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.
This paper studies the problem of repairing secret sharing schemes, i.e., schemes that encode a message into $n$ shares, assigned to $n$ nodes, so that any $n-r$ nodes can decode the message but any colluding $z$ nodes cannot infer any information about the message. In the event of node failures so that shares held by the failed nodes are lost, the system needs to be repaired by reconstructing and reassigning the lost shares to the failed (or replacement) nodes. This can be achieved trivially by a trustworthy third-party that receives the shares of the available nodes, recompute and reassign t…
The newly presented $(k+2,k)$ Hadamard minimum storage regenerating (MSR) code is the first class of high rate storage code with optimal repair property for all single node failures. In this paper, we propose a new simple repair strategy, which can considerably reduces the computation load of the node repair in contrast to the original one.
In wireless distributed storage systems, storage nodes are connected by wireless channels, which are broadcast in nature. This paper exploits this unique feature to design an efficient repair mechanism, called broadcast repair, for wireless distributed storage systems in the presence of multiple-node failures. Due to the broadcast nature of wireless transmission, we advocate a new measure on repair performance called repair-transmission bandwidth. In contrast to repair bandwidth, which measures the average number of packets downloaded by a newcomer to replace a failed node, repair-transmission…
In this letter, we provide the construction of repairable fountain codes (RFCs) for distributed storage systems that are information-theoretically secure against an eavesdropper that has access to the data stored in a subset of the storage nodes and the data downloaded to repair an additional subset of storage nodes. The security is achieved by adding random symbols to the message, which is then encoded by the concatenation of a Gabidulin code and an RFC. We compare the achievable code rates of the proposed codes with those of secure minimum storage regenerating codes and secure locally repair…
In wireless distributed storage systems, storage nodes are connected by wireless channels, which are broadcast in nature. This paper exploits this unique feature to design an efficient repair mechanism, called broadcast repair, for wireless distributed storage systems with multiple-node failures. Since wireless channels are typically bandwidth limited, we advocate a new measure on repair performance called repair-transmission bandwidth, which measures the average number of packets transmitted by helper nodes per failed node. The fundamental tradeoff between storage amount and repair-transmissi…
Reliability is essential for storing files in many applications of distributed storage systems. To maintain reliability, when a storage node fails, a new node should be regenerated by a repair process. Most of the previous results on the repair problem assume perfect (error-free) links in the networks. However, in practice, especially in a wireless network, the transmitted packets (for repair) may be lost due to, e.g., link failure or buffer overflow. We study the repair problem of distributed storage systems in packet erasure networks, where a packet loss is modeled as an erasure. The minimum…
In this paper, we find the Laplace Stieltjes transform of the probability of data loss for the k-out-of-n distributed storage system with deterministic repair times. We consider two repair models, namely the serial and parallel repair. We show that for failure rate much lower than the repair rate, mean time of data loss for the two models is the same unlike the case for exponential repair models.
We study the exact-repair tradeoff between storage and repair bandwidth in distributed storage systems (DSS). We give new inner bounds for the tradeoff region and provide code constructions that achieve these bounds.
The {\em repair locality} of a distributed storage code is the maximum number of nodes that ever needs to be contacted during the repair of a failed node. Having small repair locality is desirable, since it is proportional to the number of disk accesses during repair. However, recent publications show that small repair locality comes with a penalty in terms of code distance or storage overhead if exact repair is required. Here, we first review some of the main results on storage codes under various repair regimes and discuss the recent work on possible (information-theoretical) trade-offs betw…
In this paper, we consider heterogeneous distributed storage systems (DSSs) having flexible reconstruction degree, where each node in the system has dynamic repair bandwidth and dynamic storage capacity. In particular, a data collector can reconstruct the file at time $t$ using some arbitrary nodes in the system and for a node failure the system can be repaired by some set of arbitrary nodes. Using $min$-$cut$ bound, we investigate the fundamental tradeoff between storage and repair cost for our model of heterogeneous DSS. In particular, the problem is formulated as bi-objective optimization l…
In this paper we establish an improved outer bound on the storage-repair-bandwidth tradeoff of regenerating codes under exact repair. The result shows that in particular, it is not possible to construct exact-repair regenerating codes that asymptotically achieve the tradeoff that holds for functional repair. While this had been shown earlier by Tian for the special case of $[n,k,d]=[4,3,3]$ the present result holds for general $[n,k,d]$. The new outer bound is obtained by building on the framework established earlier by Shah et al.
