Efficient computation of algebraic immunity of symmetric Boolean functions

The computation on algebraic immunity (AI) of symmetric boolean functions includes: determining the AI of a given symmetric function and searching all symmetric functions with AI = d or AI >= d, where d <= [2/n]. In this paper we firstly showed a necessary and sufficient condition of AI of symmetric boolean functions and then proposed several efficient algorithms on computation of algebraic immunity of symmetric boolean functions. By these algorithms we could assess the vulnerability of symmetric boolean functions against algebraic/fast algebraic attacks efficiently, and find all symmetric functions having a given algebraic immunity AI(n)(f) = d, for some 0 <= d <= n.