On functions with the maximal number of bent components

TL;DR

This paper investigates functions with the maximum number of bent components, establishing bounds on their nonlinearity, analyzing specific classes of such functions, and identifying conditions under which they achieve maximal nonlinearity.

Contribution

It provides new bounds on nonlinearity for plateaued functions with maximal bent components and characterizes functions achieving these bounds, especially for even m.

Findings

Bound on nonlinearity for plateaued functions with maximal bent components.

Only specific functions attain maximal nonlinearity when m is odd.

Multiple nontrivial EA-equivalence classes with maximal bent components exist for even m.

Abstract

A function $F:\mathbb{F}_2^n\rightarrow \mathbb{F}_2^n$, $n=2m$, 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 $\mathbb{F}_2^n$ to $\mathbb{F}_2^m$, 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 $F(x) = x^{2^r}{\rm Tr^n_m}(\Lambda(x))$, where $\Lambda$ is a linearized permutation of $\mathbb{F}_{2^m}$. 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 $2^{n-1}-2^{\lfloor\frac{n+m}{2}\rfloor}$, a bound which is attained by the example $x^{2^r}{\rm Tr^n_m}(x)$, $1\le r<m$ (Pott et al. 2018). This partially solves Question 5 in Pott et al. 2018. We then analyse the functions of the form $x^{2^r}{\rm Tr^n_m}(\Lambda(x))$. We show that for odd $m$, only $x^{2^r}{\rm Tr^n_m}(x)$, $1\le r<m$, 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 $x^{2^r}{\rm Tr^n_m}(\Lambda(x))$. 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.