Secure message transmission in asynchronous networks

In the Perfectly Secure Message Transmission (PSMT) problem, a sender S and a receiver R are part of a distributed network and connected through nn node disjoint paths, also called as wires  , among which at most tt wires are controlled by a static, Byzantine adversary Atstatic, having unbounded computing power. S has a message mm, which S intends to send to R. The challenge is to design a protocol, such that at the end of the protocol, R should correctly output mm without any error (perfect reliability) and Atstatic should not get any   information about mm, whatsoever, in information theoretic sense (perfect security). The problem of Statistically Secure Message Transmission (SSMT) is same as PSMT, except that R should correctly output mm with very high probability. Sayeed and Abu-Amara (1995) [37] have given a PSMT protocol in an asynchronous network tolerating Atstatic, where S and R are connected by n=2t+1n=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 nn wires are directed from S to R, then any PSMT protocol tolerating Atstatic is possible iff n>3tn>3t. Surprisingly, we also prove that even if all the nn wires are bi-directional, then any PSMT protocol in asynchronous network tolerating Atstatic is possible iff n>3tn>3t. This is quite surprising because for synchronous networks, by the results of Dolev et al. (1993) [16], if all the wires are unidirectional (directed from S to R), then PSMT tolerating Atstatic is possible iff n>3tn>3t, where as if all the wires are bi-directional then PSMT tolerating Atstatic is possible iff n>2tn>2t. This shows that asynchrony of the network demands higher connectivity of the network for the existence of PSMT protocols  . Interestingly, we further show that n>2tn>2t wires are necessary and sufficient for the existence of any SSMT protocol in asynchronous network tolerating Atstatic, irrespective of whether the nn wires are unidirectional from S to R or the nn wires are bi-directional. By the results of Franklin and Wright (2000) [18] and Kurosawa and Suzuki (2009) [22], n>2tn>2t wires are necessary and sufficient for the existence of SSMT in synchronous networks, irrespective of whether the nn wires are unidirectional from S to R or the nn wires are bi-directional. This shows that asynchrony of the network does not demand higher connectivity of the network for SSMT protocols.

► Perfectly and statistically secure asynchronous protocols are studied.
► Asynchrony demands higher connectivity for perfectly secure protocols.
► Asynchrony does not affect connectivity for statistically secure protocols.