On vectorial functions with maximal number of bent components

Abstract: We study vectorial functions with maximal number of bent components in this paper. We first study the Walsh transform and nonlinearity of $F(x)=x<sup>{2<sup>e}h(\Tr_{2<sup>{2m}/2<sup>m}(x))$, where $e\geq0$ and $h(x)$ is a permutation over $\F_{2<sup>m}$. If $h(x)$ is monomial, the nonlinearity of $F(x)$ is shown to be at most $2<sup>{2m-1}-2<sup>{\lfloor\frac{3m}{2}\rfloor}$ and some non-plateaued and plateaued functions attaining the upper bound are found. This gives a partial answer to the open problems proposed by Pott et al. and Anbar et al. If $h(x)$ is linear, the exact nonlinearity of $F(x)$ is determined. Secondly, we give a construction of vectorial functions with maximal number of bent components from known ones, thus obtain two new classes from the Niho class and the Maiorana-McFarland class. Our construction gives a partial answer to an open problem proposed by Pott et al., and also contains vectorial functions outside the complete Maiorana-McFarland class. Finally, we show that the vectorial function $F: \F_{2^{{2m}}\rightarrow} \F_{2<sup>{2m}}$, $x\mapsto x<sup>{2<sup>m+1}+x<sup>{2<sup>i+1}$ has maximal number of bent components if and only if $i=0$.