Construction and count of 1-resilient rotation symmetric Boolean functions

Finding and constructing Boolean functions with many cryptographic properties to resist a variety of existing attacks are challenging tasks in current cryptography and information security. The key idea in this paper consists of finding a general formula for computing the number of orbits with the same length and Hamming weight by utilizing prime factorization for any integer n greater than 1. Using the property of an orthogonal array to turn the construction of 1-resilient rotation symmetric Boolean functions (RSBFs) on n variables into the solution of a linear system of equations, a complete characterization and a general construction method of this class of functions are also presented. Moreover, a formula for counting the number of functions of this class is found. Not only are the structures of all 1-resilient RSBFs that are obtained more clear, such problems regarding their construction and count are completely and exhaustively solved. In addition, our methods are simpler than existing methods. We provide the exact numbers of 1-resilient RSBFs having ten and 11 variables, which are 162091449508441568747323063140 and 403305984734393392122612918710214418571734777982178890, respectively. Finally, we use three examples to illustrate the application of our methods.