Constructions of Efficiently Implementable Boolean Functions with Provable Nonlinearity/Resiliency/Algebraic Immunity Trade-Offs
Palash Sarkar
TL;DR
This work tackles the challenge of constructing infinite families of $n$-variable Boolean functions that simultaneously balance resiliency, nonlinearity (via linear bias), and algebraic immunity, while remaining efficiently implementable with $O(n)$ two-input gates. The authors develop two basic constructions and an iterated Concat-based method to produce functions with provable trade-offs, culminating in a general theorem: for any $m_0\ge0$, $x_0\ge1$, and $a_0\ge1$, there exists an $n$-variable function with resiliency at least $m_0$, linear bias at most $2^{-x_0}$, and AI at least $a_0$, with $n$ linear in the parameters and gate complexity $O(n)$. They also present multiple classes of constructions for $m>1$, including variable augmentation and seed-based hybrids, providing scalable, efficiently implementable options with explicit bounds on $n$, $m$, $x$, and $a$, and showing clear trade-offs on the AI/LB curve. The paper situates these results in the broader cryptographic landscape, connecting to Goldreich-like local-function frameworks and prior work on resiliency/immunity trade-offs, while highlighting the practical significance of achieving provable security properties alongside low gate counts for real-world cryptosystems.
Abstract
We describe several families of efficiently implementable Boolean functions achieving provable trade-offs between resiliency, nonlinearity, and algebraic immunity. In particular, the following statement 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 $O(n)$ 2-input gates, which is essentially optimal.