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

Pott et al. (2018) showed that F(x) = x^2rTrⁿ_m(x) , n = 2m , r ≥ 1, is a nontrivial example of a vectorial function with the maximal possible number 2ⁿ-2^m of bent components. Mesnager et al. (2019) generalized this result by showing conditions on Λ (x) = x + ∑_j=1^σ α_jx^2tj, α_j ∈ F_2m, under which F(x) = x^2rTrⁿ_m(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) = Trⁿ_m(γ F(x)) , γ ∈ F_2n \ 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 D_aF(x) = F(x) + F(x+a) + F(a) = 0 , which forms a spread of F_2n. Analysing these spreads, we can infer neat conditions for functions H(x) = (F(x),G(x)) from F_2n to F_2m × 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. [ABSTRACT FROM AUTHOR]