The celebrated result of Fischer, Lynch and Paterson is the fundamental lower bound for asynchronous fault tolerant computation: any 1-crash resilient asynchronous agreement protocol must have some (possibly measure zero) probability of not terminating. In 1994, Ben-Or, Kelmer and Rabin published a proof-sketch of a lesser known lower bound for asynchronous fault tolerant computation with optimal resilience in face of a Byzantine adversary: if \(n\le 4t\) then any t-resilient asynchronous verifiable secret sharing protocol must have some non-zero probability of not terminating. Our main contribution is to revisit this lower bound and provide a rigorous and more general proof. Our second contribution is to show how to avoid this lower bound. We provide a protocol with optimal resilience that is almost surely terminating for a strong common coin functionality. Using this new primitive we provide an almost surely terminating protocol with optimal resilience for asynchronous Byzantine agreement that has a new fair validity property. To the best of our knowledge this is the first asynchronous Byzantine agreement with fair validity in the information theoretic setting.

Fischer、Lynch 和 Paterson 著名的结果是异步容错计算的基本下限：任何 1-crash 弹性异步协议协议必须有一些（可能测量为零）不终止的概率。1994 年，Ben-Or、Kelmer 和 Rabin 发表了一个鲜为人知的异步容错计算下限的证明草图，在面对拜占庭对手时具有最佳弹性：如果\(n\le 4t\)那么任何 t-resilient异步可验证秘密共享协议必须有一些非零不终止的概率。我们的主要贡献是重新审视这个下限，并提供更严格和更普遍的证明。我们的第二个贡献是展示如何避免这个下限。我们提供了一个具有最佳弹性的协议，该协议几乎肯定会终止强大的通用硬币功能。使用这个新的原语，我们为具有新的公平有效性属性的异步拜占庭协议提供了具有最佳弹性的几乎肯定终止的协议。据我们所知，这是第一个在信息论环境中具有公平有效性的异步拜占庭协议。