Public communication can facilitate low-risk coordination under surveillance

Consider a sub-population of rebels aiming at initiating a revolution. To avoid initializing a failed revolution, rebels would first strive to estimate their "power", which is often correlated with their number. However, especially in non-democratic countries, rebels avoid disclosing themselves. This poses a significant challenge for rebels: estimating their number while minimizing the risk of being identified as rebels. This paper introduces a distributed computing framework to study this question. Our main insight is that the communication pattern plays a crucial role in achieving such a task. Specifically, we distinguish between public communication, in which each message announced by an individual can be viewed by all its neighbors, and private communication, in which each message is received by one neighbor. We describe a simple protocol in the public communication model that allows rebels to estimate their number while keeping a negligible risk of being identified as rebels. The proposed protocol, inspired by historical events, can be executed covertly even under extreme conditions of surveillance. Conversely, we show that under private communication, protocols of similar simplicity are either inefficient or non-covert. These results suggest that public communication can facilitate the emergence of revolutions in non-democratic countries.