Bent Functions in C and D Outside the Completed Maiorana-McFarland Class

摘要

Two new classes of bent functions derived from the Maiorana-McFarland (M) class, so-called C and D, were introduced by Carlet [2] two decades ago. However, apart from the subclass D_o, some explicit construction methods for these functions were not provided in [2]. Assuming the possibility of specifying a bent function f that belongs to one of these two classes (apart from D_o), the most important issue is then to determine whether f is still contained in the known primary classes or lies outside their completed versions. In this article we partially solve this question by providing sufficient conditions on the permutation and related characteristic function (used to define f in these classes) so that f is provably outside the completed M class. To give some existence results, we employ recent results in [12] where some instances of bent functions in C were identified by providing specific permutations and related characteristic functions. More precisely, using our sufficient conditions that apply to both C and D, it is shown that these identified classes of C functions described in [12] do not belong to the completed M class, whereas the question (which is more difficult) whether these functions are also outside the completed partial spread class remains open. We also propose some generic methods for specifying bent functions in D outside the completed Maiorana-McFarland class.