In the problem of private information retrieval with side information, a single user wants to recover one of the @math independent messages which are stored at one or multiple servers. The user initially has a subset of messages as side information. The goal of the user is to retrieve the demand message by using minimum number of transmissions @math from the server(s) to the user under the condition that the index of the demand message should not be inferred by the server. We introduce the multi-user variant into this problem, where each user wants to retrieve one message and has a subset of messages as side information. In this paper, we study the special cases where all users want to retrieve one common message from a single server, but each user has different side information messages. We show that the optimal coding scheme can be constructed by first optimally partitioning the messages and then generating MDS codes separately in each subset of messages in the partition. We determine the @math , propose algorithms to compute @math , and construct optimal linear coding schemes with complexity polynomial in @math (but exponential in the number of side information messages).
@cite_3: The problem of cache enabled private information retrieval (PIR) is considered in which a user wishes to privately retrieve one out of @math messages, each of size @math bits from @math distributed databases. The user has a local cache of storage @math bits which can be used to store any function of the @math messages. The main contribution of this work is the exact characterization of the capacity of cache aided PIR as a function of the storage parameter @math . In particular, for a given cache storage parameter @math , the information-theoretically optimal download cost @math (or the inverse of capacity) is shown to be equal to @math . Special cases of this result correspond to the settings when @math , for which the optimal download cost was shown by Sun and Jafar to be @math , and the case when @math , i.e., cache size is large enough to store all messages locally, for which the optimal download cost is @math . The intermediate points @math can be readily achieved through a simple memory-sharing based PIR scheme. The key technical contribution of this work is the converse, i.e., a lower bound on the download cost as a function of storage @math which shows that memory sharing is information-theoretically optimal.
@cite_2: 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.
@cite_5: 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.
@cite_10: The problem of private information retrieval with private side information (PIR-PSI) is recently introduced by In this problem, @math replicated databases each store @math messages, of which @math are known to the user as side-information. The identity of these @math messages is unknown to the databases. The user wishes to retrieve, as efficiently as possible, a new desired message without revealing any information about the joint indices of the desired and side-information messages. We show that the capacity of PIR-PSI is @math . Thus, the capacity of PIR-PSI with @math messages, @math databases and side-information of @math messages is identical to the capacity of PIR with @math messages, @math databases and no side-information, which was found previously by Sun and Jafar. The achievability and converse proofs are essentially inherited from the original work of Sun and Jafar.
@cite_20: We consider the problem of private information retrieval (PIR) from @math non-colluding and replicated databases when the user is equipped with a cache that holds an uncoded fraction @math from each of the @math stored messages in the databases. We assume that the databases are unaware of the cache content. We investigate @math the optimal download cost normalized with the message size as a function of @math , @math , @math . For a fixed @math , @math , we develop an inner bound (converse bound) for the @math curve. The inner bound is a piece-wise linear function in @math that consists of @math line segments. For the achievability, we develop explicit schemes that exploit the cached bits as side information to achieve @math non-degenerate corner points. These corner points differ in the number of cached bits that are used to generate one side information equation. We obtain an outer bound (achievability) for any caching ratio by memory-sharing between these corner points. Thus, the outer bound is also a piece-wise linear function in @math that consists of @math line segments. The inner and the outer bounds match in general for the cases of very low caching ratio ( @math ) and very high caching ratio ( @math ). As a corollary, we fully characterize the optimal download cost caching ratio tradeoff for @math . For general @math , @math , and @math , we show that the largest gap between the achievability and the converse bounds is @math . Our results show that the download cost can be reduced beyond memory-sharing if the databases are unaware of the cached content.
@cite_18: In the private information retrieval (PIR) problem, a user wishes to retrieve, as efficiently as possible, one out of @math messages from @math non-communicating databases (each holds all @math messages) while revealing nothing about the identity of the desired message index to any individual database. The information theoretic capacity of PIR is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. For @math messages and @math databases, we show that the PIR capacity is @math . A remarkable feature of the capacity achieving scheme is that if we eliminate any subset of messages (by setting the message symbols to zero), the resulting scheme also achieves the PIR capacity for the remaining subset of messages.
@cite_4: 
@cite_8: 
@cite_17: 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.
@cite_7: 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.
@cite_6: A user wants to retrieve a file from a database without revealing the identity of the file retrieved at the database, which is known as the problem of private information retrieval (PIR). If it is further required that the user obtains no information about the database other than the desired file, the concept of symmetric private information retrieval (SPIR) is introduced to guarantee privacy for both parties. In this paper, the problem of SPIR is studied for a database stored among N nodes in a distributed way, by using an (N, M)-MDS storage code. The information-theoretic capacity of SPIR, defined as the maximum number of information bits of the desired file retrieved per downloaded bit, for the coded database is derived. It is shown that the SPIR capacity for coded database is 1-M N, when the amount of the shared common randomness of distributed nodes (unavailable at the user) is at least M N-M times the file size. Otherwise, the SPIR capacity for the coded database equals zero.
