On the Price of Proactivizing Round-Optimal Perfectly Secret Message Transmission

In a network of n nodes (modeled as a digraph), the goal of a perfectly secret message transmission (PSMT) protocol is to replicate sender's message m at the receiver's end without revealing any information about m to a computationally unbounded adversary that eavesdrops on any t nodes. The adversary may be mobile too that is, it may eavesdrop on a different set of t nodes in different rounds. We prove a necessary and sufficient condition on the synchronous network for the existence of r-round PSMT protocols, for any given r > 0; further, we show that round-optimality is achieved without trading-off the communication complexity; specifically, our protocols have an overall communication complexity of O(n) elements of a finite field to perfectly transmit one field element. Apart from optimality/scalability, two interesting implications of our results are: 1) adversarial mobility does not affect its tolerability: PSMT tolerating a static t-adversary is possible if and only if PSMT tolerating mobile t-adversary is possible; and 2) mobility does not affect the round optimality: the fastest PSMT protocol tolerating a static t-adversary is not faster than the one tolerating a mobile t-adversary.