On functions with the maximal number of bent components
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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 obtained as F(x) = (f_1(x),…,f_m(x),a_1(x),…, a_m(x)), where F̃(x)=(f_1(x),…,f_m(x)) is a vectorial bent function from 𝔽_2^n to 𝔽_2^m, and a_i, 1≤ i≤ m, are affine Boolean functions. A class of nontrivial examples is given in univariate form with the functions F(x) = x^2^r Tr^n_m(Λ(x)), where Λ is a linearized permutation of 𝔽_2^m. In the first part of this article it is shown that plateaued functions with 2^n-2^m bent components can have nonlinearity at most 2^n-1-2^⌊n+m/2⌋, a bound which is attained by the example x^2^r Tr^n_m(x), 1≤ r<m (Pott et al. 2018). This partially solves Question 5 in Pott et al. 2018. We then analyse the functions of the form x^2^r Tr^n_m(Λ(x)). We show that for odd m, only x^2^r Tr^n_m(x), 1≤ r<m, has maximal nonlinearity, whereas there are more of them for even m, of which we present one more infinite class explicitly. In detail, we investigate Walsh spectrum, differential spectrum and their relations for the functions x^2^r Tr^n_m(Λ(x)). Our results indicate that this class contains many nontrivial EA-equivalence classes of functions with the maximal number of bent components, if m is even, several with maximal possible nonlinearity.
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