On functions with the maximal number of bent components

Abstract: A function $$F:\mathbb{F}<em>2^n\rightarrow \mathbb{F}_2<sup>n$$, $n=2m$, can have at most $2^n-2^m$ bent component functions. Trivial examples are obtained as $$F(x) = (f_1(x),\ldots,f_m(x),a_1(x),\ldots, a_m(x))$$, where $\tilde{F}(x)=(f_1(x),\ldots,f_m(x))$ is a vectorial bent function from $\mathbb{F}_2<sup>n$ to $\mathbb{F}_2<sup>m$, and $a_i$, $1\le i\le m$, are affine Boolean functions. A class of nontrivial examples is given in univariate form with the functions $$F(x) = x^{{2^r}{\rm} Tr<sup>n_m}(\Lambda(x))$$, where $\Lambda$ is a linearized permutation of $\mathbb{F}</em>{2<sup>m}$. In the first part of this article it is shown that plateaued functions with $2^n-2^m$ bent components can have nonlinearity at most $2<sup>{n-1}-2<sup>{\lfloor\frac{n+m}{2}\rfloor}$, a bound which is attained by the example $x^{{2^r}{\rm} Tr<sup>n_m}(x)$, $1\le r&lt;m$ (Pott et al. 2018). This partially solves Question 5 in Pott et al. 2018. We then analyse the functions of the form $$x^{{2^r}{\rm} Tr<sup>n_m}(\Lambda(x))$$. We show that for odd $m$, only $x^{{2^r}{\rm} Tr<sup>n_m}(x)$, $1\le r&lt;m$, has maximal nonlinearity, whereas there are more of them for even $m$, of which we present one more infinite class explicitly. In detail, we investigate Walsh spectrum, differential spectrum and their relations for the functions $$x^{{2^r}{\rm} Tr<sup>n_m}(\Lambda(x))$$. Our results indicate that this class contains many nontrivial EA-equivalence classes of functions with the maximal number of bent components, if $m$ is even, several with maximal possible nonlinearity.