@cite_19: Private information retrieval scheme for coded data storage is considered in this paper. We focus on the case where the size of each data record is large and hence only the download cost (but not the upload cost for transmitting retrieval queries) is of interest. We prove that the tradeoff between storage cost and retrieval download cost depends on the number of data records in the system. We propose a class of linear storage codes and retrieval schemes, and derive conditions under which our schemes are error-free and private. Tradeoffs between the storage cost and retrieval costs are also obtained.
@cite_16: We consider the problem of multi-message private information retrieval (MPIR) from N non-communicating replicated databases. In MPIR, the user is interested in retrieving P messages out of M stored messages without leaking the identity of the retrieved messages. The information-theoretic sum capacity of MPIR CP is the maximum number of desired message symbols that can be retrieved privately per downloaded symbol. For the case P ≥ M 2, we determine the exact sum capacity of MPIR as CP s =1 1+M-P PN For P≤M 2, we develop lower and upper bounds for all M, P, N. These bounds match if the number of messages M is an integer multiple of the number of desired messages P, in which case, CP s = 1−1N 1−(1 N)M P. Our results indicate that joint retrieval of desired messages is more efficient than successive use of single-message retrieval schemes.
@cite_12: Private information retrieval (PIR) protocols allow a user to retrieve a data item from a database while hiding the identity of the item being retrieved. Specifically, in information-theoretic, k-server PIR protocols the database is replicated among k servers, and each server learns nothing about the item the user retrieves. The cost of such protocols is measured by the communication complexity of retrieving one out of n bits of data. For any fixed k, the complexity of the best protocols prior to our work was O(n sup 1 2k-1 ). Since then several methods were developed in an attempt to beat this bound, but all these methods yielded the same asymptotic bound. In this paper, this barrier is finally broken and the complexity of information-theoretic k-server PIR is improved to n sup O(log log k k log k) . The new PIR protocols can also be used to construct k-query binary locally decodable codes of length exp(n sup O(log log k k log k) ), compared to exp(n sup 1 k-1 ) in previous constructions. The improvements presented in this paper apply even for small values of k: the PIR protocols are more efficient than previous ones for every k spl ges 3, and the locally decodable codes are shorter for every k spl ges 4.
@cite_14: Private information retrieval (PIR) is the problem of retrieving as efficiently as possible, one out of @math messages from @math non-communicating replicated databases (each holds all @math messages) while keeping the identity of the desired message index a secret from each individual database. The information theoretic capacity of PIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. @math -private PIR is a generalization of PIR to include the requirement that even if any @math of the @math databases collude, the identity of the retrieved message remains completely unknown to them. Robust PIR is another generalization that refers to the scenario where we have @math databases, out of which any @math may fail to respond. For @math messages and @math databases out of which at least some @math must respond, we show that the capacity of @math -private and Robust PIR is @math . The result includes as special cases the capacity of PIR without robustness ( @math ) or @math -privacy constraints ( @math ).
@cite_9: 
@cite_1: 
@cite_0: 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.
@cite_13: We consider the problem of single-round private information retrieval (PIR) from @math replicated databases. We consider the case when @math databases are outdated (unsynchronized), or even worse, adversarial (Byzantine), and therefore, can return incorrect answers. In the PIR problem with Byzantine databases (BPIR), a user wishes to retrieve a specific message from a set of @math messages with zero-error, irrespective of the actions performed by the Byzantine databases. We consider the @math -privacy constraint in this paper, where any @math databases can collude, and exchange the queries submitted by the user. We derive the information-theoretic capacity of this problem, which is the maximum number of that can be retrieved privately (under the @math -privacy constraint) for every symbol of the downloaded data. We determine the exact BPIR capacity to be @math , if @math . This capacity expression shows that the effect of Byzantine databases on the retrieval rate is equivalent to removing @math databases from the system, with a penalty factor of @math , which signifies that even though the number of databases needed for PIR is effectively @math , the user still needs to access the entire @math databases. The result shows that for the unsynchronized PIR problem, if the user does not have any knowledge about the fraction of the messages that are mis-synchronized, the single-round capacity is the same as the BPIR capacity. Our achievable scheme extends the optimal achievable scheme for the robust PIR (RPIR) problem to correct the introduced by the Byzantine databases as opposed to in the RPIR problem. Our converse proof uses the idea of the cut-set bound in the network coding problem against adversarial nodes.
@cite_11: 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.
The Private Information Retrieval (PIR) problem was first studied in @cite_9 from a computational complexity perspective. Recently, the PIR problem attracted considerable attention in the information theory society and many works study this problem from an information-theoretic point of view @cite_12 @cite_16 @cite_18 @cite_14 . A single user wants to privately download one message from a database. To achieve perfect privacy in the information-theoretic sense, if the database is only stored at one server, the user has to download all messages. The problem becomes more interesting if one supposes that the database is stored in multiple servers and there is no collusion between these servers. By exploiting the advantages of replications of the database in non-colluding servers, private information retrieval can be achieved without downloading all messages and the capacity of this problem is characterized in @cite_18 . Ensuing work has studied many variations of this theme, including databases coded by erasure codes @cite_17 @cite_19 @cite_11 @cite_6 @cite_2 @cite_5 , partial colluding servers @cite_14 @cite_7 @cite_13 , side information messages available at users @cite_0 @cite_3 @cite_20 @cite_10 @cite_1 and multiple messages @cite_4 @cite_8 ,