Analysis of the number of even-variable Boolean functions with maximum algebraic immunity

Algebraic Immunity has been considered as one of significant properties for Boolean functions, and possessing maximum algebraic immunity (MAI) is a necessary criteria for Boolean functions used in stream ciphers against algebraic attacks. However, for a given number of variables, the accurate number of Boolean functions with MAI is not known, and the best known bound on the number also seems weak. In this paper, we investigate the enumeration of Boolean functions with MAI, and provide a new lower bound on the number of even-variable Boolean functions with MAI, this lower bound is better than the previous ones.