functions which covers all the previously known nontrivial ones.

[**A. A nontrivial construction** ]{}

The following theorem gives a large class of nontrivial $(n,m)$-functions with maximal number of bent components.

\[keythm-1\] Let $m, k$ be two positive integers such that $m \geq k$. Let $G(x,y): \mathbb{F}_{2^{k}}\times \mathbb{F}_{2^{k}}\rightarrow\mathbb{F}_{2^{k}}\times \mathbb{F}_{2^{m-k}}$ be defined by $(x,y)\mapsto (x^{2^i}\pi(y)+g(y), h(y))$, where $\pi(y)$ is a permutation over $\mathbb{F}_{2^{k}}$, and $g(y)$ and $h(y)$ are arbitrary functions from $\mathbb{F}_{2^{k}}$ to $\mathbb{F}_{2^{k}}$ and to $\mathbb{F}_{2^{m-k}}$, respectively. Then for any $(0,0)\neq (u,v) \in  \mathbb{F}_{2^{k}}\times \mathbb{F}_{2^{m-k}}$, the function $\langle (u,v),G(x,y)\rangle$ is a bent component of $G$ if and only if $u\neq 0$. In particular, the function $G$ admits the maximal number $2^{m}-2^{m-k}$ of bent components.

For any $(0,0)\neq (u,v) \in  \mathbb{F}_{2^{k}}\times \mathbb{F}_{2^{m-k}}$, we have $\langle (u,v), G(x,y)\rangle=\langle u,
x^{2^{i}}\pi(y)+g(y)\rangle+\langle v,h(y)\rangle={\rm Tr}^{k}_{1}(u x^{2^{i}}\pi(y))+{\rm Tr}^{k}_{1}(u g(y))+{\rm Tr}^{m-k}_{1}(vh(y)):=f(x,y)$. If $u\neq 0$, then $f(x,y)$ is in the class $\mathcal{M}$ of bent functions. Hence $G$ has at least $(2^k-1)2^{m-k}=2^m-2^{m-k}$ bent components. While according to Theorem \[number\], the number of bent components is at most $2^m-2^{m-k}$. Hence $f$ is bent if and only if $u\neq 0$, and $G$ has the maximal number of bent components.

Below we consider the differential spectrum and Walsh spectrum of the functions given 