We study the trade-off between storage overhead and inter-cluster repair bandwidth in clustered storage systems, while recovering from multiple node failures within a cluster. A cluster is a collection of $m$ nodes, and there are $n$ clusters. For data collection, we download the entire content from any $k$ clusters. For repair of $t \geq 2$ nodes within a cluster, we take help from $\ell$ local nodes, as well as $d$ helper clusters. We characterize the optimal trade-off under functional repair, and also under exact repair for the minimum storage and minimum inter-cluster bandwidth (MBR) opera…
This paper aims to go beyond resilience into the study of security and local-repairability for distributed storage systems (DSS). Security and local-repairability are both important as features of an efficient storage system, and this paper aims to understand the trade-offs between resilience, security, and local-repairability in these systems. In particular, this paper first investigates security in the presence of colluding eavesdroppers, where eavesdroppers are assumed to work together in decoding stored information. Second, the paper focuses on coding schemes that enable optimal local repa…
The problem of exact-repair regenerating codes against eavesdropping attack is studied. The eavesdropping model we consider is that the eavesdropper has the capability to observe the data involved in the repair of a subset of $\ell$ nodes. An $(n,k,d,\ell)$ secure exact-repair regenerating code is an $(n,k,d)$ exact-repair regenerating code that is secure under this eavesdropping model. It has been shown that for some parameters $(n,k,d,\ell)$, the associated optimal storage-bandwidth tradeoff curve, which has one corner point, can be determined. The focus of this paper is on characterizing su…
Modern distributed storage systems apply redundancy coding techniques to stored data. One form of redundancy is based on regenerating codes, which can minimize the repair bandwidth, i.e., the amount of data transferred when repairing a failed storage node. Existing regenerating codes mainly require surviving storage nodes encode data during repair. In this paper, we study functional minimum storage regenerating (FMSR) codes, which enable uncoded repair without the encoding requirement in surviving nodes, while preserving the minimum repair bandwidth guarantees and also minimizing disk reads. U…
In this paper, we consider the setting of exact repair linear regenerating codes. Under this setting, we derive a new outer bound on the storage-repair-bandwidth trade-off for the case when $d = k = n -1$, where $(n, k, d)$ are parameters of the regenerating code, with their usual meaning. Taken together with the achievability result of Tian et. al. [1], we show that the new outer bound derived here completely characterizes the trade-off for the case of exact repair linear regenerating codes, when $d = k = n -1$. The new outer bound is derived by analyzing the dual code of the linear regenerat…
Distributed storage systems and associated storage codes can efficiently store a large amount of data while ensuring that data is retrievable in case of node failure. The study of such systems, particularly the design of storage codes over finite fields, assumes that the physical channel through which the nodes communicate is error-free. This is not always the case, for example, in a wireless storage system. We study the probability that a subpacket is repaired incorrectly during node repair in a distributed storage system, in which the nodes communicate over an AWGN or Rayleigh fading channel…
A major issue of locally repairable codes is their robustness. If a local repair group is not able to perform the repair process, this will result in increasing the repair cost. Therefore, it is critical for a locally repairable code to have multiple repair groups. In this paper we consider robust locally repairable coding schemes which guarantee that there exist multiple distinct (not necessarily disjoint) alternative local repair groups for any single failure such that the failed node can still be repaired locally even if some of the repair groups are not available. We use linear programming…
A new system model reflecting the clustered structure of distributed storage is suggested to investigate interplay between storage overhead and repair bandwidth as storage node failures occur. Large data centers with multiple racks/disks or local networks of storage devices (e.g. sensor network) are good applications of the suggested clustered model. In realistic scenarios involving clustered storage structures, repairing storage nodes using intact nodes residing in other clusters is more bandwidth-consuming than restoring nodes based on information from intra-cluster nodes. Therefore, it is i…
