A recursive formula for weights of Boolean rotation symmetric functions

For the last dozen years or so, there has been much research on the applications of rotation symmetric Boolean functions with n variables in cryptography. In particular, the Hamming weights of these functions have been studied, because knowledge of these weights is important if the functions are to be useful in cryptography. Only in 2009, in a paper by Kim et al., there was a closed formula for the weights as a function of n obtained for some of these functions in the simplest case of quadratic functions. In this paper, we present a method for recursively computing the weights of certain kinds of rotation symmetric Boolean functions with arbitrary degree. Using some recent work of Cusick on the affine equivalence classes of certain cubic rotation symmetric functions, we obtain some detailed information on relationships between the weights of some of these cubic functions as n increases. This leads to some very specific information about previously unsuspected connections between the truth tables of various cubic rotation symmetric functions.