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

2025-10-02

AI Summary

This work addresses the challenge of simultaneously optimizing nonlinearity, resiliency order, and algebraic immunity in lightweight cryptographic Boolean functions. We propose an efficient, provably secure construction framework that achieves controllable trade-offs among these three critical cryptographic criteria. By integrating algebraic immunity analysis, linear bias control, and resilient structure design, we construct multiple families of n-variable Boolean functions where the number of variables scales linearly with the security parameters—specifically, n = O(r + d + t), with r, d, and t denoting the resiliency order, lower bound on nonlinearity, and algebraic immunity order, respectively. The resulting functions admit O(n)-size circuit implementations. To the best of our knowledge, this is the first unified construction enabling flexible, provable trade-offs among all three metrics, thereby significantly enhancing the synergy between security and efficiency. The approach is particularly suited for cryptographic algorithm design in resource-constrained environments.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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 $m_0geq 0$, $x_0geq 1$, and $a_0geq 1$, 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.

Problem

• Constructing Boolean functions with balanced cryptographic properties trade-offs
• Achieving provable resiliency, nonlinearity, and algebraic immunity simultaneously
• Ensuring efficient implementation with linear gate complexity in parameters

Innovation

• Efficiently implementable Boolean functions with provable trade-offs
• Constructed functions achieve resiliency, nonlinearity, algebraic immunity
• Implementation uses linear number of gates for scalability