On the Extremal Source Key Rates for Secure Storage over Graphs

This paper investigates secure storage codes over graphs, where multiple independent source symbols are encoded and stored at graph nodes subject to edge-wise correctness and security constraints. For each edge, a specified subset of source symbols must be recoverable from its two incident nodes, while no information about the remaining sources is revealed. To meet the security requirement, a shared source key may be employed. The ratio between the source symbol size and the source key size defines the source key rate, and the supremum of all achievable rates is referred to as the source key capacity. We study extremal values of the source key capacity in secure storage systems and provide complete graph characterizations for several fundamental settings. For the case where each edge is associated with a single source symbol, we characterize all graphs whose source key capacity equals one. We then generalize this result to the case where each edge is associated with multiple source symbols and identify a broad class of graphs that achieve the corresponding extremal capacity under a mild structural condition. In addition, we characterize all graphs for which secure storage can be achieved without using any source key.