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.

翻译：本文研究图上的安全存储编码问题，其中多个独立信源符号被编码并存储于图节点，同时满足边级正确性与安全性约束。对于每条边，必须能够从其两个关联节点恢复出指定的信源符号子集，而其余信源的信息不得泄露。为满足安全性要求，可采用共享的源密钥。信源符号尺寸与源密钥尺寸之比定义为源密钥率，所有可达速率的上确界称为源密钥容量。我们研究安全存储系统中源密钥容量的极值，并对若干基本场景给出完整的图结构刻画。针对每条边关联单个信源符号的情形，我们刻画了所有源密钥容量等于一的图结构。随后将此结果推广至每条边关联多个信源符号的情形，在温和的结构条件下识别出一大类能达到相应极值容量的图。此外，我们完整刻画了无需使用任何源密钥即可实现安全存储的所有图结构。