$e=k$, and $h$ is a permutation over $\mathbb{F}_{2^{e}}$. Hence the functions $F^{i}(x)=x^{2^{i}}h({\rm Tr}^{n}_{k}(x))$ yield new vectorial bent functions ${\rm Tr}^{n}_{k}(\alpha F^{i}(x))$, where $h$ is a permutation over $\mathbb{F}_{2^{k}}$.

One will see the functions $F(x)=x^{2^{i}}(x+x^{2^{k}})$ given by [@Pott] and $H(x)=x^{2^{i}}(x+x^{2^{k}}+\sum\limits^{\rho}_{j=1}\gamma^{(j)}(x^{2^{t_{j}}}+x^{2^{t_{j}+k}}))$ given by [@MZTZ] are subclasses of our functions $F^{i}(x)$, which means that all the known nontrivial $(n, n)$-functions with maximum number of bent components are covered by our results, to the best of our knowledge.

The rest of the paper is organized as follows. Some basic definitions are given in Section 2. In Section 3, we prove that the maximum number of bent components of $(n=2k,m)$-functions is equal to $2^m-2^{m-k}$, and when the equality holds, the algebraic degree of such functions is at most $k$. We give a characterization of the power $(n,n)$-functions admitting such large number of bent components. In Section 4, we put forward a construction of $(n,m)$-functions in bivariate representation with maximum number of bent components, and determine the [*differential spectrum*]{} and [*Walsh spectrum*]{}. We also show that the $(n,n)$-functions $F^{i}(x)=x^{2^{i}}h({\rm Tr}^{n}_{e}(x))$ has the maximal number of bent components if and only if $e=k$, and $h$ is a permutation over $\mathbb{F}_{2^{e}}$, where $n=2k$, and $e$ is a positive 