On the equal-weight symmetric Boolean functions

Abstract

Two important classes of symmetric Boolean functions are the equal-weight Boolean functions and the elementary (or homogeneous) symmetric Boolean functions. In this paper we studied the equal-weight symmetric Boolean functions. First the Walsh spectra of the equalweight symmetric Boolean functions are given. Second the sufficient and necessary condition on correlation-immunity of the equal-weight symmetric Boolean function is derived and other cryptology properties such as the nonlinearity, balance and propagation criterion are taken into account. In particular, the nonlinearity of the equal-weight symmetric Boolean functions with n(n≥10) variables is determined by their Hamming weight. Considering these properties will be helpful in further investigations of symmetric Boolean functions.

Keywords

Boolean functions / symmetric Boolean functions / Krawtchouk polynomial / nonlinearity / correlation immunity / propagation criterion