Perfectly secure message transmission over partially synchronous networks

In a distributed network, we consider two special nodes called the sender S and the receiver R that are connected by n node-disjoint (except for S and R) bi-directional wires. Out of these n wires, the adversary can control at most t wires (of its choice) in Byzantine fashion. In this setting, our goal is to design a message transmission protocol Π that assures the following two conditions hold: (1) by the end of the protocol Π, R gets the correct message m transmitted by S without any error (perfect reliability), and (2) the adversary learns no information about m, whatsoever, in information theoretic sense (perfect secrecy). Protocols that satisfy these two conditions are known as the Perfectly Secure Message Transmission (PSMT) protocols. However, out of the n wires that exist, if some number of wires say ns, fortunately, happen to be synchronous (serendipitous synchrony) then we ask under what conditions do PSMT protocols tolerating t-Byzantine faults exist. In the literature, it is known that, if either ns > 2t or n > 3t then PSMT protocols trivially exist. Therefore, we consider the case where we have at most 2t synchronous wires (i.e., ns ≤ 2t) and at most 3t wires overall (i.e., n ≤ 3t). Interestingly, we prove that in this case, no PSMT protocol exists. This concludes that, in designing PSMT protocols (tolerating the given fixed number of faults), either (serendipitous) synchronous wires alone are sufficient or we get absolutely no extra advantage of a wire being synchronous over asynchronous.