Cubic rotation symmetric boolean functions
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This paper studies degree 3 homogeneous rotation symmetric Boolean functions, that is, homogeneous Boolean functions which are invariant under any cyclic permutation of the variables. These rotation symmetric Boolean functions have been extensively studied since 1999 when Pieprzyk and Qu demonstrated their importance in cryptography and coding theory. Recently, some attention has been given to the question of when two such functions are affine equivalent and to the recursive structure of the (Hamming) weight of such functions. Here, we extend and simplify the results from Bileschi, Cusick and Padgett by relating the recursive structure of cubic monomial rotation symmetric (MRS) Boolean functions to that of the simpler Boolean functions defined by [special characters omitted] = x 1 x r x s + x 2 x r +1 x s +1 + ··· + x 1+ n-s x r + n-s x n (note this is just the sum of the first n − s + 1 terms in the definition of cubic MRS Boolean functions). Furthermore, we use the notion of patterns introduced by Cusick in 2011 to identify in detail all of the affine equivalence classes under permutations of cubic MRS Boolean functions in p k variables where p > 3 is a prime, and provide a formula for the number of these classes. We also identify all such classes when the Boolean function is in n variables, for any n , and provide a count for a large portion of these classes. The identification of these classes is then applied to the results found on the weight of cubic MRS Boolean functions to further minimize computation.