A function $F: \mathbb {F}_{2}^{n}\rightarrow \mathbb {F} _{2}^{n}$ , $n=2m$ , can have at most $2^{n}-2^{m}$ bent component functions. Trivial examples are vectorial bent functions from $\mathbb {F}_{2}^{n}$ to $\mathbb {F}_{2}^{m}$ , seen as functions on $\mathbb {F}_{2}^{n}$ . The first nontrivial example is given in univariate form as $x^{2^{r}} {\rm Tr^{n}_{m}}(x), 1\le r < m$ (Pott et al. 2018), a few more examples of similar shape are given by Mesnager et al. 2019, and finally it has been shown that the quadratic function $F(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x))$ , has $2^{n}-2^{m}$ bent components if and only if $\Lambda $ is a linearized permutation polynomial of $\mathbb {F}_{2^{m}}[x]$ (Anbar et al. 2021). In the first part of this article, an upper bound for the nonlinearity of plateaued functions with $2^{n}-2^{m}$ bent components is shown, which is attained by the example $x^{2^{r}} {\rm Tr^{n}_{m}}(x)$ . We then analyse in detail nonlinearity and differential spectrum of the class of functions $F(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x))$ , which, as will be seen, requires the study of the functions $x^{2^{r}}\Lambda (x)$ . In the last part we demonstrate that this class belongs to a larger class of functions with $2^{n}-2^{m}$ Maiorana-McFarland bent components, which also contains nonquadratic and non-plateaued functions.

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关于具有最大弯曲分量的一类函数

一个函数 $F: \mathbb {F}_{2}^{n}\rightarrow \mathbb {F} _{2}^{n}$ , $n=2m$ , 最多可以有 $2^{n}-2^{m}$弯曲组件功能。简单的例子是来自的向量弯曲函数 $\mathbb {F}_{2}^{n}$至 $\mathbb {F}_{2}^{m}$ , 被视为函数 $\mathbb {F}_{2}^{n}$ . 第一个非平凡示例以单变量形式给出 $x^{2^{r}} {\rm Tr^{n}_{m}}(x), 1\le r < m$(Pott et al. 2018)，Mesnager et al. 给出了更多类似形状的例子。2019 年，终于证明了二次函数 $F(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x))$ ， 有 $2^{n}-2^{m}$当且仅当弯曲组件 $\拉姆达$是一个线性化的置换多项式 $\mathbb {F}_{2^{m}}[x]$（安巴尔等人，2021）。在本文的第一部分，平台函数非线性的上限 $2^{n}-2^{m}$显示了弯曲的组件，这是通过示例获得的 $x^{2^{r}} {\rm Tr^{n}_{m}}(x)$ . 然后我们详细分析函数类的非线性和微分谱 $F(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x))$ ，正如将要看到的，这需要研究函数 $x^{2^{r}}\Lambda (x)$ . 在最后一部分中，我们证明了这个类属于一个更大的函数类 $2^{n}-2^{m}$Maiorana-McFarland 弯曲分量，它还包含非二次函数和非平台函数。