On quadratic binomial vectorial functions with maximal bent components

Xianhong Xie; Yi Ouyang; Shenxing Zhang

paper

On quadratic binomial vectorial functions with maximal bent components

arXiv:2604.08311

Abstract

Assume $n = 2 m \geq 2$ 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 \[ \ell(n):=\min_{Î³:~\F_2(Î³)=\F_{2^n}} \dim_{\F_2}\F_2[Ï]Î³>m, \] where $I ¨ $ is the Frobenius $(x \mapsto x^{2})$ on $\F_{2^n}$, and $g cd (d_{1}, d_{2}, 2^{m} - 1) > 1$ . 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.