Date of Submission

Spring 2024

Academic Program

Computer Science; Mathematics

Project Advisor 1

Bob McGrail

Project Advisor 2

Steven Simon

Abstract/Artist's Statement

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 $a \neq 0$. Equivalently, $F$ is APN if and only if $\mathcal{G}_F$ is a Sidon set, that is, a set in $\mathbb{F}_2^n$ where no four distinct points sum to zero. 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.

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