Communication complexity is one of the most important efficiency metrics for distributed algorithms, but considerable gaps in the communication complexity still exist for Byzantine consensus, one of the most fundamental problems in distributed computing. This paper provides three results that help close some of these gaps. We present a Byzantine agreement (BA) protocol with quadratic communication complexity with optimal resilience $f < n/2$. This result shows the tightness of the quadratic lower bound by Dolev and Reischuk, which has been an open challenge for almost forty years. Our protocol is inspired by the BA protocol under unauthenticated model with $f < n/3$ (optimal for unauthenticated model) by Berman et al. As a practical direction, we present a Byzantine fault-tolerant (BFT) replication protocol, a.k.a., blockchain, with amortized linear communication complexity with $f \leq (1/2 - \varepsilon)n$ where $\varepsilon$ is any positive constant. As BFT replication is promised to be applied to large-scale systems, linear communication complexity (clearly optimal) is a mandatory today. To get this result, we present a Byzantine consistent broadcast (BCB) protocol with linear communication complexity with the same resilience $f \leq (1/2 - \varepsilon)n$. Our new BCB protocol relies on an expander graph and a threshold signature scheme. Finally, we also show a quadratic communication lower bound of BCB with $f \ge (1/2 + \varepsilon)n$. As BCB is one the easiest consensus problems and a build block of most BFT protocols, the result implies the impossibility of linear BFT protocol under corrupt majority.

翻译：通信复杂度是分布式算法最重要的效率衡量标准之一,但对于Byzantine共识而言,通信复杂度仍存在相当大的差距,这是分布式计算中最根本的问题之一。本文提供了有助于弥合部分差距的三种结果。我们展示了拜占庭协议(BB)协议,具有四边通信复杂度,具有最佳复原力 < n/2美元。这一结果显示了Dolev和Reischuk在四边端低端连接的紧紧度,这是近40年来一个公开的挑战。我们的协议受Berman 和 Al等以 yprial communical 协议的启发,Bzartine协议以 3$(未加工模型的最佳值)为B. 。我们展示了B.k.a.b.b.b. 阻隔断层协议,以美元 平面通信的平面通信复杂度为以美元平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面。