Constructing and Counting Even-Variable Symmetric Boolean Functions with Algebraic Immunity not Less Than $d$

In this paper, we explicitly construct a large class of symmetric Boolean functions on 2k variables with algebraic immunity not less than d, where integer k is given arbitrarily and d is a given suffix of k in binary representation. If let d = k, our constructed functions achieve the maximum algebraic immunity. Remarkably, 22 k⌋+2 symmetric Boolean functions on 2k variables with maximum algebraic immunity are constructed, which is much more than the previous constructions. Based on our construction, a lower bound of symmetric Boolean functions with algebraic immunity not less than d is derived, which is 22 . As far as we know, this is the first lower bound of this kind.