it is also called the absolute trace. In this paper, $\langle,\rangle$ denotes the usual [*inner product*]{} in a vector space over $\mathbb{F}_{2}$. For any $\alpha=(\alpha_1,\ldots,\alpha_n), \beta=(\beta_1,\ldots,\beta_n)\in\mathbb{F}_2^{n}$, one has $\langle\alpha,\beta\rangle=\sum_{i=1}^n\alpha_i\beta_i$. While in the finite field $\mathbb{F}_{2^{n}}$, we take $\langle\alpha,\beta\rangle={\rm Tr}^{n}_{1}(\alpha\beta)$ for any $\alpha,\beta\in\mathbb{F}_{2^{n}}$.

For any $(n,m)$-function $F=(f_1,\ldots,f_m)$, where $f_1,\ldots,f_m\in\mathcal{B}_{n}$, all the nonzero linear combinations of $f_i, 1\leq i\leq m$ are called the components of $F$. When $F$ is viewed as a mapping from the finite field $\mathbb{F}_{2^{n}}$ to $\mathbb{F}_{2^{m}}$, the components of $F$ can be represented as ${\rm Tr}_1^m(\lambda F(x)), ~\lambda\in\mathbb{F}_{2^{m}}^*.$

A Boolean function $f\in\mathcal{B}_{n}$ can be uniquely represented by a multivariate polynomial as $$\begin{aligned}
f(x_1,\ldots,x_n)\!=\!\sum_{I\subseteq
\{1,2,\ldots,n\}}a_I\prod_{i\in I}x_i,\end{aligned}$$ where the number of variables in the highest order term with nonzero coefficient is called the algebraic degree of $f$. While for a general $(n,m)$-function $F$, the highest algebraic degree of its coordinate functions is called the algebraic degree of $F$. The function $F$ is called [*quadratic*]{} if its algebraic degree is no more than 2.

The [*Walsh transform*]{} of a Boolean function $f\in \mathcal{B}_{n}$ at a point $\lambda \in \mathbb{F}_{2^{n}}$ is defined by $$W_{f}(\lambda)\!=\!\sum\limits_{x\in \mathbb{F}_{2^{n}}}(\!-\!1)^{f(x)\!+\!{\rm Tr}^{n}_{1}(\lambda x)}.$$ A function $f\in \mathcal{B}_{n}$ is called bent if $|W_{f}(\lambda)|=2^{\frac{n}{2}}$ for all $\lambda \in \mathbb{F}_{2^{n}}$. It 