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 $γ_F$ in $2n$ variables ($γ_F(a,b)=1$ if $a\neq{\bf 0}$ and equation $F(x)+F(x+a)=b$ has solutions) has the form $γ_F(a,b) = Φ_F(a) \cdot b + φ_F(a) + 1$ for appropriate functions $Φ_F:\mathbb{F}_2^n\to \mathbb{F}_2^n$ and $φ_F:\mathbb{F}_2^n\to \mathbb{F}_2$. We summarize the known results and prove new ones regarding properties of $Φ_F$ and $φ_F$. For instance, we prove that degree of $Φ_F$ 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 $Φ_F$ is $n-2$. We show that this conjecture is based on two other conjectures of independent interest.