This paper presents private information retrieval (PIR) schemes for coded storage with colluding servers, which are not restricted to maximum distance separable (MDS) codes. PIR schemes for general linear codes are constructed and the resulting PIR rate is calculated explicitly. It is shown that codes with transitive automorphism groups yield the highest possible rates obtainable with the proposed scheme. This rate coincides with the known asymptotic PIR capacity for MDS-coded storage systems without collusion. While many PIR schemes in the literature require field sizes that grow with the number of servers and files in the system, we focus especially on the case of a binary base field, for which Reed- Muller codes serve as an important and explicit class of examples.
RM codes have previously been considered for PIR in other settings @cite_3 . There, the system model is different from the present paper, in that the coding is between different files and the primary goal is to minimize storage overhead for a given PIR scheme, along the same lines as in @cite_10 . In our work, coding is between different blocks of the same file, and the goal is to minimize the download overhead for fixed storage codes.
@cite_10 @cite_3
2007725949 2963175488 2739960748 1619839175 2565803816 2963413537 2591083178 2524023279 2100816365 2950878776 2584331985 2007636263 2005598127 2128268793 2105695025 2752936485 2962865070 2296973760 2006551665 2155418912
Private information retrieval (PIR) systems allow a user to retrieve a record from a public database without revealing to the server which record is being retrieved. The literature on PIR considers only replication-based systems, wherein each storage node stores a copy of the entire data. However, systems based on erasure codes are gaining increasing popularity due to a variety of reasons. This paper initiates an investigation into PIR in erasure-coded systems by establishing its capacity and designing explicit codes and algorithms. The notion of privacy considered here is information-theoretic, and the metric optimized is the amount of data downloaded by the user during PIR. In this paper, we present four main results. First, we design an explicit erasure code and PIR algorithm that requires only one extra bit of download to provide perfect privacy. In contrast, all existing PIR algorithms require a download of at least twice the size of the requisite data. Second, we derive lower bounds proving the necessity of downloading at least one additional bit. This establishes the precise capacity of PIR with respect to the metric of download. These results are also applicable to PIR in replication-based systems, which are a special case of erasure codes. Our third contribution is a negative result showing that capacity-achieving codes necessitate super-linear storage overheads. This motivates the fourth contribution of this paper: an erasure code and PIR algorithm that requires a linear storage overhead, provides high reliability to the data, and is a small factor away from the capacity. We consider the problem of providing privacy, in the private information retrieval (PIR) sense, to users requesting data from a distributed storage system (DSS). The DSS uses an (n, k) Maximum Distance Separable (MDS) code to store the data reliably on unreliable storage nodes. Some of these nodes can be spies which report to a third party, such as an oppressive regime, which data is being requested by the user. An information theoretic PIR scheme ensures that a user can satisfy its request while revealing, to the spy nodes, no information on which data is being requested. A user can achieve PIR by downloading all the data in the DSS. However, this is not a feasible solution due to its high communication cost. We construct PIR schemes with low download communication cost. When there is b = 1 spy node in the DSS, we construct PIR schemes with download cost 1 1−R per unit of requested data (R = k n is the code rate), achieving the information theoretic limit for linear schemes. The proposed schemes are universal since they depend on the code rate, but not on the generator matrix of the code. When there are 2 ≤ b ≤ n − k spy nodes, we devise linear PIR schemes that have download cost equal to b + k per unit of requested data. We consider the problem of private information retrieval (PIR) over a distributed storage system. The storage system consists of N non-colluding databases, each storing an MDS-coded version of M messages. In the PIR problem, the user wishes to retrieve one of the available messages without revealing the message identity to any individual database. We derive the information-theoretic capacity of this problem, which is defined as the maximum number of bits of the desired message that can be privately retrieved per one bit of downloaded information. We show that the PIR capacity in this case is C = (1 + K N + K2 N2 + ··· + KM−1 NN−1)−1 = (1 + R c + R2 c + ··· + RM c −1)−1 = 1−R c R c M, where R c is the rate of the (N, K) code used. The capacity is a function of the code rate and the number of messages only regardless of the explicit structure of the storage code. The result implies a fundamental tradeoff between the optimal retrieval cost and the storage cost. The result generalizes the achievability and converse results for the classical PIR with replicating databases to the case of coded databases. Private information retrieval (PIR) protocols allow a user to retrieve a data item from a database without revealing any information about the identity of the item being retrieved. Specifically, in information-theoretic k-server PIR, the database is replicated among k non-communicating servers, and each server learns nothing about the item retrieved by the user. The cost of PIR protocols is usually measured in terms of their communication complexity, which is the total number of bits exchanged between the user and the