This short note revisits the problem of designing secure minimum storage regenerating (MSR) codes for distributed storage systems. A secure MSR code ensures that a distributed storage system does not reveal the stored information to a passive eavesdropper. The eavesdropper is assumed to have access to the content stored on @math number of storage nodes in the system and the data downloaded during the bandwidth efficient repair of an additional @math number of storage nodes. This note combines the Gabidulin codes based precoding [18] and a new construction of MSR codes (without security requirements) by Ye and Barg [27] in order to obtain secure MSR codes. Such optimal secure MSR codes were previously known in the setting where the eavesdropper was only allowed to observe the repair of @math nodes among a specific subset of @math nodes [7, 18]. The secure coding scheme presented in this note allows the eavesdropper to observe repair of any @math ouf of @math nodes in the system and characterizes the secrecy capacity of linear repairable MSR codes.
We consider the @math -eavesdropper model introduced in @cite_28 . Let @math nodes in the DSS are indexed by the set @math . For @math and @math such that @math , an @math -eavesdropper can directly access the content stored on any @math storage nodes indexed by the set @math . Additionally, the eavesdropper observes the data downloaded during the repair of any @math storage nodes indexed by the set @math . The nodes indexed by the sets @math and @math are referred to as storage-eavesdropped and download-eavesdropped nodes, respectively. Note that a download-eavesdropped node may reveal more information compared to a storage-eavesdropped node as the content stored on a node is a function of the data downloaded during its repair. In this document we focus on coding schemes that are information theoretically secure against an @math -eavesdropper. We formalize this notion in the following definition.
@cite_28
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In this paper, we revisit the problem of characterizing the secrecy capacity of minimum storage regenerating (MSR) codes under the passive @math -eavesdropper model, where the eavesdropper has access to data stored on @math nodes and the repair data for an additional @math nodes. We study it from the information-theoretic perspective. First, some general properties of MSR codes as well as a simple and generally applicable upper bound on secrecy capacity are given. Second, a new concept of stable MSR codes is introduced, where the stable property is shown to be closely linked with secrecy capacity. Finally, a comprehensive and explicit result on secrecy capacity in the linear MSR scenario is present, which generalizes all related works in the literature and also predicts certain results for some unexplored linear MSR codes.