Affine equivalence classes of 2-rotation symmetric cubic Boolean functions

A Boolean function in 2n variables that is generated by applying all even powers of the cyclic permutation of the variables to a single cubic monomial is called a cubic monomial rotation symmetric 2-function (or cubic 2-MRS function for short). In 2015, Cusick and Johns gave a complete description of the affine equivalence classes for such functions with generating monomial x₁x_rx_s with 1<r<s and r and s not both odd. In this paper we develop the theory further by determining the smallest sets that act on the set of all these cubic 2-MRS functions to give the affine equivalence classes. Also, suppose f and g are two of these functions such that there exists a permutation σ of the variables where σ(f) = g. We give a complete description of all such permutations σ and an exact count of their number.