On vectorial functions with maximal number of bent components

TL;DR

This paper investigates vectorial functions with the maximum number of bent components, analyzing their Walsh transform and nonlinearity, and introduces new constructions that partially answer open problems in cryptography.

Contribution

It provides new bounds on nonlinearity for certain vectorial functions, constructs new classes with maximal bent components, and addresses open problems in the field.

Findings

Nonlinearity of specific functions is bounded and some attain the bound.

New classes of functions with maximal bent components are constructed.

Characterization of functions with maximal bent components in specific cases.

Abstract

We study vectorial functions with maximal number of bent components in this paper. We first study the Walsh transform and nonlinearity of $F(x)=x^{2^e}h(\Tr_{2^{2m}/2^m}(x))$, where $e\geq0$ and $h(x)$ is a permutation over $\F_{2^m}$. If $h(x)$ is monomial, the nonlinearity of $F(x)$ is shown to be at most $ 2^{2m-1}-2^{\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 $F: \F_{2^{2m}}\rightarrow \F_{2^{2m}}$, $x\mapsto x^{2^m+1}+x^{2^i+1}$ has maximal number of bent components if and only if $i=0$.