Homogeneous rotation symmetric (invariant under cyclic permutation of the variables) Boolean functions have been extensively studied in recent years due to their applications in cryptography. In this paper we give an explicit formula for the number of homogeneous rotation symmetric functions over the finite field G F ( p m ) using Polya’s enumeration theorem, which completely solves the open problem proposed by Yuan Li in 2008 . This result simplifies the proof and the nonexplicit counting formula given by Shaojing Fu et al. over the field G F ( p ) . This paper also gives an explicit count for n -variable balanced rotation symmetric Boolean functions with n = p q , where p and q are distinct primes. Previous work only gave an explicit count for the case where n is prime and lower bounds for the case where n is a prime power.
L. K.V., Dr. M. Sethumadhavan, and Cusick, T. W., “Counting rotation symmetric functions using Polya’s theorem”, Discrete Applied Mathematics, vol. 169, pp. 162 - 167, 2014.