Probabilistic Indistinguishability and the Quality of Validity in Byzantine Agreement

Lower bounds and impossibility results in distributed computing are both intellectually challenging and practically important. Hundreds if not thousands of proofs appear in the literature, but surprisingly, the vast majority of them apply to deterministic algorithms only. Probabilistic distributed problems have been around for at least four decades and receive a lot of attention with the emergence of blockchain systems. Nonetheless, we are aware of only a handful of randomized lower bounds. In this paper we provide a formal framework to reason about randomized distributed algorithms. We generalize the notion of indistinguishability, the most useful tool in deterministic lower bounds, to apply to a probabilistic setting. The power of this framework is applied to completely characterize the quality of decisions in the randomized multi-valued Consensus problem in an asynchronous environment with Byzantine faults. That is, we provide a tight bound on the probability of honest parties deciding on a possibly bogus value and prove that, in a precise sense, no algorithm can do better.