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 Mathematics , 2010, Abstract: We consider the problem of optimally allocating a given total storage budget in a distributed storage system. A source has a data object which it can code and store over a set of storage nodes; it is allowed to store any amount of coded data in each node, as long as the total amount of storage used does not exceed the given budget. A data collector subsequently attempts to recover the original data object by accessing each of the nodes independently with some constant probability. By using an appropriate code, successful recovery occurs when the total amount of data in the accessed nodes is at least the size of the original data object. The goal is to find an optimal storage allocation that maximizes the probability of successful recovery. This optimization problem is challenging because of its discrete nature and nonconvexity, despite its simple formulation. Symmetric allocations (in which all nonempty nodes store the same amount of data), though intuitive, may be suboptimal; the problem is nontrivial even if we optimize over only symmetric allocations. Our main result shows that the symmetric allocation that spreads the budget maximally over all nodes is asymptotically optimal in a regime of interest. Specifically, we derive an upper bound for the suboptimality of this allocation and show that the performance gap vanishes asymptotically in the specified regime. Further, we explicitly find the optimal symmetric allocation for a variety of cases. Our results can be applied to distributed storage systems and other problems dealing with reliability under uncertainty, including delay tolerant networks (DTNs) and content delivery networks (CDNs).
 Mathematics , 2012, Abstract: We study the problem of storing a data object in a set of data nodes that fail independently with given probabilities. Our problem is a natural generalization of a homogenous storage allocation problem where all the nodes had the same reliability and is naturally motivated for peer-to-peer and cloud storage systems with different types of nodes. Assuming optimal erasure coding (MDS), the goal is to find a storage allocation (i.e, how much to store in each node) to maximize the probability of successful recovery. This problem turns out to be a challenging combinatorial optimization problem. In this work we introduce an approximation framework based on large deviation inequalities and convex optimization. We propose two approximation algorithms and study the asymptotic performance of the resulting allocations.
 Mathematics , 2013, Abstract: 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 coding scheme is sufficient to achieve any admissible rate for any exact repair distributed storage system. Equalities of certain joint entropies of random variables induced by concatenation scheme is shown. These equalities of joint entropies are new tools to simplify the linear programming bound and to obtain stronger converse results for exact repair distributed storage systems.
 Computer Science , 2012, Abstract: Distributed storage codes have recently received a lot of attention in the community. Independently, another body of work has proposed integrity checking schemes for cloud storage, none of which, however, is customized for coding-based storage or can efficiently support repair. In this work, we bridge the gap between these two currently disconnected bodies of work. We propose NC-Audit, a novel cryptography-based remote data integrity checking scheme, designed specifically for network coding-based distributed storage systems. NC-Audit combines, for the first time, the following desired properties: (i) efficient checking of data integrity, (ii) efficient support for repairing failed nodes, and (iii) protection against information leakage when checking is performed by a third party. The key ingredient of the design of NC-Audit is a novel combination of SpaceMac, a homomorphic message authentication code (MAC) scheme for network coding, and NCrypt, a novel chosen-plaintext attack (CPA) secure encryption scheme that is compatible with SpaceMac. Our evaluation of a Java implementation of NC-Audit shows that an audit costs the storage node and the auditor a modest amount computation time and lower bandwidth than prior work.
 Mathematics , 2012, Abstract: This paper studies the design of codes for distributed storage systems (DSS) that enable local repair in the event of node failure. This paper presents locally repairable codes based on low degree multivariate polynomials. Its code construction mechanism extends work on Noisy Interpolating Set by Dvir et al. \cite{dvir2011}. The paper presents two classes of codes that allow node repair to be performed by contacting 2 and 3 surviving nodes respectively. It further shows that both classes are good in terms of their rate and minimum distance, and allow their rate to be bartered for greater flexibility in the repair process.
 Mathematics , 2014, Abstract: We study the secrecy of a distributed storage system for passwords. The encoder, Alice, observes a length-n password and describes it using two hints, which she then stores in different locations. The legitimate receiver, Bob, observes both hints. The eavesdropper, Eve, sees only one of the hints; Alice cannot control which. We characterize the largest normalized (by n) exponent that we can guarantee for the number of guesses it takes Eve to guess the password subject to the constraint that either the number of guesses it takes Bob to guess the password or the size of the list that Bob must form to guarantee that it contain the password approach 1 as n tends to infinity.
 Computer Science , 2014, Abstract: This paper studies the problem of code symbol availability: a code symbol is said to have $(r, t)$-availability if it can be reconstructed from $t$ disjoint groups of other symbols, each of size at most $r$. For example, $3$-replication supports $(1, 2)$-availability as each symbol can be read from its $t= 2$ other (disjoint) replicas, i.e., $r=1$. However, the rate of replication must vanish like $\frac{1}{t+1}$ as the availability increases. This paper shows that it is possible to construct codes that can support a scaling number of parallel reads while keeping the rate to be an arbitrarily high constant. It further shows that this is possible with the minimum distance arbitrarily close to the Singleton bound. This paper also presents a bound demonstrating a trade-off between minimum distance, availability and locality. Our codes match the aforementioned bound and their construction relies on combinatorial objects called resolvable designs. From a practical standpoint, our codes seem useful for distributed storage applications involving hot data, i.e., the information which is frequently accessed by multiple processes in parallel.
 Mathematics , 2011, Abstract: 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.
 Computer Science , 2014, Abstract: We consider the problem of synchronizing data in distributed storage networks under an edit model that includes deletions and insertions. We present two modifications of MDS, regenerating and locally repairable codes that allow updates in the parity-check values to be performed with one round of communication at low bit rates and using small storage overhead. Our main contributions are novel protocols for synchronizing both hot and semi-static data and protocols for data deduplication applications, based on intermediary permutation, Vandermonde and Cauchy matrix coding.
 Mathematics , 2010, Abstract: We consider the problem of distributing a file in a network of storage nodes whose storage budget is limited but at least equals to the size file. We first generate $T$ encoded symbols (from the file) which are then distributed among the nodes. We investigate the optimal allocation of $T$ encoded packets to the storage nodes such that the probability of reconstructing the file by using any $r$ out of $n$ nodes is maximized. Since the optimal allocation of encoded packets is difficult to find in general, we find another objective function which well approximates the original problem and yet is easier to optimize. We find the optimal symmetric allocation for all coding redundancy constraints using the equivalent approximate problem. We also investigate the optimal allocation in random graphs. Finally, we provide simulations to verify the theoretical results.
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