Rotation symmetric Boolean functions-Count and cryptographic properties
Discrete Applied Mathematics
Note: On the weight and nonlinearity of homogeneous rotation symmetric Boolean functions of degree 2
Discrete Applied Mathematics
9-variable Boolean functions with nonlinearity 242 in the generalized rotation symmetric class
Information and Computation
Enumeration of 9-variable rotation symmetric boolean functions having nonlinearity 240
INDOCRYPT'06 Proceedings of the 7th international conference on Cryptology in India
Search for Boolean Functions With Excellent Profiles in the Rotation Symmetric Class
IEEE Transactions on Information Theory
Affine equivalence of cubic homogeneous rotation symmetric functions
Information Sciences: an International Journal
A recursive formula for weights of Boolean rotation symmetric functions
Discrete Applied Mathematics
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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. Some of the cryptographic applications are described in a 2002 paper of Cusick and St驴nic驴, 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 x 1 x 2 x 3. 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.