Probabilistic Indistinguishability and the Quality of Validity in Byzantine Agreement

TL;DR

This paper develops a formal framework for analyzing randomized distributed algorithms, specifically proving bounds on decision quality in Byzantine consensus protocols within asynchronous environments.

Contribution

It introduces a generalized indistinguishability concept for probabilistic algorithms and characterizes decision quality bounds in Byzantine consensus.

Findings

Established a lower bound on honest parties' decision probability

Provided a protocol that matches the proven bound

Generalized indistinguishability for probabilistic settings

Abstract

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.