Settling The Round Complexity of Byzantine Agreement Against a Full-Information, Adaptive Adversary

2026-07-15

AI Summary

This work investigates the round complexity lower bound of randomized synchronous Byzantine agreement under a strongly adaptive adversary with full information. By integrating multi-round concentration inequalities, crash-scheduling techniques, and a refined analysis of adversarial corruption strategies, the paper establishes a new lower bound of Ω(t²/(n log(n+1))), improving upon the previous best-known bound of Ω(t/√(n log n)). This result significantly narrows the theoretical gap between lower and upper bounds, especially when t ≪ n, where it approaches the current best upper bound up to only a log²n factor, thereby nearly achieving tightness.

Abstract

We prove that every randomized synchronous Byzantine Agreement protocol in the full-information, strongly adaptive adversary model, secure against $t$ corrupt parties, has worst-case expected round complexity \[ Ω\!\left(\frac{t^2}{n\log(n+1)}\right). \] This improves upon the seminal $Ω(\frac{t}{\sqrt{n\log n}})$ bound of [Bar-Joseph, Ben-Or 98]. Our result matches the recent upper bound of $O\left(\min\left\{\frac{t^2\log n}{n},\frac{t}{\log n}\right\}\right)$ of [Dufoulon, Pandurangan 25], up to a $\log^2 n$ factor in the $t\ll n$ regime. Our proof takes inspiration from the recent works of [Etesami, Mahloujifar, Mahmoody 20] and [Haitner, Karidi-Heller 26]. Specifically, we prove a multi-round concentration lemma showing that any transcript event of probability $p$ can be forced with probability one by corrupting $O(\sqrt{n\log(\frac1p)})$ parties in expectation. From there, tools from [Chor, Merritt, Shmoys 89] allow us to lower-bound the probability of the protocol not concluding in $R$ rounds by $\frac{1}{n^{O(R)}}$, using a crash schedule involving at most $R$ parties. The combination of these techniques yields the desired bound.

Problem

• Byzantine Agreement
• round complexity
• adaptive adversary
• full-information model
• lower bound

Innovation

• Byzantine Agreement
• round complexity
• adaptive adversary
• lower bound
• concentration lemma