Consensus Capacity of Noisy Broadcast Channels

TL;DR

This work analyzes Byzantine consensus over memoryless broadcast channels, formalizing a consensus-capacity problem and identifying a natural common channel $W_{V|X}$ as the key structural requirement for consensus. It derives a precise capacity formula, showing that consensus is possible iff the embedded common channel has positive point-to-point capacity, and providing a capacity expression $C_{Byz}=\max_{P_U} \min_{P_{X|U}:W_{V|X}(.|x)=W_{V|X}(.|u)} \min(I(U;Y),I(U;Z))$, with an intuitive interpretation in terms of adversarial input choices that preserve common-channel outputs. The paper analyzes a concrete two-step binary erasure broadcast channel to illustrate the spectrum of behaviors: $C_{Byz}=1-pq$ when $p<1$, and $C_{Byz}=0$ when $p=1$, while the common-channel capacity is $1-p$ and the common-message capacity upper and lower bounds bound the achievable rates. Beyond the two-receiver case, the authors extend to general broadcast channels, discuss randomness considerations, and outline extensions to multiple receivers and open problems on alternative error regimes and interactive settings.

Abstract

We study communication with consensus over a broadcast channel - the receivers reliably decode the sender's message when the sender is honest, and their decoder outputs agree even if the sender acts maliciously. We characterize the broadcast channels which permit this byzantine consensus and determine their capacity. We show that communication with consensus is possible only when the broadcast channel has embedded in it a natural ''common channel'' whose output both receivers can unambiguously determine from their own channel outputs. Interestingly, in general, the consensus capacity may be larger than the point-to-point capacity of the common channel, i.e., while decoding, the receivers may make use of parts of their output signals on which they may not have consensus provided there are some parts (namely, the common channel output) on which they can agree.

Figures (3)

• Figure 1: For any input $x^n$, the outputs of the decoders $g_{\sB}$ and $g_{\sC}$ must agree; furthermore, if $x^n$ is the codeword for some message $m$, then the decoders must output $m$ (both conditions need to hold with high probability).
• Figure 2: Two-step binary erasure broadcast channel. The channel erases in two steps -- with probability $1-p$, both receivers (simultaneously) receive the input symbol unerased; with the remaining probability $p$, the input symbol is passed further through independent binary erasure channels which erase with probability $q$ and whose unerased output symbols acquire a $\tilde{\space}$. The characteristic graph has three connected components (unless $p=1$ when there is only one connected component). The common channel is a binary erasure channel with erasure probability $p$ and erasure symbol $\Delta$.
• Figure 3: An example to show that $C_\textup{Byz}$ could be strictly in between the point-to-point capacity of the common channel and the common message capacity of the broadcast channel. For all values of $p$ except $p=0.5$, $C_\textup{Byz}=1$. For $p=0.5$, when the common channel is trivial, $C_\textup{Byz}=0$. Here $\cU=\{0,1\}\subsetneq\cX$ (except for $p=0.5$ when $\cU$ is a singleton).

Theorems & Definitions (36)

• Remark 1
• Definition 1: Common Channel, Common Channel Output Functions
• Example 1: Two-step binary erasure broadcast channel
• Claim 1
• Remark 2
• Claim 2
• Lemma 3
• Definition 2: Effective Input Alphabet $\cU$ and representation channel $\widetilde{P}_{U|X}$
• Theorem 4
• Remark 3