On constructions and properties of (n, m)-functions with maximal number of bent components

Abstract

For any positive integers $n=2k$ and m such that $m\ge k,$ in this paper we show that the maximal number of bent components of any (n, m)-function is equal to $2^{m}-2^{m-k},$ and for those attaining the equality, their algebraic degree is at most k. It is easily seen that all (n, m)-functions of the form $G(x)=(F(x),0),$ with F(x) being any vectorial bent (n, k)-function, have the maximal number of bent components. Those simple functions G are called trivial in this paper. We show that for a power (n, n)-function, it has the maximal number of bent components if and only if it is trivial. We also consider the (n, n)-function of the form $F(x)=xh(\mathrm{Tr}^{n}_{e}(x)),$ where $h: \mathbb {F}_{2^{e}} \rightarrow \mathbb {F}_{2^{e}},$ and show that F has the maximal number of bent components if and only if $e=k,$ and h is a permutation over $\mathbb {F}_{2^{e}}.$ It essentially shows that all previously known nontrivial functions with maximal number of bent components are subclasses of the class described by F. Based on the Maiorana–McFarland class, we present constructions of large numbers of (n, m)-functions with maximal number of bent components for any integer m in bivariate representation. We also determine the differential spectra and Walsh spectra of the constructed functions. It turns out that our constructions can also provide new plateaued vectorial functions.