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 communication-intensive, often requiring a large number of messages to be exchanged, especially in the presence of Byzantine faults, making efficiency a central challenge in the design of practical agreement protocols. In this paper, we study the problem of Strong Byzantine Agreement and establish tight upper and lower bounds on communication complexity, parameterized by the actual number of Byzantine faults. Specifically, for a system of n parties tolerating up to t Byzantine faults, out of which only f t are actually faulty, we obtain the following results: In the partially synchronous setting, we present the first Byzantine Agreement protocol that achieves adaptive communication complexity of O (n + t f) words, which is asymptotically optimal. Our protocol has an optimal resilience of t < n/3. In the asynchronous setting, we prove a lower bound of (n + t²) on the expected number of messages, and design an almost matching protocol with an optimal resilience that solves agreement with O ( (n + t²) n) words. Our main technical contribution in the asynchronous setting is the utilization of a bipartite expander graph that allows for low-cost information dissemination.
Constantinescu et al. (Mon,) studied this question.