A new system model reflecting the clustered structure of distributed storage is suggested to investigate bandwidth requirements for repairing failed storage nodes. Large data centers with multiple racks/disks or local networks of storage devices (e.g. sensor network) are good applications of the suggested clustered model. In realistic scenarios involving clustered storage structures, repairing storage nodes using intact nodes residing in other clusters is more bandwidth-consuming than restoring nodes based on information from intra-cluster nodes. Therefore, it is important to differentiate bet…
We study the repair problem of distributed storage systems in erasure networks where the packets transmitted from surviving nodes to the new node might be lost. The fundamental storage-bandwidth tradeoff is calculated by multicasting analysis in erasure networks. The optimal tradeoff bound can be asymptotically achieved when the number of transmission (packets) goes to infinity. For a limited number of transmission, we study the probability of successful regenerating. Then, we investigate two approaches of increasing the probability of successful regenerating, namely, by connecting more surviv…
The optimal tradeoff between node storage and repair bandwidth is an important issue for distributed storage systems (DSSs). As for realistic DSSs with clusters, when repairing a failed node, it is more efficient to download more data from intra-cluster nodes than from cross-cluster nodes. Therefore, it is meaningful to differentiate the repair bandwidth from intra-cluster and cross-cluster. For cluster DSSs the tradeoff has been considered with special repair assumptions where all the alive nodes are utilized to repair a failed node. In this paper, we investigate the optimal tradeoff for clus…
The high repair cost of (n,k) Maximum Distance Separable (MDS) erasure codes has recently motivated a new class of codes, called Regenerating Codes, that optimally trade off storage cost for repair bandwidth. In this paper, we address bandwidth-optimal (n,k,d) Exact-Repair MDS codes, which allow for any failed node to be repaired exactly with access to arbitrary d survivor nodes, where k<=d<=n-1. We show the existence of Exact-Repair MDS codes that achieve minimum repair bandwidth (matching the cutset lower bound) for arbitrary admissible (n,k,d), i.e., k<n and k<=d<=n-1. Our approach is based…
The max-flow outer bound is achievable by regenerating codes for functional repair distributed storage system. However, the capacity of exact repair distributed storage system is an open problem. In this paper, the linear programming bound for exact repair distributed storage systems is formulated. A notion of symmetrical sets for a set of random variables is given and equalities of joint entropies for certain subsets of random variables in a symmetrical set is established. Concatenation coding scheme for exact repair distributed storage systems is proposed and it is shown that concatenation c…
In order to provide high data reliability, distributed storage systems disperse data with redundancy to multiple storage nodes. Regenerating codes is a new class of erasure codes to introduce redundancy for the purpose of improving the data repair performance in distributed storage. Most of the studies on regenerating codes focus on the single-failure recovery, but it is not uncommon to see two or more node failures at the same time in large storage networks. To exploit the opportunity of repairing multiple failed nodes simultaneously, a cooperative repair mechanism, in the sense that the node…
We present a precise characterization of linear functional-repair storage codes in terms of {\em admissible states/}, with each state made up from a collection of vector spaces over some fixed finite field. To illustrate the usefulness of our characterization, we provide several applications. We first describe a simple construction of functional-repair storage codes for a family of code parameters meeting the cutset bound outside the MBR and MSR points; these codes are conjectured to have optimal rate with respect to their repair locality. Then, we employ our characterization to develop a cons…
Erasure-correcting codes, that support local repair of codeword symbols, have attracted substantial attention recently for their application in distributed storage systems. This paper investigates a generalization of the usual locally repairable codes. In particular, this paper studies a class of codes with the following property: any small set of codeword symbols can be reconstructed (repaired) from a small number of other symbols. This is referred to as cooperative local repair. The main contribution of this paper is bounds on the trade-off of the minimum distance and the dimension of such c…
In a distributed storage system, the storage costs of different storage nodes, in general, can be different. How to store a file in a given set of storage nodes so as to minimize the total storage cost is investigated. By analyzing the min-cut constraints of the information flow graph, the feasible region of the storage capacities of the nodes can be determined. The storage cost minimization can then be reduced to a linear programming problem, which can be readily solved. Moreover, the tradeoff between storage cost and repair-bandwidth is established.
