On the Complexity of Asynchronous Agreement Against Powerful Adversaries

TL;DR

This paper establishes that achieving both perfect correctness and termination in asynchronous agreement algorithms against a powerful adversary inherently requires exponential running time, highlighting fundamental complexity limits.

Contribution

It introduces a new model of a strongly adaptive adversary and proves that any algorithm meeting perfect correctness and termination under this model must run exponentially slowly.

Findings

Randomized algorithms extend to defeat a strongly adaptive adversary.

Exponential running time is inherent for algorithms with perfect correctness against such adversaries.

Provides insights into the trade-offs between correctness, termination, and efficiency in asynchronous agreement.

Abstract

We introduce new techniques for proving lower bounds on the running time of randomized algorithms for asynchronous agreement against powerful adversaries. In particular, we define a \emph{strongly adaptive adversary} that is computationally unbounded and has a limited ability to corrupt a dynamic subset of processors by erasing their memories. We demonstrate that the randomized agreement algorithms designed by Ben-Or and Bracha to tolerate crash or Byzantine failures in the asynchronous setting extend to defeat a strongly adaptive adversary. These algorithms have essentially perfect correctness and termination, but at the expense of exponential running time. In the case of the strongly adaptive adversary, we show that this dismally slow running time is \emph{inherent}: we prove that any algorithm with essentially perfect correctness and termination against the strongly adaptive adversary must have exponential running time. We additionally interpret this result as yielding an enhanced understanding of the tools needed to simultaneously achieving perfect correctness and termination as well as fast running time for randomized algorithms tolerating crash or Byzantine failures.