AI 中文总结
该研究针对一般敌手，基于有限射影几何构造敌手结构，得出同步与异步网络中多类共识问题的通信下界，还设计非终止可靠广播协议并证明下界紧性。
AI 中文摘要
我们研究针对一般（非阈值）敌手的n方容错共识问题。我们描述了一个基于有限射影几何的无限族Z_proj^{n,d}，它是满足Q^d条件的n方敌手结构，会使针对L位输入的交互式一致性的无差错R轮协议需要Ω(Ln^{2+1/d})位的期望通信量。同样，Z_proj^{n,d}会使无差错拜占庭协议和广播的通信成本达到Ω(Ln^{1+1/d})位。在所有情况下，下界都是Ω(L_out · n^{1+1/d})位，其中L_out是输出长度。该族Z_proj^{n,d}还会使异步网络中的可靠广播和拜占庭协议的期望通信成本达到Ω(Ln^{1+1/d})位。此外，存在一个相关的族Z_2-proj^{n,d}，它是满足Q^d条件的敌手结构，会使核心集协议的成本达到Ω(Ln^{2+1/d})位。这些异步下界针对发送-遗漏敌手成立，即使协议使用密码学也如此。其基础是：如果一个非故障方的仲裁组就某个输出达成一致并终止，那么它们终止前发送的消息必须足以让仲裁组外的各方也以相同输出终止。令人惊讶的是，如果我们不要求各方在输出后终止（停止发送消息），那么这些下界就不再成立。我们通过设计一种非终止的、可容忍遗漏的可靠广播协议来证明这一点，该协议可针对任意参数δ>1调整为成本(1 + 1/(δ - 1))Ln + O(δn²log(δn))位，这本身就具有独立意义。最后，我们展示如何在O(Ln^{1+1/d} + n²logn)位的成本下实现终止（假设Q^d条件成立），从而证明我们的异步下界是紧的。
英文摘要
We study $n$-party fault-tolerant consensus against general (non-threshold) adversaries. We describe an infinite family $Z_\mathsf{proj}^{n,d}$ of $Q^d$-satisfying $n$-party adversary structures based on finite projective geometry which cause error-free $R$-round protocols for interactive consistency on $L$-bit inputs to require $Ω(Ln^{2+1/d})$ bits of expected communication. Likewise, $Z_\mathsf{proj}^{n,d}$ causes error-free byzantine agreement and broadcast to cost $Ω(Ln^{1+1/d})$ bits. In every case, the lower bound is $Ω(L_{\mathsf{out}} \cdot n^{1+1/d})$ bits, where $L_{\mathsf{out}}$ is the output length. The family $Z_\mathsf{proj}^{n,d}$ also causes reliable broadcast and byzantine agreement to cost $Ω(Ln^{1+1/d})$ bits of expected communication in asynchronous networks. Moreover, there exists a related family $Z_\mathsf{2\textsf-proj}^{n,d}$ of $Q^d$-satisfying adversary structures that make core set agreement cost $Ω(Ln^{2+1/d})$ bits. These asynchronous lower bounds hold against send-omission adversaries, even if the protocol uses cryptography. Their basis is that if a quorum of non-faulty parties agree on an output and terminate, then the messages they sent before terminating must suffice for the parties outside the quorum to also terminate with the same output. Surprisingly, if we do not require the parties to terminate (stop sending messages) after they output, then these bounds no longer hold. We show this by designing a non-terminating omission-tolerant reliable broadcast protocol that can for any parameter $δ> 1$ be tuned to cost $(1 + \frac{1}{δ- 1})Ln + O(δn^2\log(δn))$ bits, which is of independent interest. Lastly, we show how to get termination with $O(Ln^{1+1/d} + n^2\log n)$ bits (assuming the $Q^d$ condition), and thus prove our asynchronous lower bounds tight.
Comments33 pages, 1 figure, full version of a DISC 2026 paper
