The Capacity of Private Information Retrieval From Coded Databases

Abstract

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...

Private information retrieval from MDS coded data with colluding servers: Settling a conjecture by Freij-Hollanti et al

“…From [5], we know that the capacity of MDS-PIR with (K, N, K c ) = (2, 4, 3) is 4/7. Therefore, the upper bound follows.…”

“…The capacity achieving scheme of Banawan and Ulukus [5] improved upon a scheme proposed earlier by Tajeddine and Rouayheb in [7]. Tajeddine and Rouayheb also proposed an achievable scheme for MDS-TPIR for the T = 2 setting.…”

Private Polynomial Computation for Noncolluding Coded Databases

“…denotes the set of all possible queries made by the user. Although this seems to be intuitively true, a proof of this property is still required and can be found in [13].…”

“…In [16], it is shown that the MDS-PIR capacity [13] can be achieved using Protocol 1 for a special class of [n, k] codes. In particular, to achieve the MDS-PIR capacity using Protocol 1, the [n, k] storage code should possess a specific underlying structure as given by the following theorem.…”

Survey on secure search over encrypted data on the cloud

“…In a system consisting of n replicated servers, the user sends homomorphically encrypted queries 32,43 to the n servers to retrieve data blocks from the servers that collectively make up the result. In a system consisting of n replicated servers, the user sends homomorphically encrypted queries 32,43 to the n servers to retrieve data blocks from the servers that collectively make up the result.…”