On a Class of Functions With the Maximal Number of Bent Components
Nurdagül Anbar, Tekgül Kalaycı, Wilfried Meidl, László Mérai · IEEE Transactions on Information Theory · 2022
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 vectorial bent functions from$\mathbb {F}_{2}^{n}$to$\mathbb {F}_{2}^{m}$, seen as functions on$\mathbb {F}_{2}^{n}$. The first nontrivial example is given in univariate form as$x^{2^{r}} {\rm Tr^{n}_{m}}(x), 1\le r < m$(Pott et al. 2018), a few more examples of similar shape are given by Mesnager et al. 2019, and finally it has been shown that the quadratic function$F(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x))$, has$2^{n}-2^{m}$bent components if and only if$\Lambda $is a linearized permutation polynomial of$\mathbb {F}_{2^{m}}[x]$(Anbar et al. 2021). In the first part of this article, an upper bound for the nonlinearity of plateaued functions with$2^{n}-2^{m}$bent components is shown, which is attained by the example$x^{2^{r}} {\rm Tr^{n}_{m}}(x)$. We then analyse in detail nonlinearity and differential spectrum of the class of functions$F(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x))$, which, as will be seen, requires the study of the functions$x^{2^{r}}\Lambda (x)$. In the last part we demonstrate that this class belongs to a larger class of functions with$2^{n}-2^{m}$Maiorana-McFarland bent components, which also contains nonquadratic and non-plateaued functions.