On Designs and Multiplier Groups Constructed from Almost Perfect Nonlinear Functions

Cubic bent functions outside the completed Maiorana-McFarland class

“…, for bent functions 2-ranks have been extensively studied in [37,38]; • Γ-rank(f ) is the 2-rank of M (dev(G f )), Γ-ranks were mostly studied in the context of inequivalence of vectorial mappings [17,18]; • SNF(f ) is the Smith normal form of the incidence matrix M (dev(G f )), given by the multiset…”

“…, Γ-ranks were mostly studied in the context of inequivalence of vectorial mappings [17,18]; • SNF(f ) is the Smith normal form of the incidence matrix M (dev(G f )), given by the multiset…”

On design-theoretic aspects of Boolean and vectorial bent functions

“…A translation design of a function F : F n 2 → F m 2 (not necessarily perfect nonlinear) is defined as the development dev(A) of a certain set A, which is constructed from the function F and has a nice combinatorial structure [9,10]. The classical choice of a set A for a Boolean bent function f : F n 2 → F 2 is either the support D f , a (2 n , 2 n−1 ± 2 n/2−1 , 2 n−2 ± 2 n/2−1 ) difference set, or the graph G f , a 2 n , 2, 2 n , 2 n−1 relative difference set, while for the vectorial function F : F n 2 → F m 2 one considers only the graph G F , which is a (2 n , 2 m , 2 n , 2 n−m ) relative difference set.…”