Assume $n=2m\geq 2$ and let $F(x)=x^{d_1}+x^{d_2}$ be a binomial vectorial function over $\F_{2^n}$ possessing the maximal number (i.e. $2^n-2^m$) of bent components. Suppose the $2$-adic Hamming weights $\wt_2(d_1)$ and $\wt_2(d_2)$ are both at most $2$, we prove that $F(x)$ is affine equivalent to either $x^{2^m+1}$ or $x^{2^i}(x+x^{2^m})$, provided that \[ \ell(n):=\min_{γ:~\F_2(γ)=\F_{2^n}} \dim_{\F_2}\F_2[σ]γ>m, \] where $σ$ is the Frobenius $(x\mapsto x^2)$ on $\F_{2^n}$, and $\gcd(d_1,d_2,2^m-1)>1$. Under this condition, we also establish two bounds on the nonlinearity and the differential uniformity of $F$ by means of the cardinality of its image set.

翻译：假设 $n=2m\geq 2$ 且令 $F(x)=x^{d_1}+x^{d_2}$ 为定义在 $\F_{2^n}$ 上且具有最大数量（即 $2^n-2^m$）弯曲分量的二项向量函数。假设 $2$ 进制汉明重量 $\wt_2(d_1)$ 和 $\wt_2(d_2)$ 均不超过 $2$，我们证明当 \[ \ell(n):=\min_{γ:~\F_2(γ)=\F_{2^n}} \dim_{\F_2}\F_2[σ]γ>m, \] 其中 $σ$ 为 $\F_{2^n}$ 上的 Frobenius 映射 $(x\mapsto x^2)$，且 $\gcd(d_1,d_2,2^m-1)>1$ 时，$F(x)$ 仿射等价于 $x^{2^m+1}$ 或 $x^{2^i}(x+x^{2^m})$ 中的一种。在此条件下，我们还通过 $F$ 的像集基数建立了其非线性度和差分均匀性的两个界。