On the number of rotation symmetric functions over GF(p)

Rotation symmetric functions which are invariant under circular translation of indices are widely used as components of different cryptosystems. In this paper, we do some enumeration on rotation symmetric functions over $GF\left(p\right)$ . First, we prove that the construction and enumeration of balanced rotation symmetric functions over $GF\left(p\right)$ are equivalent to solving a specified equation system and enumerating the solutions, and then a lower bound on the number of balanced rotation symmetric functions over $GF\left(p\right)$ is given. Furthermore, we give a formula to count homogeneous rotation symmetric functions which solves the open problem in Li (2008).