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.