Recursive weights for some Boolean functions

This paper studies degree 3 Boolean functions in n variables x ₁, ⋯ , x _n 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. Let wt(f) denote the (Hamming) weight of the Boolean function f. We extend and simplify results of Bileschi, Cusick and Padgett, who gave an algorithm for finding a recursion for the sequence {wt(f ⁿ _r,s) : n = 3, 4, ⋯}, where f ⁿ _r,s = x ₁x _rx _s + x ₂x _r+1x _s+1 + ⋯ + x _nx _r-1x _s-1 is the cubic rotation symmetric function generated by the monomial x ₁x _rx _s : We show that the weights for the function g ⁿ _r,s = x ₁x _rx _s + x ₂x _r+1x _s+1 + ⋯ + x _1+n-sx _r+n-sx _n (note this is just the sum of the first n - s + 1 terms in the definition of f ⁿ _r,s) satisfy the same recursion as the weights for f ⁿ _r,s . We prove the recursion for the weights of g ⁿ _r,s by a method based on the work by Bileschi-Cusick-Padgett, but we are able to avoid a lot of the complications in their proofs. In particular, we do not need the technical notion of "g terms" or the elaborate proof of similarity of certain matrices.