This paper considers the good-case latency of Byzantine Reliable Broadcast (BRB), i.e., the time taken by correct processes to deliver a message when the initial sender is correct. This time plays a crucial role in the performance of practical distributed systems. Although significant strides have been made in recent years on this question, progress has mainly focused on either asynchronous or randomized algorithms. By contrast, the good-case latency of deterministic synchronous BRB under a majority of Byzantine faults has been little studied. In particular, it was not known whether a good-case latency below the worst-case bound of \(t+1\) rounds could be obtained. This work answers this open question positively and proposes a deterministic synchronous Byzantine reliable broadcast that achieves a good-case latency of \(\textsf{max} (2,t+3-c)\) rounds (or equivalently \(\textsf{max} (2,f+t+3-n)\)), where t is the upper bound on the number of Byzantine processes, \(f\le t\) the number of effectively Byzantine processes, and \(c=n-f\) the number of effectively correct processes. The proposed algorithm does not put any constraint on t, and assumes an authenticated setting, in which individual processes can sign the messages they send, and verify the authenticity of the signatures they receive.

本文考虑了拜占庭可靠广播（BRB）的良好情况延迟，即当初始发送者正确时正确进程传递消息所花费的时间。这个时间对于实际分布式系统的性能起着至关重要的作用。尽管近年来在这个问题上取得了重大进展，但进展主要集中在异步或随机算法上。相比之下，大多数拜占庭故障下确定性同步 BRB 的良好情况延迟却很少被研究。特别是，尚不清楚是否可以获得低于\(t+1\)轮最坏情况界限的好情况延迟。这项工作积极回答了这个悬而未决的问题，并提出了一种确定性同步拜占庭可靠广播，可以实现\(\textsf{max} (2,t+3-c)\)轮（或等效\(\textsf{ max} (2,f+t+3-n)\) )，其中t是拜占庭进程数量的上限，\(f\le t\)是有效拜占庭进程的数量，\(c= nf\)有效正确进程的数量。所提出的算法不对t施加任何限制，并假设经过身份验证的设置，其中各个进程可以对它们发送的消息进行签名，并验证它们接收的签名的真实性。