On $2k$-Variable Symmetric Boolean Functions with Maximum Algebraic Immunity $k$

Algebraic immunity of Boolean function f is defined as the minimal degree of a nonzero g such that fg = 0 or (f + 1)g = 0. Given a positive even integer n, it is found that the weight distribution of any n-variable symmetric Boolean function with maximum algebraic immunity n 2 is determined by the binary expansion of n. Based on the foregoing, all n-variable symmetric Boolean functions with maximum algebraic immunity are constructed. The amount is (2wt(n) + 1)22 .