2n − 1\) elements in \(\mathbb{Z}/n\) contains a zero-sum subsequence of length \(n\). While algebraic techniques have predominated in deriving many deep generalizations of this theorem over the past sixty years, here we introduce topological approaches to zero-sum problems which have proven fruitful in other combinatorial contexts. Our main result (1) is a topological criterion for determining when any \(\mathbb{Z}/n\)- coloring of an n-uniform hypergraph contains a zero-sum hyperedge. In addition to applications for Kneser hypergraphs, for complete hypergraphs our methods recover Olson’s generalization of the EGZ theorem for arbitrary finite groups. Furthermore, we (2) give a fractional generalization of the EGZ theorem with applications to balanced set families and (3) provide a constrained EGZ theorem which imposes combinatorial restrictions on zero-sum sequences in the original result.

We define a cap in the affine geometry \(\mathrm{AG}(n,2)\) to be a subset in which any collection of \(4\) points is in general position. In this paper, we classify, up to affine equivalence, all caps in \(\mathrm{AG}(n,2)\) of size \(k \leq 9\). As a result, we obtain a complete characterization of caps in dimension \(n \leq 6\), in particular complete and maximal caps. Since the EvenQuads card deck is a model for \(\mathrm{AG}(6,2)\), as a consequence, we determine the probability that an arbitrary \(k\)-card layout contains a quad.