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 $F(x)=x^{2^e}h(\textrm{Tr}_{2^{2m}/2^m}(x))$, where $e\ge 0$ and h(x) is a permutation over ${\mathbb {F}}_{2^m}$. If h(x) is monomial, the nonlinearity of F(x) is shown to be at most $2^{2\,m-1}-2^{\lfloor \frac{3\,m}{2}\rfloor }$ and some non-plateaued and plateaued functions attaining the upper bound are found. 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 quadratic vectorial function that is not equivalent to the known functions of the form xh(x), and also contains vectorial functions outside the completed Maiorana-McFarland class. Finally, we show that the vectorial function $F: {\mathbb {F}}_{2^{2m}}\rightarrow {\mathbb {F}}_{2^{2m}}$, $x\mapsto x^{2^m+1}+x^{2^i+1}$ has maximal number of bent components if and only if $i=0$.