The Dolev-Reischuk lower bound establishes that any deterministic Byzantine Agreement (BA) protocol for $n$ processors tolerating $f$ faults requires $Ω(f^2+n)$ messages. But what exactly does this quadratic cost pay for? Even the minimal requirement that every correct processor \emph{receive at least one message} already necessitates $Ω(f^2 + n)$ messages. This raises a fundamental question: is the Dolev-Reischuk bound about the difficulty of \emph{reaching univalency} -- the point at which the protocol's outcome is determined -- or merely about \emph{disseminating} the outcome to all processors afterward? We resolve this question by showing that reaching univalency does \emph{not} require quadratic communication. Specifically, we introduce $ε$-BA, a relaxation allowing an $ε$-fraction of correct processors to output incorrectly, and prove it can be solved deterministically with $O(n \log n)$ communication complexity when $f < n(1/3 - ε)$. Crucially, any $ε$-BA protocol can serve as the first phase of a full BA protocol: after $ε$-BA, a single all-to-all exchange and majority vote completes BA. Since the outcome is already determined after $ε$-BA, this demonstrates that the quadratic cost in Dolev-Reischuk stems entirely from dissemination, rather than from reaching univalency. We also define Extractable BA for authenticated settings, capturing when processors collectively hold enough signed messages to determine the agreed value, and show it can be solved with communication complexity $O(f \log f)$.

翻译：Dolev-Reischuk下界确立了任何容忍$f$个故障的确定性拜占庭协议（BA）协议都需要$Ω(f^2+n)$条消息。但这一二次成本究竟支付了什么？即使是最低要求——每个正确的处理器\emph{至少接收一条消息}——也已经需要$Ω(f^2 + n)$条消息。这引发了一个根本性问题：Dolev-Reischuk下界是关于\emph{达到单值性}（即协议结果确定的时刻）的难度，还是仅仅关于之后将结果\emph{传播}给所有处理器？我们通过证明达到单值性\emph{并不}需要二次通信来解决这个问题。具体而言，我们引入了$ε$-BA，这是一种允许$ε$比例的正确处理器输出错误结果的松弛协议，并证明当$f < n(1/3 - ε)$时，可以以$O(n \log n)$的通信复杂度确定性求解。关键的是，任何$ε$-BA协议都可以作为完整BA协议的第一阶段：在$ε$-BA之后，通过一次全对全交换和多数表决即可完成BA。由于结果在$ε$-BA之后已经确定，这表明Dolev-Reischuk中的二次成本完全源于传播阶段，而非达到单值性。我们还为认证设置定义了可提取BA，以刻画处理器何时共同持有足够的签名消息来确定一致值，并证明其可以以$O(f \log f)$的通信复杂度求解。