论文标题
关于二次APN功能相关布尔函数的属性的注释
A note on the properties of associated Boolean functions of quadratic APN functions
论文作者
论文摘要
令$ f $为$ n $变量的二次APN函数。相关的布尔函数$γ_f$ in $ 2n $变量（$γ_f（a，b）= 1 $如果$ a \ neq {\ bf 0} $和方程$ f（x） + f（x） + f（x + a）= b $ solutions Solutions solutions solutions solutions solutions solutions solutions solutions solutions $γ_f（a，b）= a，b）= $ + cd（a） $φ_F：\ mathbb {f} _2^n \ to \ mathbb {f} _2^n $和$φ_f：\ mathbb {f} _2^n \ to \ mathbb {f} _2 $。我们总结了已知结果，并证明了有关$φ_F$和$φ_F$的属性的新结果。例如，我们证明$φ_F$的度量是$ n $或更小或等于$ n-2 $。根据计算实验，我们制定了一个猜想，即$φ_F$的任何组件函数的度量均为$ n-2 $。我们表明，这种猜想是基于另外两个独立利益的猜想。
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.
