Relations between Different Constructions of Bent Functions and Their Enumerations

Bent functions have important applications in cryptography and coding theory, but there are many open problems in bent functions. No formal method is known to construct all bent functions, and their number is also unknown for variables of more than eight by now. Different constructions of bent functions are proposed using different techniques, some are extremely brief. In this paper we study the relation and enumeration of some constructions of bent functions. We prove that Steinbach et al.'s construction include all the bent functions of Rothaus's construction and its extension. The comparison shows that Boolean differential equation is a powerful tool which constructs a relative large subset of all bent functions.