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

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