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In this paper, we explicitly construct a large class of symmetric Booleanfunctions on $2k$ variables with algebraic immunity not less than $d$, whereinteger $k$ is given arbitrarily and $d$ is a given suffix of $k$ in binaryrepresentation. If let $d = k$, our constructed functions achieve the maximumalgebraic immunity. Remarkably, $2^{\lfloor \log_2{k} \rfloor + 2}$ symmetricBoolean functions on $2k$ variables with maximum algebraic immunity areconstructed, which is much more than the previous constructions. Based on ourconstruction, a lower bound of symmetric Boolean functions with algebraicimmunity not less than $d$ is derived, which is $2^{\lfloor \log_2{d} \rfloor +2(k-d+1)}$. As far as we know, this is the first lower bound of this kind.

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