Weights of Boolean cubic monomial rotation symmetric functions

This paper studies degree 3 Boolean functions in n variables x1, ..., xn which are rotation symmetric, that is, invariant under any cyclic shift of the indices of the variables. These rotation symmetric functions have been extensively studied in the last dozen years or so because of their importance in cryptography. Some of the cryptographic applications are described in a 2002 paper of Cusick and Stănică, which gave a recursion for the truth table and a nonhomogeneous recursion for the (Hamming) weight of the homogeneous cubic rotation symmetric function generated by the monomial x1x2x3. Until now, this was the only investigation of the recursive structure of such functions. Here we provide an algorithm for finding a recursion for the truth table of any cubic rotation symmetric Boolean function generated by a monomial, as well as a homogeneous recursion for its weight as n increases; in doing so we greatly reduce the computational complexity of a problem that appeared to be exponential in the number of variables, as well as provide a new way of studying the structure of the functions. The method makes some computations practically accessible that were previously entirely unfeasible. Once the weights have been computed for the initial small values of n, the further weights can be computed from the recursion, without looking at the truth table at all.