servers. However, another important cost parameter is the storage overhead, which is the ratio between the total number of bits stored on all the servers and the number of bits in the database. Since single-server information-theoretic PIR is impossible, the storage overhead of all existing PIR protocols is at least 2 (or k, in the case of k-server PIR). In this work, we show that information-theoretic PIR can be achieved with storage overhead arbitrarily close to the optimal value of 1, without sacrificing the communication complexity. Specifically, we prove that all known k-server PIR protocols can be efficiently emulated, while preserving both privacy and communication complexity but significantly reducing the storage overhead. To this end, we distribute the n bits of the database among s + r servers, each storing n s coded bits (rather than replicas). Notably, our coding scheme remains the same, regardless of the specific k-server PIR protocol being emulated. For every fixed k, the resulting storage overhead (s +r) s approaches 1 as s grows; explicitly we have equation. Moreover, in the special case k = 2, the storage overhead is only 1 + 1 s. In order to achieve these results, we introduce and study a new kind of binary linear codes, called here k-server PIR codes. Finally, we show how such codes can be constructed from multidimensional cubic, from Steiner systems, and from one-step majority-logic decodable codes. We propose an information-theoretic private information retrieval (PIR) scheme for distributed storage systems where data is stored using a linear systematic code of rate R> 1 2. The proposed scheme generalizes the PIR scheme for data stored using maximum distance separable codes recently proposed by Tajeddine and El Rouayheb for the scenario of a single spy node. We further propose an algorithm to optimize the communication price of privacy (cPoP) using the structure of the underlying linear code. As an example, we apply the proposed algorithm to several distributed storage codes, showing that the cPoP can be significantly reduced by exploiting the structure of the distributed storage code. We present a general framework for private information retrieval (PIR) from arbitrary coded databases that allows one to adjust the rate of the scheme to the suspected number of colluding servers. If the storage code is a generalized Reed--Solomon code of length @math and dimension @math , we design PIR schemes that achieve a PIR rate of @math while protecting against any @math colluding servers, for any @math . This interpolates between the previously studied cases of @math and @math and achieves PIR capacity in both of these cases asymptotically as the number of files in the database grows. The notion of a Private Information Retrieval (PIR) code was recently introduced by Fazeli, Vardy and Yaakobi [1] who showed that this class of codes permit PIR at reduced levels of storage overhead in comparison with rephcated-server PIR. In the present paper, the construction of an (n, k) τ-server binary linear PIR code having parameters n = lΣ i=0 (m i ), k = (m i ) and τ = 2l for any integer m ≥ l ≥ 0 is presented. These codes are obtained through homogeneous-polynomial evaluation and correspond to the binary. Projective Reed Muller (PRM) code. The construction can be extended to yield PIR codes for any τ = ∊ 2l, 2l − 1 | l ∊ Z, l ≥ 0 and any value of k, through a combination of single-symbol puncturing and shortening of the PRM code. Each of these code constructions above, have smaller storage overhead in comparison with known short block length codes in [1]. For the particular case of τ = 3,4, we show that the codes constructed here are optimal, systematic PIR codes by providing an improved lower bound on the block length n k, τ) of a systematic PIR code. It follows from a result by Vardy and Yaakobi [2], that these codes also yield optimal, systematic primitive multi-set n, k, τ) B batch codes for τ = 3,4. The PIR code constructions presented here also yield upper bounds on the generahzed Hamming weights of binary PRM codes. We consider the problem of private information retrieval (PIR) over a distributed storage system. The storage system consists of @math non-colluding databases, each storing a coded version of @math messages. In the PIR problem, the user wishes to retrieve one of the available messages without revealing the message identity to any individual database. We derive the information-theoretic capacity of this problem, which is defined as the maximum number of bits of the desired message that can be privately retrieved per one bit of downloaded information. We show that the PIR capacity in this case is @math , where @math is the rate of the @math code used. The capacity is a function of the code rate and the number of messages only regardless of the explicit structure of the storage code. The result implies a fundamental tradeoff between the optimal retrieval cost and the storage cost. The result generalizes the achievability and converse results for the classical PIR with replicating databases to the case of coded databases. In this letter, we propose a novel forward error correction (FEC) coding method using a modified Reed-Muller (RM) code for providing accurate dimming control in on-off keying (OOK) based visible light communication (VLC) systems. The proposed FEC scheme has a property that produces exactly same occurrence probability of 1's and 0's. Therefore, it can contain minimal compensation symbols in supporting multiple dimming target values. Moreover, VLC with our proposed scheme provides more coding gain than VLC scheme with the conventional FEC codes [the first-order RM and Reed-Solomon (RS) codes]. Simulation results for various dimming target values show that the proposed scheme leads to at least 2 dB or more signal-to-noise ratio (SNR) gain in 1 bit error rate (BER) than two FEC codes. Petabyte-scale distributed storage