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

A function F : F-2(n) -> F-2(n), n = 2m, can have at most 2(n) - 2(m) bent component functions. Trivial examples are vectorial bent functions from F-2(n) to F-2(m), seen as functions on F-2(n). The first nontrivial example is given in univariate form as x(2r) Tr-m(n)(x), 1 <= r < m (Pon 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(2r) Tr-m(n) (Lambda(x)), has - 2(n) - 2(m) bent components if and only if Lambda is a linearized permutation polynomial of F-2m [x] (Anhar 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(2r) Tr-m(n) (x). We then analyse in detail nonlinearity and differential spectrum of the class of functions F(x) = x(2r) Tr-m(n)(Lambda(x)), which, as will be seen, requires the study of the functions x(2r) 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.