Security in Locally Repairable Storage

In this paper we extend the notion of locally repairable codes to secret sharing schemes. The main problem that we consider is to find optimal ways to distribute shares of a secret among a set of storage-nodes (participants) such that the content of each node (share) can be recovered by using contents of only few other nodes, and at the same time the secret can be reconstructed by only some allowable subsets of nodes. As a special case, an eavesdropper observing some set of specific nodes (such as less than certain number of nodes) does not get any information. In other words, we propose to study a locally repairable distributed storage system that is secure against a passive eavesdropper that can observe some subsets of nodes. We provide a number of results related to such systems including upper-bounds and achievability results on the number of bits that can be securely stored with these constraints. In particular, we provide conditions under which a locally repairable code can be turned into a secret sharing scheme and extend the results of secure repairable storage to cooperative repair and storage on networks. Additionally, we consider perfect secret sharing schemes over general access structures under locality constraints and give an example of a perfect secret sharing scheme that can have small locality. Lastly, we provide a lower bound on the size of a share compared to the size of the secret that shows how locality affects the sizes of shares in a perfect scheme.