We study the problem of Private Information Retrieval (PIR) in the presence of prior side information. The problem setup includes a database of K 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 M 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 K − M messages, and in the second scenario it is ⌈ K M +1 ⌉ messages, which should be compared to K 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. 1 We assume that this side information subset does not contain the desired message. Otherwise, the problem is degenerate.
We study Private Information Retrieval with Side Information (PIR-SI) in the single-server multi-message setting. In this setting, a user wants to download D messages from a database of K ≥ D messages, stored on a single server, without revealing any information about the identities of the demanded messages to the server. The goal of the user is to achieve information-theoretic privacy by leveraging the side information about the database. The side information consists of a random subset of M messages in the database which could have been obtained in advance from other users or from previous interactions with the server. The identities of the messages forming the side information are initially unknown to the server. Our goal is to characterize the capacity of this setting, i.e., the maximum achievable download rate.In our previous work, we have established the PIR-SI capacity for the special case in which the user wants a single message, i.e., D = 1 and showed that the capacity can be achieved through the Partition and Code (PC) scheme. In this paper, we focus on the case when the user wants multiple messages, i.e., D > 1. Our first result is that if the user wants more messages than what they have as side information, i.e., D > M , then the capacity is D K−M , and it can be achieved using a scheme based on the Generalized Reed-Solomon (GRS) codes. In this case, the user must learn all the messages in the database in order to obtain the desired messages. Our second result shows that this may not be necessary when D ≤ M , and the capacity in this case can be higher. We present a lower bound on the capacity based on an achievability scheme which we call Generalized Partition and Code (GPC).
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