Constructing Even-Variable Symmetric Boolean Functions with High Algebraic Immunity

Abstract

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, 2^{⌊log ₂k⌋+2} symmetric Boolean functions on 2k variables with maximum algebraic immunity are constructed, which are 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 2^{⌊ log ₂d⌋+2(k-d+1)}. As far as we know, this is the first lower bound of this kind.