On Symmetric Boolean Functions With High Algebraic Immunity on Even Number of Variables

In this paper, we put forward an efficient method to study the symmetric Boolean functions with high algebraic immunity on even number of variables. We obtain some powerful necessary conditions for symmetric Boolean functions to achieve high algebraic immunity by studying the weight support of some specific types of Boolean functions of low degrees. With these results, we prove that the algebraic immunity of a large class of symmetric correlation immune Boolean functions, namely the symmetric palindromic functions, is not high. Besides, we construct all symmetric Boolean functions with maximum algebraic immunity and give a description for those with submaximum algebraic immunity. We also determine the Hamming weight, degrees and nonlinearity of the symmetric Boolean functions with maximum algebraic immunity.