Constructions of Efficiently Implementable Boolean Functions with Provable Nonlinearity/Resiliency/Algebraic Immunity Trade-Offs

We describe several families of efficiently implementable Boolean functions achieving provable trade-offs between resiliency, nonlinearity, and algebraic immunity. In concrete terms, the following result holds for each of the function families that we propose. Given integers $m_0\geq 0$, $x_0\geq 1$, and $a_0\geq 1$, it is possible to construct an $n$-variable function which has resiliency at least $m_0$, linear bias (which is an equivalent method of expressing nonlinearity) at most $2^{-x_0}$ and algebraic immunity at least $a_0$; further, $n$ is linear in $m_0$, $x_0$ and $a_0$, and the function can be implemented using $O(n)$ gates.