Affine equivalence for cubic rotation symmetric Boolean functions with variables

Rotation symmetric Boolean functions have been extensively studied in the last fifteen years or so because of their importance in cryptography and coding theory. Until recently, very little was known about the basic question of when two such functions are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in a 2009 paper of Kim, Park and Hahn. The much more complicated analogous problem for cubic functions was solved for permutations using a new concept of patterns in a 2011 paper of Cusick, and it is conjectured that, as in the quadratic case, this solution actually applies for all affine transformations. The patterns method enables a detailed analysis of the affine equivalence classes for various special classes of cubic rotation symmetric functions in n variables. Here the case of functions generated by a single monomial and having p q variables, where p and q are primes, is examined in detail, and in particular, a formula for the number of classes is proved. This is significant because it is the first time that a complete enumeration of the number of classes has been found when the number of variables is divisible by two distinct primes.