integer dividing $k$, $h$ is an arbitrary map from $\mathbb{F}_{2^{e}}$ to $\mathbb{F}_{2^{e}}$. This yields new vectorial bent function ${\rm Tr}^{n}_{k}(\alpha x^{2^{i}}h({\rm Tr}^{n}_{k}(x)))$, where $h$ is a permutation over $\mathbb{F}_{2^{k}}$, $\alpha$ is a primitive element in $\mathbb{F}_{2^{n}}$. Moreover, we compare our new construction with previous ones, and prove that all the known nontrivial results are covered by ours. A few concluding remarks are given in Section 5.

Preliminaries
=============

Let $\mathbb{F}_{2^{n}}$ be the finite field consisting of $2^{n}$ elements, then the group of units of $\mathbb{F}_{2^{n}}$, denoted by $\mathbb{F}^{\ast}_{2^{n}}$, is a cyclic group of order $2^{n}-1$. Throughout this paper we always identify $\mathbb{F}_{2^{n}}$ with the vector space $\mathbb{F}^{n}_{2}$ over $\mathbb{F}_{2}$. Any function $F: \mathbb{F}_{2^{n}}\rightarrow\mathbb{F}_{2^{m}}$ is called an $(n,m)$-function ([*cryptographic function*]{}, [*vectorial Boolean function*]{} or [*substitution box*]{}). Usually $(n,1)$-functions are called Boolean functions in $n$ variables, the set of which is denoted by $\mathcal{B}_{n}$. As cryptographic functions, the main cryptographic criteria for Boolean functions are algebraic degree, [*nonlinearity*]{} and [*algebraic immunity*]{}, etc. (see [@Carletb1]), and the main cryptographic criteria for vectorial Boolean functions are algebraic degree, nonlinearity and [*differential uniformity*]{}, etc. (see [@Carletb2]).

The trace function ${\rm Tr}^{n}_{m}: \mathbb{F}_{2^{n}}\rightarrow \mathbb{F}_{2^{m}}$, where $m|n$, is defined as $${\rm Tr}^{n}_{m}(x)\!=\!x\!+\!x^{2^{m}}\!+\!x^{2^{2m}}\!+\!\cdots\!+\!x^{2^{(n/m\!-\!1)m}}, ~\forall ~x\in \mathbb{F}_{2^{n}}.$$ When $m=1$, 