In the PSMT problem, a sender S and a receiver R are part of a distributed network and connected through n node disjoint paths, also called as wires among which at most t are controlled by an all powerful Byzantine adversary A t. S has a message m, which S intends to send to R. The challenge is to design a protocol, such that at the end, R should correctly output m without any error (perfect reliability) and At should not get any information about m, what so ever, in information theoretic sense (perfect security). The problem of USMT is same as PSMT, except that R should output m with a small probability of error. Sayeed et al. [15] have given a PSMT protocol in an asynchronous network tolerating At, where S and R are connected by n = 2t + 1 wires. However, we show that their protocol does not provide perfect security. We then prove that in an asynchronous network, if all the n wires are directed from S to R, then any PSMT protocol tolerating At is possible iff n > 3t. Surprisingly, we further prove that even if all the n wires are bi-directional, then any PSMT protocol in asynchronous network tolerating At is possible iff n > 3t. This is quite interesting because for synchronous networks, by the results of Dolev et al. [6], if all the wires are unidirectional (directed from S to R), then PSMT tolerating A t is possible iff n > 3t, where as if all the wires are bi-directional then PSMT tolerating At is possible iff n > 2t. This shows that synchrony of the network affects the connectivity requirement for PSMT protocols. However, we show that n > 2t wires are necessary and sufficient for the existence of any USMT protocol in asynchronous network tolerating At, irrespective of whether the n wires are unidirectional from S to R or the n wires are bi-directional. © 2009 Springer.