On quadratic binomial vectorial functions with maximal bent components

Abstract

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 \[ \ell(n):=\min_{\gamma:~\F_2(\gamma)=\F_{2^n}} \dim_{\F_2}\F_2[\sigma]\gamma >m, \] where $\sigma$ 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.