Round-Optimal Perfectly Secret Message Transmission With Linear Communication Complexity

Consider an arbitrary network of n nodes, up to any t of which are eavesdropped on by an adversary. A sender S wishes to send a message m to a receiver R such that the adversary learns nothing about m (unless it eavesdrops on one among {S, R}). We prove a necessary and sufficient condition on the (synchronous) network for the existence of r-round protocols for perfect communication, for any given r > 0. Our results/protocols are easily adapted to asynchronous networks too and are shown to be optimal in asynchronous "rounds". Further, we show that round-optimality is achieved without trading-off the communication complexity; specifically, our protocols have an overall message complexity of O(n) elements of a finite field to perfectly transmit one field element. Interestingly, optimality (of protocols) also implies: (a) when the shortest path between S and R has O(n) nodes, perfect secrecy is achieved for "free", because any (insecure routing) protocol would also take O(n) rounds and send O(n) messages (one message along each edge in the shortest path) for transmission and (b) it is well-known that (t + 1) vertex disjoint paths from S to R are necessary for a protocol to exist; a consequent folklore is that the length of the (t + 1)th ranked (disjoint shortest) path would dictate the round complexity of protocols; we show that the folklore is false; round-optimal protocols can be substantially faster than the aforementioned length.