On a class of functions with the maximal number of bent components

Official URL: http://dx.doi.org/10.1109/TIT.2022.3174672

A function F : Fn2 → Fn2, n = 2m, can have at most 2n - 2m bent component functions. Trivial examples are vectorial bent functions from Fn2 to Fm2, seen as functions on Fn2. The first nontrivial example is given in univariate form as x2r Trn m(x), 1 ≤ 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) = x2r Trn m(Λ(x)), has 2n - 2m bent components if and only if Λ is a linearized permutation polynomial of F2m[x] (Anbar et al. 2021). In the first part of this article, an upper bound for the nonlinearity of plateaued functions with 2n-2m bent components is shown, which is attained by the example x2r Trn m(x). We then analyse in detail nonlinearity and differential spectrum of the class of functions F(x) = x2r Trn m(Λ(x)), which, as will be seen, requires the study of the functions x2r Λ(x). In the last part we demonstrate that this class belongs to a larger class of functions with 2n - 2m Maiorana-McFarland bent components, which also contains nonquadratic and non-plateaued functions.