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. An @math maximum distance separable (MDS) array code of length @math , dimension @math , and sub-packetization @math is formed of @math matrices over a finite field @math , with every column of the matrix stored on a separate node in the distributed storage system and viewed as a coordinate of the codeword. Repair of a failed node (recovery of one erased column) can be performed by accessing a set of @math surviving (helper) nodes. The code is said to have the optimal access property if the amount of data accessed at each of the helper nodes meets a lower bound on this quantity. For optimal-access MDS codes with @math , the sub-packetization @math satisfies the bound @math . In our previous work (IEEE Trans. Inf. Theory, vol. 63, no. 4, 2017), for any @math and @math , we presented an explicit construction of optimal-access MDS codes with sub-packetization @math . In this paper, we take up the question of reducing the sub-packetization value @math to make it to approach the lower bound. We construct an explicit family of optimal-access codes with @math , which differs from the optimal value by at most a factor of @math . These codes can be constructed over any finite field @math as long as @math , and afford low-complexity encoding and decoding procedures. We also define a version of the repair problem that bridges the context of regenerating codes and codes with locality constraints (LRC codes), which we call group repair with optimal access . In this variation, we assume that the set of @math nodes is partitioned into @math repair groups of size @math , and require that the amount of accessed data for repair is the smallest possible whenever the @math helper nodes include all the other @math nodes from the same group as the failed node. For this problem, we construct a family of codes with the group optimal access property. These codes can be constructed over any field @math of size @math , and also afford low-complexity encoding and decoding procedures. We address the problem of securing distributed storage systems against passive eavesdroppers that can observe a limited number of storage nodes. An important aspect of these systems is node failures over time, which demand a repair mechanism aimed at maintaining a targeted high level of system reliability. If an eavesdropper observes a node that is added to the system to replace a failed node, it will have access to all the data downloaded during repair, which can potentially compromise the entire information in the system.We are interested in determining the secrecy capacity of distributed storage systems under repair dynamics, i.e., the maximum amount of data that can be securely stored and made available to a legitimate user without revealing any information to any eavesdropper. We derive a general upper bound on the secrecy capacity and show that this bound is tight for the bandwidth-limited regime which is of importance in scenarios such as peer-to-peer distributed storage systems. We also provide a simple explicit code construction that achieves the capacity for this regime. We study a generalization of the setting of regenerating codes, motivated by applications to storage systems consisting of clusters of storage nodes. There are @math clusters in total, with @math nodes per cluster. A data file is coded and stored across the @math nodes, with each node storing @math symbols. For availability of data, we require that the file be retrievable by downloading the entire content from any subset of @math clusters. Nodes represent entities that can fail. We distinguish between intra-cluster and inter-cluster bandwidth (BW) costs during node repair. Node-repair in a cluster is accomplished by downloading @math symbols each from any set of @math other clusters, dubbed remote helper clusters, and also up to @math symbols each from any set of @math surviving nodes, dubbed local helper nodes, in the host cluster. We first identify the optimal trade-off between storage-overhead and inter-cluster repair-bandwidth under functional repair, and also present optimal exact-repair code constructions for a class of parameters. The new trade-off is strictly better than what is achievable via space-sharing existing coding solutions, whenever @math . We then obtain sharp lower bounds on the necessary intra-cluster repair BW to achieve optimal trade-off. Under functional repair, random linear network codes (RLNCs) simultaneously optimize usage of both inter- and intra-cluster repair BW; simulation results based on RLNCs suggest optimality of the bounds on intra-cluster repair-bandwidth. Our bounds reveal the interesting fact that, while it is beneficial to increase the number of local helper nodes @math in order to improve the storage-vs-inter-cluster-repair-BW trade-off, increasing @math not only increases intra-cluster BW in the host-cluster, but also increases the intra-cluster BW in the remote helper clusters. We also analyze resilience of the clustered storage system against passive eavesdropping by providing file-size bounds and optimal code constructions.