On Bent-based and Linear Based Cryptographic Functions

Abstract

In this paper we show that several classes of already known cryptographic Boolean functions are either bent-based (BB) functions or linear-based (LB) functions. In particular, we show that all nonlinear resilient functions with maximum resiliency order, i.e. (n, n−3, 2, 2n-2) functions, are either BB or LB. We provide an explicit count for the functions in both classes. We also show that all symmetric bent functions that achieve the maximum possible nonlinearity are bent-based. for n even, we have 4 bent-based symmetric bent functions and for n odd, we also have 4 bent-based symmetric functions. Furthermore, we prove that there are no BB homogeneous functions with algebraic degree >2 and we provide a count for LB homogeneous functions. Some of the results obtained are extended to functions over GF(p).