All Byzantine Agreement Problems are Expensive

TL;DR

This paper proves that all non-trivial Byzantine agreement problems require at least quadratic message complexity, extending the classical Dolev-Reischuk bound to a broader class of agreement variants.

Contribution

It establishes a universal quadratic lower bound for the communication complexity of any non-trivial Byzantine agreement problem, resolving a long-standing open question.

Findings

Proves Omega(t^2) message lower bound for Byzantine broadcast.

Introduces a novel indistinguishability argument for the lower bound.

Extends the lower bound to all non-trivial agreement problems.

Abstract

Byzantine agreement, arguably the most fundamental problem in distributed computing, operates among n processes, out of which t < n can exhibit arbitrary failures. The problem states that all correct (non-faulty) processes must eventually decide (termination) the same value (agreement) from a set of admissible values defined by the proposals of the processes (validity). Depending on the exact version of the validity property, Byzantine agreement comes in different forms, from Byzantine broadcast to strong and weak consensus, to modern variants of the problem introduced in today's blockchain systems. Regardless of the specific flavor of the agreement problem, its communication cost is a fundamental metric whose improvement has been the focus of decades of research. The Dolev-Reischuk bound, one of the most celebrated results in distributed computing, proved 40 years ago that, at least for Byzantine broadcast, no deterministic solution can do better than Omega(t^2) exchanged messages in the worst case. Since then, it remained unknown whether the quadratic lower bound extends to seemingly weaker variants of Byzantine agreement. This paper answers the question in the affirmative, closing this long-standing open problem. Namely, we prove that any non-trivial agreement problem requires Omega(t^2) messages to be exchanged in the worst case. To prove the general lower bound, we determine the weakest Byzantine agreement problem and show, via a novel indistinguishability argument, that it incurs Omega(t^2) exchanged messages.