We consider distributed storage (DS) for a wireless network where mobile devices arrive and depart according to a Poisson random process. Content is stored in a number of mobile devices, using an erasure correcting code. When requesting a piece of content, a user retrieves the content from the mobile devices using device-to-device communication or, if not possible, from the base station (BS), at the expense of a higher communication cost. We consider the repair problem when a device that stores data leaves the network. In particular, we introduce a repair scheduling where repair is performed (…
Distributed storage codes have important applications in the design of modern storage systems. In a distributed storage system, every storage node has a probability to fail and once an individual storage node fails, it must be reconstructed using data stored in the surviving nodes. Computation load and network bandwidth are two important issues we need to concern when repairing a failed node. The traditional maximal distance separable (MDS) storage codes have low repair complexity but high repair bandwidth. On the contrary, minimal storage regenerating (MSR) codes have low repair bandwidth but…
We study a generalization of the setting of regenerating codes, motivated by applications to storage systems consisting of clusters of storage nodes. There are $n$ clusters in total, with $m$ nodes per cluster. A data file is coded and stored across the $mn$ nodes, with each node storing $α$ symbols. For availability of data, we require that the file be retrievable by downloading the entire content from any subset of $k$ 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 a…
The problem of exact repair of a failed node in multi-hop networked distributed storage systems is considered. Contrary to the most of the current studies which model the repair process by the direct links from surviving nodes to the new node, the repair is modeled by considering the multi-hop network structure, and taking into account that there might not exist direct links from all the surviving nodes to the new node. In the repair problem of these systems, surviving nodes may cooperate to transmit the repair traffic to the new node. In this setting, we define the total number of packets tra…
In distributed storage systems reliability is achieved through redundancy stored at different nodes in the network. Then a data collector can reconstruct source information even though some nodes fail. To maintain reliability, an autonomous and efficient protocol should be used to repair the failed node. The repair process causes traffic and consequently transmission cost in the network. Recent results found the optimal trafficstorage tradeoff, and proposed regenerating codes to achieve the optimality. We aim at minimizing the transmission cost in the repair process. We consider the network to…
We introduce the Fixed Cluster Repair System (FCRS) as a novel architecture for Distributed Storage Systems (DSS), achieving a small repair bandwidth while guaranteeing a high availability. Specifically we partition the set of servers in a DSS into $s$ clusters and allow a failed server to choose any cluster other than its own as its repair group. Thereby, we guarantee an availability of $s-1$. We characterize the repair bandwidth vs. storage trade-off for the FCRS under functional repair and show that the minimum repair bandwidth can be improved by an asymptotic multiplicative factor of $2/3$…
High availability of containerized applications requires to perform robust storage of applications' state. Since basic replication techniques are extremely costly at scale, storage space requirements can be reduced by means of erasure or repairing codes. In this paper we address storage regeneration using repair codes, a robust distributed storage technique with no need to fully restore the whole state in case of failure. In fact, only the lost servers' content is replaced. To do so, new cleanslate storage units are made operational at a cost for activating new storage servers and a cost for t…
Cooperative regenerating codes are designed for repairing multiple node failures in distributed storage systems. In contrast to the original repair model of regenerating codes, which are for the repair of single node failure, data exchange among the new nodes is enabled. It is known that further reduction in repair bandwidth is possible with cooperative repair. Currently in the literature, we have an explicit construction of exact-repair cooperative code achieving all parameters corresponding to the minimum-bandwidth point. We give a slightly generalized and more flexible version of this coope…
Self-Repairing Codes (SRC) are codes designed to suit the need of coding for distributed networked storage: they not only allow stored data to be recovered even in the presence of node failures, they also provide a repair mechanism where as little as two live nodes can be contacted to regenerate the data of a failed node. In this paper, we propose a new instance of self-repairing codes, based on constructions of spreads coming from projective geometry. We study some of their properties to demonstrate the suitability of these codes for distributed networked storage.
