Multivalued Consensus: General Adversaries Require More Communication

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.