On the number of rotation symmetric Boolean functions

Rotation symmetric Boolean functions (RSBFs) have been used as components of different cryptosystems. This class of functions are invariant under circular translation of indices. In this paper, we investigated balanced RSBFs and 1st order correlation immune RSBFs. Based on constructive techniques, we give an accurate enumeration formula for n-variable balanced RSBFs when n is a power of a prime. Furthermore, an original and efficient method to enumerate all n-variable (n prime) 1st order correlationimmune functions is presented. The exact number of 1st order correlation immune RSBFs with 11 variables is 6925047156550478825225250374129764511077684773805520800 and the number of 13 variables has 189 digits. Then for more variables, we also provide a significant lower bound on the number of 1st order correlation immune RSBFs.