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 @math -server PIR, the database is replicated among @math 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, and 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 @math . In this work, we show that information-theoretic PIR can be achieved with storage overhead arbitrarily close to the optimal value of @math , without sacrificing the communication complexity. Specifically, we prove that all known @math -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 @math bits of the database among @math servers, each storing @math coded bits (rather than replicas). For every fixed @math , the resulting storage overhead @math approaches @math as @math grows; explicitly we have @math . Moreover, in the special case @math , the storage overhead is only @math . In order to achieve these results, we introduce and study a new kind of binary linear codes, called here @math -server PIR codes. We then show how such codes can be constructed, and we establish several bounds on the parameters of @math -server PIR codes. Finally, we briefly discuss extensions of our results to nonbinary alphabets, to robust PIR, and to @math -private PIR.
There are several previous works which construct coded schemes for the purpose of fast or private retrieval. The first work we know of for the purpose of coded private retrieval is the recent work by @cite_23 . The authors showed how to encode files in multiple servers with very low communication complexity. However, their constructions require an exponentially large number of servers which may depend on the number of files or their size. In another recent work @cite_24 , studied the tradeoff between storage overhead and communication complexity, though only for setups in which the size of each file is relatively large. A similar approach to ours was studied by @cite_36 , where the authors also partitioned the database into several parts in order to avoid repetition and thereby reduce the storage overhead. However, their construction works only for the PIR scheme using the multiplicity codes by @cite_0 and they didn't encode the parts of the database as we study in this work.
@cite_24 @cite_0 @cite_36 @cite_23
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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. 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 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. 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.