In this report, we study the problem of Byzantine fault-tolerant distributed set intersection and the importance of redundancy in solving this problem. Specifically, consider a distributed system with $n$ agents, each of which has a local set. There are up to $f$ agents that are Byzantine faulty. The goal is to find the intersection of the sets of the non-faulty agents. We derive the Byzantine set intersection problem from the Byzantine optimization problem. We present the definition of $2f$-redundancy, and identify the necessary and sufficient condition if the Byzantine set intersection problem can be solved if a certain redundancy property is satisfied, and then present an equivalent condition. We further extend our results to arbitrary communication graphs in a decentralized setting. Finally, we present solvability results for the Byzantine optimization problem, inspired by our findings on Byzantine set intersection. The results we provide are for synchronous and asynchronous systems both.

翻译：在本报告中，我们研究了拜占庭容错分布式集合交集问题以及冗余性在解决该问题中的重要性。具体而言，考虑一个包含 $n$ 个智能体的分布式系统，每个智能体拥有一个本地集合。其中最多有 $f$ 个智能体存在拜占庭故障。目标是找出非故障智能体集合的交集。我们从拜占庭优化问题推导出拜占庭集合交集问题。我们给出了 $2f$-冗余性的定义，并确定了在满足特定冗余性质时拜占庭集合交集问题可解的充要条件，随后给出了一个等价条件。我们进一步将结果推广至去中心化环境下的任意通信图。最后，受拜占庭集合交集研究结果的启发，我们给出了拜占庭优化问题的可解性结果。所提供的结果同时适用于同步系统和异步系统。