Analysis of (n, n)-Functions Obtained From the Maiorana-McFarland Class

Pott et al. (2018) showed that F(x) = x2r Trnm(x), n = 2m, r ≥ 1, is a nontrivial example of a vectorial function with the maximal possible number 2n-2m of bent components. Mesnager et al. (2019) generalized this result by showing conditions on Λ (x) = x + Σj=1σ αjx2tj, αj ∈ F 2m, under which F(x) = x2r Trnm(Λ (x)) has the maximal possible number of bent components. We simplify these conditions and further analyse this class of functions. For all related vectorial bent functions F(x) = Trnm(γF(x)), γ ∈ F2n F 2m, which as we will point out belong to the Maiorana-McFarland class, we describe the collection of the solution spaces for the linear equations DaF(x) = F(x) + F(x+a) + F(a) = 0, which forms a spread of F2n. Analysing these spreads, we can infer neat conditions for functions H(x) = (F(x),G(x)) from F2n to F2m× F 2m to exhibit small differential uniformity (for instance for Λ (x) = x and r=0 this fact is used in the construction of Carlet's, Pott-Zhou's, Taniguchi's APN-function). For some classes of H(x) we determine differential uniformity and with a method based on Bezout's theorem nonlinearity.