Further study on the maximum number of bent components of vectorial functions

Abstract

In 2018, Pott et al. have studied in (IEEE Trans Inf Theory 64(1):403-411, 2018) the maximum number of bent components of vectorial functions. They have presented many nice results and suggested several open problems in this context. This paper is in the continuation of their study in which we solve two open problems raised by Pott et al. and partially solve an open problem raised by the same authors. Firstly, we prove that for a vectorial function, the property of having the maximum number of bent components is invariant under the so-called CCZ equivalence. Secondly, we prove the non-existence of APN plateaued functions having the maximum number of bent components. In particular, quadratic APN functions cannot have the maximum number of bent components. Finally, we present some sufficient conditions that the vectorial function defined from $\mathbb {F}_{2^{2k}}$ to $\mathbb {F}_{2^{2k}}$ by its univariate representation:

$$\begin{aligned} \alpha x^{2^i}\left( x+x^{2^k}+\sum \limits _{j=1}^{\rho }\gamma ^{(j)}x^{2^{t_j}} +\sum \limits _{j=1}^{\rho }\gamma ^{(j)}x^{2^{t_j+k}}\right) \end{aligned}$$

has the maximum number of bent components, where $\rho \le k$. Further, we show that the differential spectrum of the function $x^{2^i}(x+x^{2^k}+x^{2^{t_1}}+x^{2^{t_1+k}}+x^{2^{t_2}}+x^{2^{t_2+k}})$ (where $i,t_1,t_2$ satisfy some conditions) is different from the binomial function $F^i(x)= x^{2^i}(x+x^{2^k})$ presented in the article of Pott et al.