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.

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