Bounds and Protocols for Graph-Based Distributed Secret Sharing

Distributed Secret Sharing is a (multi) secret sharing model in which the shares are distributed over storage nodes of a network and each participant is able to reconstruct a specific secret by accessing a subset of the storage nodes. In this work, we provide new Distributed (multi) Secret Sharing Protocols for a specific class of access structures, namely those that can be described with a graph. The protocols improve on previous results allowing a faster encoding and decoding phase while maintaining optimal storage requirements. Moreover, our protocols can manage any kind of graph, while previous protocols have been designed only for complete graphs, and we provide a complete characterization of graph-based protocols. We also prove some tight bounds on the size of the information held in the storage nodes and communication complexity by using an information-theoretic approach. Finally, we also introduce a computationally secure technique for the general case that allows improvements in the size of the needed disk space if secrecy is computational, that is, if the scheme is robust against resource-bounded adversaries.