systems are currently transitioning to erasure codes to achieve higher storage efficiency. Classical codes like Reed-Solomon are highly sub-optimal for distributed environments due to their high overhead in single-failure events. Locally Repairable Codes (LRCs) form a new family of codes that are repair efficient. In particular, LRCs minimize the number of nodes participating in single node repairs during which they generate small network traffic. Two large-scale distributed storage systems have already implemented different types of LRCs: Windows Azure Storage and the Hadoop Distributed File System RAID used by Facebook. The fundamental bounds for LRCs, namely the best possible distance for a given code locality, were recently discovered, but few explicit constructions exist. In this work, we present an explicit and optimal LRCs that are simple to construct. Our construction is based on grouping Reed-Solomon (RS) coded symbols to obtain RS coded symbols over a larger finite field. We then partition these RS symbols in small groups, and re-encode them using a simple local code that offers low repair locality. For the analysis of the optimality of the code, we derive a new result on the matroid represented by the code generator matrix. In Private Information Retrieval (PIR), one wants to download a file from a database without revealing to the database which file is being downloaded. Much attention has been paid to the case of the database being encoded across several servers, subsets of which can collude to attempt to deduce the requested file. With the goal of studying the achievable PIR rates in realistic scenarios, we generalize results for coded data from the case of all subsets of servers of size t colluding, to arbitrary subsets of the servers. We investigate the effectiveness of previous strategies in this new scenario, and present new results in the case where the servers are partitioned into disjoint colluding groups. In this paper, we consider the security of exact-repair regenerating codes operating at the minimum-storage-regenerating (MSR) point. The security requirement (introduced in Shah et. al.) is that no information about the stored data file must be leaked in the presence of an eavesdropper who has access to the contents of l 1 nodes as well as all the repair traffic entering a second disjoint set of l 2 nodes. We derive an upper bound on the size of a data file that can be securely stored that holds whenever l 2 ≤ d - k + 1. This upper bound proves the optimality of the product-matrix-based construction of secure MSR regenerating codes by Shah et. al. In this letter, we propose a new forward error correction (FEC) coding method using a coset constructed from a bent function and a Reed-Muller (RM) code for providing accurate dimming control in on-off keying-based visible light communication (VLC) systems. By partitioning the coset made from bent function and the first-order RM codes, the new FEC scheme can be made to be suitable for dimming close to 0 or 100 . Therefore, it can contain minimal compensation symbols in supporting multiple dimming target values, which causes improved coding gain when the total amount of frame data is fixed. Simulation results for various dimming target values show that the proposed scheme leads to about 1-dB signal-to-noise ratio gain in 10-2 bit error rate compared with other three FEC codes. Erasure codes, such as Reed-Solomon (RS) codes, are increasingly being deployed as an alternative to data-replication for fault tolerance in distributed storage systems. While RS codes provide significant savings in storage space, they can impose a huge burden on the I O and network resources when reconstructing failed or otherwise unavailable data. A recent class of erasure codes, called minimum-storage-regeneration (MSR) codes, has emerged as a superior alternative to the popular RS codes, in that it minimizes network transfers during reconstruction while also being optimal with respect to storage and reliability. However, existing practical MSR codes do not address the increasingly important problem of I O overhead incurred during reconstructions, and are, in general, inferior to RS codes in this regard. In this paper, we design erasure codes that are simultaneously optimal in terms of I O, storage, and network bandwidth. Our design builds on top of a class of powerful practical codes, called the product-matrix-MSR codes. Evaluations show that our proposed design results in a significant reduction the number of I Os consumed during reconstructions (a 5× reduction for typical parameters), while retaining optimality with respect to storage, reliability, and network bandwidth. Several works have developed vector-linear maximum-distance separable (MDS) storage codes that minimize the total communication cost required to repair a single coded symbol after an erasure, referred to as repair bandwidth (BW). Vector codes allow communicating fewer sub-symbols per node, instead of the entire content. This allows non trivial savings in repair BW. In sharp contrast, classic codes, like Reed-Solomon (RS), used in current storage systems, are deemed to suffer from naive repair, i.e. downloading the entire stored message to repair one failed node. This mainly happens because they are scalar-linear. In this work, we present a simple framework that treats scalar codes as vector-linear. In some cases, this allows significant savings in repair BW. We show that vectorized scalar codes exhibit properties that simplify the design of repair schemes. Our framework can be seen as a finite field analogue of real interference alignment. Using our simplified framework, we design a scheme that we call clique-repair which provably identifies the best linear repair strategy for any scalar 2-parity MDS code, under some conditions on the sub-field chosen for vectorization. We specify optimal repair schemes for specific (5,3)- and (6,4)-Reed-Solomon (RS) codes. Further, we present a repair strategy for the RS code currently deployed in the Facebook Analytics Hadoop cluster that leads to 20 of repair BW savings over naive repair which is the repair scheme currently used for this code. We study the problem of Private Information Retrieval (PIR) in the presence of prior side information. The problem setup includes a database of @math independent messages possibly replicated on several servers, and a user that needs to retrieve one of these messages. In addition, the user has some prior side information in the form of a subset of @math messages, not containing the desired message and unknown to the servers. This problem is motivated by practical settings in which the user can obtain side information opportunistically from other users or has previously downloaded some messages using classical PIR schemes. The objective of the user is to retrieve the required message without revealing its identity while minimizing the amount of data downloaded from the servers. We focus on achieving information-theoretic privacy in two scenarios: (i) the user wants to protect jointly its demand and side information; (ii) the user wants to protect only the information about its demand, but not the side information. To highlight the role of side information, we focus first on the case of a single server (single database). In the first scenario, we prove that the minimum download cost is @math messages, and in the second scenario it is @math messages, which should be compared to @math messages, the minimum download cost in the case of no side information. Then, we extend some of our results to the case of the database replicated on multiple servers. Our proof techniques relate PIR with side information to the index coding problem. We leverage this connection to prove converse results, as well as to design achievability schemes. 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. This paper considers a distributed storage system, where multiple storage nodes can be reconstructed simultaneously at a centralized location. This centralized multi-node repair (CMR) model is a generalization of regenerating codes that allow for bandwidth-efficient repair of a single failed node. This work focuses on the trade-off between the amount of data stored and repair bandwidth in this CMR model. In particular, repair bandwidth bounds are derived for the minimum storage multi-node repair (MSMR) and the minimum bandwidth multi-node repair (MBMR) operating points. The tightness of these bounds are analyzed via code constructions. The MSMR point is characterized through codes achieving this point under functional repair for general set of CMR parameters, as well as with codes enabling exact repair for certain CMR parameters. The MBMR point, on the other hand, is characterized with exact repair codes for all CMR parameters for systems that satisfy a certain entropy accumulation property. Finally, the model proposed here is utilized for the secret sharing problem, where the codes for the multi-node repair problem is used to construct communication efficient secret sharing schemes with the property of bandwidth efficient share repair. Petabyte-scale distributed storage systems are currently transitioning to erasure codes to achieve higher storage efficiency. Classical codes like Reed-Solomon are highly suboptimal for distributed environments due to their high overhead in single-failure events. Locally Repairable Codes (LRCs) form a new family of codes that are repair efficient. In particular, LRCs minimize the number of nodes participating in single node repairs during which they generate small network traffic. Two large-scale distributed storage systems have already implemented different types of LRCs: Windows Azure Storage and the Hadoop Distributed File System RAID used by Facebook. The fundamental bounds for LRCs, namely the best possible distance for a given code locality, were recently discovered, but few explicit constructions exist. In this work, we present an explicit and simple to implement construction of optimal LRCs, for code parameters previously established by existence results. For the analysis of the optimality of our code, we derive a new result on the matroid represented by the code's generator matrix. Linear erasure codes with local repairability are desirable for distributed data storage systems. An [n,k,d] linear code having all-symbol (r,δ)-locality, denoted as (r,δ)_a, is considered optimal if it has the actual highest minimum distance of any code of the given parameters n,k,r and δ. A minimum distance bound is given in . The existing results on the existence and the construction of optimal (r, δ)_a linear codes are limited to only two small regions within this special case, namely, i) m=0 and ii) m≥ (v+δ-1)>(δ-1) and δ=2, where m=n mod(r+δ-1) and v=k mod r. This paper investigates the properties and existence conditions for optimal (r,δ)_a linear codes with general r and δ. First, a structure theorem is derived for general optimal (r,δ)_a codes which helps illuminate some of their structure properties. Next, the entire problem space with arbitrary n, k, r and δ is divided into eight different cases (regions) with regard to the specific relations of these parameters. For two cases, it is rigorously proved that no (r,δ)_a linear code can achieve the minimum distance bound in . For four other cases the optimal (r,δ)_a codes are shown to exist over a field of size q≥( n k-1 ), deterministic constructions are proposed. Our new constructive algorithms not only cover more cases, but for the same cases where previous algorithms exist, the new constructions require a smaller field, which translates to potentially lower computational complexity. Our findings substantially enriches the knowledge on optimal (r,δ)_a linear codes, leaving only two cases in which the construction of optimal codes are not yet known.