We study vectorial functions with maximal number of bent components in this paper. We first study the Walsh transform and nonlinearity of $F(x)=x^{2^e}h(\Tr_{2^{2m}/2^m}(x))$, where $e\geq0$ and $h(x)$ is a permutation over $\F_{2^m}$. If $h(x)$ is monomial, the nonlinearity of $F(x)$ is shown to be at most $ 2^{2m-1}-2^{\lfloor\frac{3m}{2}\rfloor}$ and some non-plateaued and plateaued functions attaining the upper bound are found. This gives a partial answer to the open problems proposed by Pott et al. and Anbar et al. If $h(x)$ is linear, the exact nonlinearity of $F(x)$ is determined. Secondly, we give a construction of vectorial functions with maximal number of bent components from known ones, thus obtain two new classes from the Niho class and the Maiorana-McFarland class. Our construction gives a partial answer to an open problem proposed by Pott et al., and also contains vectorial functions outside the complete Maiorana-McFarland class. Finally, we show that the vectorial function $F: \F_{2^{2m}}\rightarrow \F_{2^{2m}}$, $x\mapsto x^{2^m+1}+x^{2^i+1}$ has maximal number of bent components if and only if $i=0$.

翻译：本文研究具有最大数量弯曲分量的向量函数。我们首先研究函数 $F(x)=x^{2^e}h(\Tr_{2^{2m}/2^m}(x))$ 的沃尔什变换和非线性度，其中 $e\geq0$，$h(x)$ 是 $\F_{2^m}$ 上的置换。若 $h(x)$ 为单项式，则 $F(x)$ 的非线性度至多为 $2^{2m-1}-2^{\lfloor\frac{3m}{2}\rfloor}$，并发现了一些达到该上界的非平坦函数和平坦函数。这为Pott等人及Anbar等人提出的公开问题提供了部分解答。若 $h(x)$ 为线性函数，则确定了 $F(x)$ 的精确非线性度。其次，我们通过已知函数构造了具有最大数量弯曲分量的向量函数，从而从Niho类和Maiorana-McFarland类中得到了两个新类。该构造部分解答了Pott等人提出的公开问题，且所构造的向量函数包含完全Maiorana-McFarland类之外的函数。最后，我们证明向量函数 $F: \F_{2^{2m}}\rightarrow \F_{2^{2m}}$，$x\mapsto x^{2^m+1}+x^{2^i+1}$ 具有最大数量弯曲分量当且仅当 $i=0$。