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

Pott et al. (2018) showed that F(x) = x(2r) Tr-m(n)(x), n = 2m, r >= 1, is a nontrivial example of a vectorial function with the maximal possible number 2(n) - 2(m) of bent components. Mesnager et al. (2019) generalized this result by showing conditions on Lambda(x) = x + Sigma(sigma)(j=1) alpha(j)x(2tj), alpha(j) is an element of F-2m, under which F(x) = x(2r) Tr-m(n)(Lambda(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) = Tr-m(n)(gamma F(x)), gamma is an element of 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 DaF (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 x F-2m to exhibit small differential uniformity (for instance for Lambda(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.