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

TL;DR

This paper investigates the properties of Boolean functions associated with quadratic APN functions, providing new theoretical insights and conjectures about their structure and degree, supported by computational experiments.

Contribution

It introduces new theoretical results on the degrees of functions related to quadratic APN functions and formulates conjectures based on computational evidence.

Findings

Degree of $oldsymbol{ ext{ extit{ extbf{Phi}}}_F}$ is either $n$ or at most $n-2$

Conjecture that all component functions of $oldsymbol{ ext{ extit{ extbf{Phi}}}_F}$ have degree $n-2$

The conjecture is supported by two other independent conjectures.

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 $a\neq{\bf 0}$ and equation $F(x)+F(x+a)=b$ has solutions) has the form $\gamma_F(a,b) = \Phi_F(a) \cdot b + \varphi_F(a) + 1$ for appropriate functions $\Phi_F:\mathbb{F}_2^n\to \mathbb{F}_2^n$ and $\varphi_F:\mathbb{F}_2^n\to \mathbb{F}_2$. We summarize the known results and prove new ones regarding properties of $\Phi_F$ and $\varphi_F$. 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.