On Minimal Connectivity Requirement for Secure Message Transmission in Asynchronous Networks

• Authors: Ashish Choudhary;Arpita Patra;B. V. Ashwinkumar;Kannan Srinathan;C. Pandu Rangan
• Venue: ICDCN '09 Proceedings of the 10th International Conference on Distributed Computing and Networking
• Year: 2009

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 ${\mathcal{A}_t}$. 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 ${\mathcal{A}_t}$ 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 ${\mathcal A}_t$, 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 ${\mathcal{A}_t}$ 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 ${\mathcal{A}_t}$ 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 ${\mathcal{A}_t}$ is possible iff n 3t , where as if all the wires are bi-directional then PSMT tolerating ${\mathcal{A}_t}$ 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 ${\mathcal{A}_t}$, irrespective of whether the n wires are unidirectional from S to R or the n wires are bi-directional.