  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 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.
  In this paper we provide 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 apply this framework to prove a result of independent interest.
Namely, we completely characterize the quality of decisions that protocols for
a randomized multi-valued Consensus problem can guarantee in an asynchronous
environment with Byzantine faults. 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. Finally, we show that the bound is tight by
providing a protocol that matches it.

Distributed, Parallel, and Cluster Computing