Probabilistic Perfectly Reliable and Secure Message Transmission - Possibility, Feasibility and Optimality

In this paper, we study the problem of probabilistic perfectly reliable message transmission (PPRMT) and probabilistic perfectly secure message transmission (PPSMT) between a sender S and a receiver R in a synchronous network, where S and R are connected by n vertex disjoint paths called wires. Some of these n wires may be corrupted by a mixed adversary (tb, to, tf , tp), having unbounded computing power, who can corrupt tb, to, tf , tp wires in Byzantine, omission, failstop and passive fashion respectively. In spite of the presence of such an adversary, S wants to send a message to R such that after exchanging messages in phases, R should correctly receive S’s message with high probability (probability ≥ 1 − δ, for any 0 < δ < 1/2) and the adversary should have no information about S’s message. We prove the necessary and sufficient condition for the existence of PPRMT and PPSMT protocols tolerating mixed adversary. We give the lower bound on the communication complexity of PPRMT and PPSMT protocols in the presence of mixed adversary and design bit-optimal PPRMT protocol and PPSMT protocols. We also compare our results with the existing results for perfectly reliable message transmission (PRMT), perfectly secure message transmission (PSMT), PPRMT and PPSMT and show that randomness and probabilistic approaches lead to improved communication, phase and computational complexities. Moreover results on mixed adversaries reveal higher level of fault tolerance available in the underlying network.