Achieving agreement among distributed parties is a fundamental task in modern systems, underpinning applications such as consensus in blockchains, coordination in cloud infrastructure, and fault tolerance in critical services. However, this task can be intensive, often requiring a large number of messages to be exchanged as well as many rounds of communication, especially in the presence of Byzantine faults. This makes efficiency a central challenge in the design of practical agreement protocols. In this paper, we study the problem of Binary Agreement and give protocols that are simultaneously optimal in both message and round complexity, parameterized by the actual number of Byzantine faults. In contrast to previous works, we demonstrate that optimal message complexity can be achieved without sacrificing latency. Concretely, for a system of $n$ parties tolerating up to $t$ Byzantine faults, out of which only $f \leq t$ are actually faulty, we give the following results: When $t = Ω(n)$, in the synchronous (resp. partially synchronous) setting, with optimal resiliency $t < n/2$ (resp. $t < n/3$), we describe a deterministic protocol with optimal communication complexity $O(n \cdot (f+1))$ and optimal round complexity $O(f + 1)$. Building upon this previous result, when $t = o(n)$, for both the synchronous and partially synchronous setting, we describe a deterministic protocol with near-optimal communication complexity $\widetilde{O}(n + t\cdot f)$ and near-optimal round complexity $\widetilde{O}(f+1)$. Our approach relies on a novel use of dispersers to efficiently disseminate a value. For the asynchronous setting, we show a $Ω(n + t^2)$ lower bound in expectation and provide a randomized protocol with near-optimal $\widetilde{O}(n + t^2)$ communication complexity and $O(1)$ round complexity in expectation.

翻译：在分布式系统中达成共识是一项基础性任务，支撑着区块链共识、云基础设施协调以及关键服务容错等应用。然而，该任务通常需要大量消息交换与多轮通信，尤其在存在拜占庭故障时更为显著，这使得效率成为实际共识协议设计的核心挑战。本文研究二元共识问题，并提出在消息复杂度与轮次复杂度上同时达到最优的协议，其性能参数化于实际发生的拜占庭故障数量。与先前工作不同，我们证明无需牺牲延迟即可实现最优消息复杂度。具体而言，对于包含n个参与方、最多容忍t个拜占庭故障（其中实际故障数f ≤ t）的系统，我们取得以下成果：当t = Ω(n)时，在同步（及部分同步）设置下，在最优容错条件t < n/2（及t < n/3）下，我们提出确定性协议，其通信复杂度达到最优O(n·(f+1))，轮次复杂度达到最优O(f+1)。基于此结果，当t = o(n)时，针对同步与部分同步设置，我们提出确定性协议，其通信复杂度接近最优$\widetilde{O}(n + t·f)$，轮次复杂度接近最优$\widetilde{O}(f+1)$。我们的方法依赖于对分散器的新颖运用以实现高效值传播。对于异步设置，我们证明了期望下Ω(n + t²)的下界，并提出了随机化协议，其通信复杂度接近最优$\widetilde{O}(n + t²)$，期望轮次复杂度为O(1)。