Constant Phase Bit Optimal Protocols for Perfectly Reliable and Secure Message Transmission Tolerating Static and Mobile Mixed Adversary

We study the fundamental problem of perfectly secure message transmission (PSMT), where two non-faulty players, the sender S and the receiver R are part of a synchronous network modeled as an undirected graph, a part of which may be under the influence of a malicious unbounded computational powerful adversary who can disrupt the communication in variety of ways. S intends to transmit a message m to R using some protocol such that after interacting in phases as per the protocol R should correctly obtain S’s message while adversary should be completely ignorant about the same. The problem of perfectly reliable message transmission (PRMT), requires only correctness of the message received by R. However, unlike most of the literature on the problem, we assume the adversary modeling the faults is mixed (and can corrupt different set of nodes in Byzantine, failstop and passive fashion simultaneously) and thus versatile enough to disrupt the information flow in the network in variety of ways. The mixed adversary can be either static or mobile. The communication complexity of multiphase interactive PRMT and PSMT protocols is one of their most important complexity measure. In this work we look into the optimal communication complexity of PRMT and PSMT problem considering static and mobile mixed adversary. We observe that (a) Under Byzantine Adversary, the communication complexity lower bounds for PRMT and PSMT problems are “same” for both mobile and static case; (b) Under Mixed Adversary, the communication complexity lower bounds for PSMT problem are “same” for both mobile and static case; (c) However under mixed adversary, it is curious and interesting to note that communication complexity lower bounds are “different” for mobile and static case. So, we prove the lower bound for PRMT problem considering mobile mixed adversary and provide a three phase protocol whose communication complexity matches with the bound. Our other contribution includes a worst case four phase communication optimal PSMT protocol tolerating static mixed adversary which is a clear improvement over the existing log tf phase protocol. Then we present an elegant worst case nine phase communication optimal PSMT protocol tolerating mobile mixed adversary which is first of it’s kind. Finally, we also show that the connectivity requirement for PRMT and PSMT tolerating static mixed adversary remains same for PRMT and PSMT tolerating mobile mixed adversary. This proves that as far as the possibility and optimality in communication of PSMT problem against mixed adversary is concerned, the additional power of mobility of adversary has no effect.