An Analysis of the 𝒞 Class of Bent Functions

Abstract

Two (so-called 𝒞, D) classes of permutation-based bent Boolean functions were introduced by Carlet [4] two decades ago, but without specifying some explicit construction methods for their construction (apart from the subclass 𝒟₀). In this article, we look in more detail at the 𝒞 class, and derive some existence and nonexistence results concerning the bent functions in the 𝒞 class for many of the known classes of permutations over 𝔽_{2 ⁿ} . Most importantly, the existence results induce generic methods of constructing bent functions in class 𝒞 which possibly do not belong to the completed Maiorana-McFarland class. The question whether the specific permutations and related subspaces we identify in this article indeed give bent functions outside the completed Maiorana-McFarland class remains open.