The Capacity of Private Information Retrieval With Partially Known Private Side Information

Abstract

We consider the problem of private information retrieval (PIR) of a single message out of K messages from N replicated and non-colluding databases where a cacheenabled user (retriever) of cache-size M possesses side information in the form of full messages that are partially known to the databases. In this model, the user and the databases engage in a two-phase scheme, namely, the prefetching phase where the user acquires side information and the retrieval phase where the user downloads desired information. In the prefetching phase, the user receives m n full messages from the nth database, under the cache memory size constraint N n=1 m n ≤ M . In the retrieval phase, the user wishes to retrieve a message such that no individual database learns anything about the identity of the desired message. In addition, the identities of the side information messages that the user did not prefetch from a database must remain private against that database. Since the side information provided by each database in the prefetching phase is known by the providing database and the side information must be kept private against the remaining databases, we coin this model as partially *

The Capacity of Private Information Retrieval With Private Side Information Under Storage Constraints

“…which is the optimal normalized download cost of the PIR with PSI problem as shown in [32]. We can further generalize the result to the PIR with partially known PSI as shown in [33]. Note further that for M > S,…”

“…Lemma 1 differs from its counterpart in [31, Lemma 1] in two aspects; first, the left hand side is D(r)−L(1−r) in [31] as the user requests to download the uncached bits only, and second, [31, Lemma 1] constructs K − 1 distinct lower bounds by changing k, in contrast to only one bound here. In addition, we note that a similar argument to Lemma 1 can be implied from [32] and [33]. The main difference between Lemma 1 and [32,33] is that W H refers to parts of messages here, while in [32,33], W H refers to full messages.…”

“…The result is somewhat pessimistic since the user cannot further reduce the download cost by using the cache content. This has motivated subsequent works which have considered the case where the databases are completely unaware or partially unaware of the cache content [30][31][32][33]36,39]. Within this sub-branch of literature, references [30,32,33] have considered PIR with PSI.…”

“…For general number of databases, the optimal normalized download cost is characterized in [32] as D * (M) = 1 + 1 N + · · · + 1 N K−1−M . In [33], a more practical scenario is considered where each database is aware of the identities of the messages cached from that database only and unaware of the remaining identities of messages cached from other databases, which is coined as PIR with partially known PSI. Interestingly, the optimal normalized download cost for PIR with partially known PSI is the same as the optimal normalized download cost for PIR with PSI.…”

The Capacity of Private Information Retrieval from Decentralized Uncoded Caching Databases

“…Thus, in order to formulate a meaningful PIR problem, we allow the user (retriever) access the data center as well as the databases in the retrieval phase. Finally, we remark about another sub-branch of PIR literature that considers caching: [30][31][32][33]39,52]; there the user (retriever) itself has a cache memory where it stores a subset of the bits available in the databases. That problem is unrelated to the setting here even though it is also referred to as PIR with caching; in essence, it is PIR with side information.…”