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 a 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 thiswhere 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.while browsing restricted contents from the internet, see [1,5]. In the seminal paper Chor et. al. [1], the classical problem of private information retrieval (PIR) is introduced. In the classical PIR setting, a user requests to download a certain message (or file) from N non-communicating databases without leaking the identity of the message to any individual database. The contents of these databases are identical, i.e., they are repetition coded. A trivial solution for this seemingly challenging task is to download all of the contents of the databases. However, this solution is highly impractical, in particular for large number of messages which is the case in modern storage systems. The aim of the PIR problem is to design efficient retrieval schemes that maximize the ratio of the desired information bits to the total downloaded bits under the privacy constraint.In the classical PIR problem, the user prepares N queries each directed to a specific database. The queries are designed such that they do not reveal any information about the identity of the desired message. Upon receiving these queries, databases respond truthfully with answering strings. Based on the collected answer strings, the user reconstructs the desired message. In the original formulation of the problem in the computer science literature [1], the messages are assumed to have a size of one bit. In this formulation, the performance metric was the sum of lengths of the answer strings (download cost) and the size of the queries (upload cost). The information-theoretic reformulation of the problem assumes that the messages are of arbitrarily large size and hence the upload cost can be neglected with respect to the download cost [6]. The pioneering work [7] derives the exact capacity of the classical PIR problem. The capacity is defined as the maximum number of bits of the desired message per bit of total download. The achievable scheme is based on an interesting relationship between PIR and blind interference alignment introduced for wireless networks in [8] as...

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 C P s 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 C P s =

We consider the problem of single-round private information retrieval (PIR) from N replicated databases. We consider the case when B 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 M messages with zero-error, irrespective of the actions performed by the Byzantine databases. We consider the T -privacy constraint in this paper, where any T 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 correct symbols that can be retrieved privately (under the Tprivacy constraint) for every symbol of the downloaded data. We determine the exact BPIR capacity to be C = N −2BThis capacity expression shows that the effect of Byzantine databases on the retrieval rate is equivalent to removing 2B databases from the system, with a penalty factor of N −2B N , which signifies that even though the number of databases needed for PIR is effectively N − 2B, the user still needs to access the entire N 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 errors introduced by the Byzantine databases as opposed to erasures in the RPIR problem. Our converse proof uses the idea of the cut-set bound in the network coding problem against adversarial nodes.

We consider the problem of private information retrieval (PIR) from N non-colluding and replicated databases when the user is equipped with a cache that holds an uncoded fraction r from each of the K stored messages in the databases. We assume that the databases are unaware of the cache content. We investigate D * (r) the optimal download cost normalized with the message size as a function of K, N , r. For a fixed K, N , we develop an inner bound (converse bound) for the D * (r) curve. The inner bound is a piece-wise linear function in r that consists of K line segments. For the achievability, we develop explicit schemes that exploit the cached bits as side information to achieve K − 1 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 r that consists of K line segments. The inner and the outer bounds match in general for the cases of very low caching ratio (r ≤ 1 1+N +N 2 +···+N K−1 ) and very high caching ratio (r ≥ K−2 (N +1)K+N 2 −2N −2 ). As a corollary, we fully characterize the optimal download cost caching ratio tradeoff for K = 3. For general K, N , and r, we show that the largest gap between the achievability and the converse bounds is 1 6 . Our results show that the download cost can be reduced beyond memory-sharing if the databases are unaware of the cached content.

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