Probabilistic Indistinguishability and the Quality of Validity in Byzantine Agreement

This paper provides a formal framework for reasoning about randomized distributed algorithms. We generalize the notion of indistinguishability, the most useful tool in deterministic lower bounds, to apply to a probabilistic setting. We use the new notion to prove a lower bound on the probability at which it can be guaranteed that honest parties will not decide on a possibly bogus value. Moreover, we show that the bound is tight by providing a protocol that matches the bound. This completely characterizes the quality of decisions that protocols for a randomized multi-valued Consensus problem can guarantee in an asynchronous environment with Byzantine faults. Proving lower bounds for probabilistic protocols can be rather subtle. Probabilistic protocols have been around for at least four decades and are receiving a lot of attention with the emergence of blockchain systems. Nonetheless, we are aware of only a handful of randomized lower bounds. The formal tool that we develop in order to carry out our lower bound proof is of independent interest. It provides a rigorous yet accessible means to prove lower bounds for probabilistic protocols in general.