Secure communication in minimal connectivity models

Abstract

Problems of secure communication and computation have been studied extensively in network models. In this work, we ask what is possible in the information-theoretic setting when the adversary is very strong (Byzantine) and the network connectivity is very low (minimum needed for crash-tolerance). For some natural models, our results imply a sizable gap between the connectivity required for perfect security and for probabilistic security. Our results also have implications to the commonly studied simple channel model and to general secure multiparty computation.