The ranks of Maiorana-McFarland bent functions

Abstract

In this paper, the ranks of a special family of Maiorana-McFarland bent functions are discussed. The upper and lower bounds of the ranks are given and those bent functions whose ranks achieve these bounds are determined. As a consequence, the inequivalence of some bent functions are derived. Furthermore, the ranks of the functions of this family are calculated when t 6.

Cubic bent functions outside the completed Maiorana-McFarland class

“…Proof. The first and the second claims hold, since the statements (4) and (5) were proven in [37,38] for 2-ranks, and by Theorem 1 we know, that 2-ranks and Γ-ranks coincide for all non-constant Boolean functions. Finally, the third claim follows from (5) and the definition of the primary construction.…”

“…, 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…”