Pott et al. (2018) showed that $\mathcal {F}(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(x)$ , $n = 2m$ , $r\ge 1$ , is a nontrivial example of a vectorial function with the maximal possible number $2^{n}-2^{m}$ of bent components. Mesnager et al. (2019) generalized this result by showing conditions on $\Lambda (x) = x + \sum _{j=1}^\sigma \alpha _{j}x^{2^{t_{j}}}$ , $\alpha _{j}\in {\mathbb F} _{2^{m}}$ , under which $\mathcal {F}(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x))$ has the maximal possible number of bent components. We simplify these conditions and further analyse this class of functions. For all related vectorial bent functions $F(x) = {\rm Tr^{n}_{m}}(\gamma \mathcal {F}(x))$ , $\gamma \in {\mathbb F}_{2^{n}}\setminus {\mathbb F} _{2^{m}}$ , which as we will point out belong to the Maiorana-McFarland class, we describe the collection of the solution spaces for the linear equations $\mathcal {D}_{a}F(x) = F(x) + F(x+a) + F(a) = 0$ , which forms a spread of ${\mathbb F}_{2^{n}}$ . Analysing these spreads, we can infer neat conditions for functions $H(x) = (F(x),G(x))$ from ${\mathbb F}_{2^{n}}$ to ${\mathbb F}_{2^{m}}\times {\mathbb F} _{2^{m}}$ to exhibit small differential uniformity (for instance for $\Lambda (x) = x$ and $r=0$ this fact is used in the construction of Carlet’s, Pott-Zhou’s, Taniguchi’s APN-function). For some classes of $H(x)$ we determine differential uniformity and with a method based on Bezout’s theorem nonlinearity.

中文翻译：

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分析 （n, n)-从 Maiorana-McFarland 类获得的函数

波特等人。(2018) 表明 $\mathcal {F}(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(x)$ , $n = 200 万美元 , $r\ge 1$ , 是具有最大可能数的向量函数的重要示例 $2^{n}-2^{m}$ 弯曲的组件。消息人士等。(2019) 通过显示条件来概括这个结果 $\Lambda (x) = x + \sum _{j=1}^\sigma \alpha _{j}x^{2^{t_{j}}}$ , $\alpha _{j}\in {\mathbb F} _{2^{m}}$ ，其中 $\mathcal {F}(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x))$ 具有最大可能数量的弯曲组件。我们简化这些条件并进一步分析此类函数。对于所有相关的矢量弯曲函数 $F(x) = {\rm Tr^{n}_{m}}(\gamma \mathcal {F}(x))$ , $\gamma \in {\mathbb F}_{2^{n}}\setminus {\mathbb F} _{2^{m}}$ ，正如我们将指出的那样属于 Maiorana-McFarland 类，我们描述了线性方程的解空间的集合 $\mathcal {D}_{a}F(x) = F(x) + F(x+a) + F(a) = 0$ ，这形成了一个传播 ${\mathbb F}_{2^{n}}$ . 分析这些价差，我们可以推断出函数的简洁条件 $H(x) = (F(x),G(x))$ 从 ${\mathbb F}_{2^{n}}$ 至 ${\mathbb F}_{2^{m}}\times {\mathbb F} _{2^{m}}$ 表现出小的差异均匀性（例如 $\Lambda (x) = x$ 和 $r=0$ 这个事实被用于 Carlet 的、Pott-Zhou 的、Taniguchi 的 APN 函数的构造）。对于某些类 $H(x)$ 我们确定微分均匀性，并使用基于 Bezout 定理非线性的方法。