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CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
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IEEE Transactions on Information Theory
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In Secure Message Transmission (SMT) problem, a sender $\cal S$ is connected to a receiver $\cal R$ through N node disjoint bidirectional paths in the network, t of which are controlled by an adversary with unlimited computational power. $\cal{S}$ wants to send a message m to $\cal{R}$ in a reliable and private way. It is proved that SMT is possible if and only if N≥2t+1. In Reliable Message Transmission (RMT) problem, the network setting is the same and the goal is to provide reliability for communication, only. In this paper we focus on 1-round δ-RMT and (0,δ)-SMT where the chance of protocol failure (receiver cannot decode the sent message) is at most δ, and in the case of SMT, privacy is perfect. We propose a new approach to the construction of 1-round δ-RMT and (0, δ)-SMT for all connectivities N≥2t+1, using list decodable codes and message authentication codes. Our concrete constructions use folded Reed-Solomon codes and multireceiver message authentication codes. The protocols have optimal transmission rates and provide the highest reliability among all known comparable protocols. Important advantages of these constructions are, (i) they can be adapted to all connectivities, and (ii) have simple and direct security (privacy and reliability) proofs using properties of the underlying codes, and δ can be calculated from parameters of the underlying codes. We discuss our results in relation to previous work in this area and propose directions for future research.