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, $2^{\lfloor \log_2{k} \rfloor + 2}$ 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 $2^{\lfloor \log_2{d} \rfloor + 2(k-d+1)}$. As far as we know, this is the first lower bound of this kind.

💡 Research Summary

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The paper addresses the construction and enumeration of symmetric Boolean functions on an even number of variables that possess high algebraic immunity (AI). Algebraic immunity, defined as the minimum degree of a non‑zero annihilator of a Boolean function (or its complement), is a crucial metric for resisting algebraic attacks on stream ciphers. While general Boolean functions with maximum AI are known to be abundant, the situation for symmetric functions—especially those with an even number of variables—has been far less explored.

Main Contributions

1. Sufficient Condition (Theorem 3.3).
   For n = 2k variables, let d be a binary suffix of k (i.e., the low‑order bits of k). The authors partition the integer interval