In coding for distributed storage systems, efficient data reconstruction and repair through accessing a predefined number of arbitrarily chosen storage nodes is guaranteed by regenerating codes. Traditionally, code parameters, specially the number of helper nodes participating in a repair process, are predetermined. However, depending on the state of the system and network traffic, it is desirable to adapt such parameters accordingly in order to minimize the cost of repair. In this work a class of regenerating codes with minimum storage is introduced that can simultaneously operate at the opti…
We present the construction of a family of erasure correcting codes for distributed storage that achieve low repair bandwidth and complexity at the expense of a lower fault tolerance. The construction is based on two classes of codes, where the primary goal of the first class of codes is to provide fault tolerance, while the second class aims at reducing the repair bandwidth and repair complexity. The repair procedure is a two- step procedure where parts of the failed node are repaired in the first step using the first code. The downloaded symbols during the first step are cached in the memory…
The reliability of erasure-coded distributed storage systems, as measured by the mean time to data loss (MTTDL), depends on the repair bandwidth of the code. Repair-efficient codes provide reliability values several orders of magnitude better than conventional erasure codes. Current state of the art codes fix the number of helper nodes (nodes participating in repair) a priori. In practice, however, it is desirable to allow the number of helper nodes to be adaptively determined by the network traffic conditions. In this work, we propose an opportunistic repair framework to address this issue. I…
In this paper, three outer bounds on the normalized storage-repair bandwidth (S-RB) tradeoff of regenerating codes having parameter set $\{(n,k,d),(α,β)\}$ under the exact-repair (ER) setting are presented. The first outer bound is applicable for every parameter set $(n,k,d)$ and in conjunction with a code construction known as {\em improved layered codes}, it characterizes the normalized ER tradeoff for the case $(n,k=3,d=n-1)$. It establishes a non-vanishing gap between the ER and functional-repair (FR) tradeoffs for every $(n,k,d)$. The second bound is an improvement upon an existing bound …
In a realistic distributed storage environment, storage nodes are usually placed in racks, a metallic support designed to accommodate electronic equipment. It is known that the communication (bandwidth) cost between nodes within a rack is much lower than the communication (bandwidth) cost between nodes within different racks. In this paper, a new model, where the storage nodes are placed in two racks, is proposed and analyzed. In this model, the storage nodes have different repair costs to repair a node depending on the rack where they are placed. A threshold function, which minimizes the amou…
In this paper, a new repair scheme for a modified construction of MDS codes is studied. The obtained repair scheme has optimal bandwidth for multiple failed nodes under the cooperative repair model. In addition, the repair scheme has relatively low access property, where the number of data accessed is less than two times the optimal value.
