Asynchronous Byzantine Agreement in Incomplete Networks

Abstract

The Byzantine agreement problem is considered to be a core problem in distributed systems. For example, Byzantine agreement is often used to build a blockchain, a totally ordered log of records. Blockchains are asynchronous distributed systems, fault-tolerant against Byzantine nodes. In the literature, the asynchronous byzantine agreement problem is studied in a fully connected network model where every node can directly send messages to every other node. This assumption is questionable in many real-world environments. In the reality, nodes might need to communicate by means of an incomplete network, and Byzantine nodes might not forward messages. Furthermore, Byzantine nodes might not behave correctly and, for example, corrupt messages. Therefore, in order to truly understand Byzantine Agreement, we need both ingredients: asynchrony and incomplete communication networks. In this paper, we study the asynchronous Byzantine agreement problem in incomplete networks. A classic result by Danny Dolev proved that in a distributed system with n nodes in the presence of f Byzantine nodes, the vertex connectivity of the system communication graph should be at least (2f +1). While Dolev's result was for synchronous deterministic systems, we demonstrate that the same bound also holds for asynchronous randomized systems. We show that the bound is tight by presenting a randomized algorithm, and a matching lower bound. This algorithm is based on a protocol which allows other Byzantine agreement algorithms to be implemented in incomplete networks.