论文标题
关于布尔和矢量弯曲功能的设计理论方面
On design-theoretic aspects of Boolean and vectorial bent functions
论文作者
论文摘要
$（n，m）$ - 弯曲功能的设计方法有两种构造方法，称为翻译和加法设计。 In this paper we analyze, which equivalence relation for Boolean bent functions, i.e. $(n,1)$-bent functions, and vectorial bent functions, i.e. $(n,m)$-bent functions with $2\le m\le n/2$, is coarser: extended-affine equivalence or isomorphism of associated translation and addition designs.首先，我们观察到，类似于布尔弯曲功能，矢量$（n，m）$的扩展效果 - 弯曲函数和添加设计的同构函数与所有$ n $和$ n $和$ m \ le n/2 $都是相同的概念。此外，我们表明，$ n $变量的扩展不相等的布尔弯曲功能，其翻译设计是同构的，所有$ n \ ge6 $都存在。这意味着，布尔弯曲函数的翻译设计的同构是比扩展的易期等效性更粗糙的对等关系。但是，我们在少数变量中没有观察到相同的现象。我们在六个变量中对所有矢量弯曲函数进行了分类，并表明与布尔案例相反，一个人不能从扩展的不等性vectorial $（6，m）$ -M \ in \ in \ in \ in \ {2,3 \} $中表现出同构的翻译设计。
There are two construction methods of designs from $(n,m)$-bent functions, known as translation and addition designs. In this paper we analyze, which equivalence relation for Boolean bent functions, i.e. $(n,1)$-bent functions, and vectorial bent functions, i.e. $(n,m)$-bent functions with $2\le m\le n/2$, is coarser: extended-affine equivalence or isomorphism of associated translation and addition designs. First, we observe that similar to the Boolean bent functions, extended-affine equivalence of vectorial $(n,m)$-bent functions and isomorphism of addition designs are the same concepts for all even $n$ and $m\le n/2$. Further, we show that extended-affine inequivalent Boolean bent functions in $n$ variables, whose translation designs are isomorphic, exist for all $n\ge6$. This implies, that isomorphism of translation designs for Boolean bent functions is a coarser equivalence relation than extended-affine equivalence. However, we do not observe the same phenomenon for vectorial bent functions in a small number of variables. We classify and enumerate all vectorial bent functions in six variables and show, that in contrast to the Boolean case, one cannot exhibit isomorphic translation designs from extended-affine inequivalent vectorial $(6,m)$-bent functions with $m\in\{ 2,3 \}$.