The high repair cost of (n,k) Maximum Distance Separable (MDS) erasure codes has recently motivated a new class of codes, called Regenerating Codes, that optimally trade off storage cost for repair bandwidth. On one end of this spectrum of Regenerating Codes are Minimum Storage Regenerating (MSR) codes that can match the minimum storage cost of MDS codes while also significantly reducing repair bandwidth. In this paper, we describe Exact-MSR codes which allow for any failed nodes (whether they are systematic or parity nodes) to be regenerated exactly rather than only functionally or informatio…
We consider the design of regenerating codes for distributed storage systems that enjoy the property of local, exact and uncoded repair, i.e., (a) upon failure, a node can be regenerated by simply downloading packets from the surviving nodes and (b) the number of surviving nodes contacted is strictly smaller than the number of nodes that need to be contacted for reconstructing the stored file. Our codes consist of an outer MDS code and an inner fractional repetition code that specifies the placement of the encoded symbols on the storage nodes. For our class of codes, we identify the tradeoff b…
We introduce a new class of exact Minimum-Bandwidth Regenerating (MBR) codes for distributed storage systems, characterized by a low-complexity uncoded repair process that can tolerate multiple node failures. These codes consist of the concatenation of two components: an outer MDS code followed by an inner repetition code. We refer to the inner code as a Fractional Repetition code since it consists of splitting the data of each node into several packets and storing multiple replicas of each on different nodes in the system. Our model for repair is table-based, and thus, differs from the random…
Repair of multiple partially failed cache nodes is studied in a distributed wireless content caching system, where $r$ out of a total of $n$ cache nodes lose part of their cached data. Broadcast repair of failed cache contents at the network edge is studied; that is, the surviving cache nodes transmit broadcast messages to the failed ones, which are then used, together with the surviving data in their local cache memories, to recover the lost content. The trade-off between the storage capacity and the repair bandwidth is derived. It is shown that utilizing the broadcast nature of the wireless …
We consider a set up where a file of size M is stored in n distributed storage nodes, using an (n,k) minimum storage regenerating (MSR) code, i.e., a maximum distance separable (MDS) code that also allows efficient exact-repair of any failed node. The problem of interest in this paper is to minimize the repair bandwidth B for exact regeneration of a single failed node, i.e., the minimum data to be downloaded by a new node to replace the failed node by its exact replica. Previous work has shown that a bandwidth of B=[M(n-1)]/[k(n-k)] is necessary and sufficient for functional (not exact) regene…
In this paper, we propose locally repairable codes (LRCs) with optimal minimum distance for distributed storage systems (DSS). A two-layer encoding structure is employed to ensure data reconstruction and the designated repair locality. The data is first encoded in the first layer by any existing maximum distance separable (MDS) codes, and then the encoded symbols are divided into non-overlapping groups and encoded by an MDS array code in the second layer. The encoding in the second layer provides enough redundancy for local repair, while the overall code performs recovery of the data based on …
Regenerating codes are a class of recently developed codes for distributed storage that, like Reed-Solomon codes, permit data recovery from any subset of k nodes within the n-node network. However, regenerating codes possess in addition, the ability to repair a failed node by connecting to an arbitrary subset of d nodes. It has been shown that for the case of functional-repair, there is a tradeoff between the amount of data stored per node and the bandwidth required to repair a failed node. A special case of functional-repair is exact-repair where the replacement node is required to store data…
In multiple descriptions on distributed storage, a source is stored in a shared fashion on multiple servers. When a subset of servers are contacted, the source should be estimated with a certain maximum distortion depending on the number of servers. The problem considered in this paper is how to restore the system operation when one of the servers fail and a new server replaces it, that is, repair. The requirement is that the distortions in the restored system should be no more than in the original system. The question is how many extra bits are needed for repair. We find the optimum solution …
The problem of securing data against eavesdropping in distributed storage systems is studied. The focus is on systems that use linear codes and implement exact repair to recover from node failures.The maximum file size that can be stored securely is determined for systems in which all the available nodes help in repair (i.e., repair degree $d=n-1$, where $n$ is the total number of nodes) and for any number of compromised nodes. Similar results in the literature are restricted to the case of at most two compromised nodes. Moreover, new explicit upper bounds are given on the maximum secure file …
Distributed storage systems in the presence of a wiretapper are considered. A distributed storage system (DSS) is parameterized by three parameters (n, k,d), in which a file stored across n distributed nodes, can be recovered from any k out of n nodes. If a node fails, any d out of (n-1) nodes help in the repair of the failed node. For such a (n,k,d)-DSS, two types of wiretapping scenarios are investigated: (a) Type-I (node) adversary which can wiretap the data stored on any l<k nodes; and a more severe (b) Type-II (repair data) adversary which can wiretap the contents of the repair data that …
We study security in partial repair in wireless caching networks where parts of the stored packets in the caching nodes are susceptible to be erased. Let us denote a caching node that has lost parts of its stored packets as a sick caching node and a caching node that has not lost any packet as a healthy caching node. In partial repair, a set of caching nodes (among sick and healthy caching nodes) broadcast information to other sick caching nodes to recover the erased packets. The broadcast information from a caching node is assumed to be received without any error by all other caching nodes. A…
