AI 中文总结
研究针对全信息、自适应对手的拜占庭协议轮复杂度，通过证明多轮集中引理等，得出最坏情况期望轮复杂度为\(\Omega\!\left(\frac{t^2}{n\log(n + 1)}\right)\)，改进了前人结果并与近期上界匹配。
AI 中文摘要
我们证明，在全信息、强自适应对手模型中，针对\(t\)个腐败方安全的每个随机同步拜占庭协议，其最坏情况期望轮复杂度为\(\Omega\!\left(\frac{t^2}{n\log(n + 1)}\right)\)。这改进了[Bar-Joseph, Ben-Or 98]中的开创性\(\Omega(\frac{t}{\sqrt{n\log n}})\)界。我们的结果与[Dufoulon, Pandurangan 25]最近的\(O\left(\min\left\{\frac{t^2\log n}{n},\frac{t}{\log n}\right\}\right)\)上界在\(t\ll n\)时相差\(\log^2 n\)因子。我们的证明借鉴了[Etesami, Mahloujifar, Mahmoody 20]和[Haitner, Karidi-Heller 26]的工作。具体来说，我们证明了一个多轮集中引理，表明任何概率为\(p\)的转录事件可以通过期望中破坏\(O(\sqrt{n\log(\frac{1}{p})})\)个方以概率1强制发生。从那里，[Chor, Merritt, Shmoys 89]的工具允许我们通过使用涉及最多\(R\)个方的崩溃调度，将协议在\(R\)轮内不结束的概率下界为\(\frac{1}{n^{O(R)}}\)。这些技术的组合产生了所需的界。
英文摘要
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.
