Perfectly Secure Message Transmission in Directed Networks Tolerating Threshold and Non Threshold Adversary

In this paper we study Perfectly Secure Message Transmission (PSMT) between a sender S and a receiver R, connected in a directed synchronous network through multiple parallel edges (called wires), each of which are directed from S to R or vice-versa. The unreliability of the network is modeled by an Byzantine adversary with infinite power. Thus the security we gain is information theoretic. We investigate the problem with two different adversarial flavors: (i)threshold: number of nodes corrupted by a Byzantine adversary is parametrized by t and (ii) non-threshold: the behavior of the adversary is captured by an adversary structure which is the snapshots of adversary at different executions. In [3], the authors proved the necessity and sufficiency of at least max(3t− 2u + 1, 2t + 1) wires from S to R for any PSMT protocol in such a network, where u is the number of wires from R to S. However, their PSMT protocol was exponential both in terms of number of phases and communication complexity. In addition, they also presented a polynomial phase PSMT protocol with n′ = max(3t − u + 1, 2t + 1) wires from S to R. In this paper, we significantly improve the exponential phase protocol and present an elegant and efficient three phase PSMT protocol with polynomial communication complexity (and computational complexity) with n = max(3t − 2u + 1, 2t + 1) wires from S to R. Also with n′ = max(3t − u + 1, 2t + 1) wires from S to R, we are able to further improve the communication complexity of our three ? A brief version of this paper appeared in [17] which permits full expanded version to appear elsewhere ?? Work Supported by Project No. CSE/05-06/076/DITX/CPAN on Protocols for Secure Communication and Computation Sponsored by Department of Information Technology, Government of India. ? ? ? Work Supported by Project No. CSE/05-06/076/DITX/CPAN on Protocols for Secure Communication and Computation Sponsored by Department of Information Technology, Government of India. 3 A phase is a send from S to R or R to S phase PSMT protocol. This is a significant improvement over the protocols presented in [3]. Our second contribution in this paper is the first ever characterization for any two phase PSMT protocol between S and R in such a network. Finally, we also characterize PSMT protocol in directed networks tolerating non-threshold adversary. In [5], the authors have given the characterization for PSMT against non-threshold adversary. However, in their characterization, they have only considered the paths from S to R, excluding the feedback paths (i.e paths from R to S) and hence their characterization holds good for single phase protocols. We characterize multiphase PSMT considering feedback paths.