On a Class of Functions With the Maximal Number of Bent Components

A function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F: \mathbb {F}_{2}^{n}\rightarrow \mathbb {F} _{2}^{n}$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n=2m$ </tex-math></inline-formula> , can have at most <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{n}-2^{m}$ </tex-math></inline-formula> bent component functions. Trivial examples are vectorial bent functions from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{2}^{n}$ </tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{2}^{m}$ </tex-math></inline-formula> , seen as functions on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{2}^{n}$ </tex-math></inline-formula> . The first nontrivial example is given in univariate form as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x^{2^{r}} {\rm Tr^{n}_{m}}(x), 1\le r &lt; m$ </tex-math></inline-formula> (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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x))$ </tex-math></inline-formula> , has <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{n}-2^{m}$ </tex-math></inline-formula> bent components if and only if <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Lambda $ </tex-math></inline-formula> is a linearized permutation polynomial of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{2^{m}}[x]$ </tex-math></inline-formula> (Anbar et al. 2021). In the first part of this article, an upper bound for the nonlinearity of plateaued functions with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{n}-2^{m}$ </tex-math></inline-formula> bent components is shown, which is attained by the example <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x^{2^{r}} {\rm Tr^{n}_{m}}(x)$ </tex-math></inline-formula> . We then analyse in detail nonlinearity and differential spectrum of the class of functions <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x))$ </tex-math></inline-formula> , which, as will be seen, requires the study of the functions <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x^{2^{r}}\Lambda (x)$ </tex-math></inline-formula> . In the last part we demonstrate that this class belongs to a larger class of functions with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{n}-2^{m}$ </tex-math></inline-formula> Maiorana-McFarland bent components, which also contains nonquadratic and non-plateaued functions.