by $\delta_{F}(a,b)$ the number of solutions $x\in\mathbb{F}^{n}_{2}$ such that $F(x+a)+F(x)=b$, for any $a\in\mathbb{F}^{n}_{2}$ and $b\in\mathbb{F}^{m}_{2}$. The multiset $$\{\ast ~\delta_{F}(a,b):a\in \mathbb{F}^{n}_{2}\backslash \{0\}, b\in\mathbb{F}^{m}_{2}~ \ast \},$$ is called the [*differential spectrum*]{} of $F$, and $F$ is said to be differentially $\delta$-uniform, where $\delta=\max_{a\in \mathbb{F}^{n}_{2}\backslash \{0\}, b\in\mathbb{F}^{m}_{2}}\delta_{F}(a,b)$. In particular, $F$ is called [*almost perfect nonlinear*]{} (${\rm APN}$ for short) if $\delta=2$. For cryptographic applications, it is usually required that the differential uniformity of $F$ should be as low as possible (see for instance [@Pt] and the references therein).

Two $(n,m)$-functions $F$ and $G$ are called [*extended affine equivalent*]{} (EA-equivalent) if there exist some affine permutation $L_1$ over $\mathbb{F}_{2^n}$ and some affine permutation $L_2$ over $\mathbb{F}_{2^m}$, and some affine function $A$ such that $F=L_2\circ G\circ L_1+A$. They are called [*Carlet-Charpin-Zinoviev equivalent*]{} (CCZ-equivalent) if there exists some affine automorphism $L=(L_1, L_2)$ of $\mathbb{F}_{2^n}\times \mathbb{F}_{2^m}$, where $L_1: \mathbb{F}_{2^n}\times \mathbb{F}_{2^m}\rightarrow \mathbb{F}_{2^n}$ and $L_2: \mathbb{F}_{2^n}\times \mathbb{F}_{2^m}\rightarrow \mathbb{F}_{2^m}$ are affine functions, such that $y=G(x)$ if and only if $L_2(x, y)=F\circ L_1(x, y)$. It is well known that EA-equivalence is a special kind of CCZ-equivalence, and that CCZ-equivalence preserves the extended Walsh spectrum and the differential spectrum (but not for algebraic degree) [@CCZ].

$(n,m)$-functions with Maximal Number of Bent Components
