A note on the properties of associated Boolean functions of quadratic APN functions

Abstract

Let $F$ be a quadratic APN function of $n$ variables. The associated Boolean function ${\gamma }_F$ in $2n$ variables (${\gamma }_F(a,b)=1$ if $a\ne 0$ and equation $F(x)+F(x+a)=b$ has solutions) has the form ${\gamma }_F(a,b)={\Phi }_F(a)\cdot b+{\phi }_F(a)+1$ for appropriate functions ${\Phi }_F:{\mathbb{F}}_2^n\to {\mathbb{F}}_2^n$ and ${\phi }_F:{\mathbb{F}}_2^n\to {\mathbb{F}}_2$. We summarize the known results and prove new ones regarding properties of ${\Phi }_F$ and ${\phi }_F$. For instance, we prove that degree of ${\Phi }_F$ is either $n$ or less or equal to $n-2$. Based on computation experiments, we formulate a conjecture that degree of any component function of ${\Phi }_F$ is $n-2$. We show that this conjecture is based on two other conjectures of independent interest.