Fundamental Limits of Cache-Aided Private Information Retrieval With Unknown and Uncoded Prefetching

Abstract

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.

Cross Subspace Alignment and the Asymptotic Capacity of $X$ -Secure $T$ -Private Information Retrieval

“…In particular, capacity is known for disjoint colluding sets [4]. The rapidly growing body of literature in this area has produced capacity results for PIR under a rich variety of constraints [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. However, the capacity for the natural setting of secure storage remains unknown, and relatively unexplored.…”

Single-server Multi-message Private Information Retrieval with Side Information

“…Hence, Eqn. (16) guarantees that the total number of side information messages used by any N coding subspaces is not larger than M . Additionally, if the size of one coding subspace is equal to or less than N , the number of side information messages that can be used in such coding subspace can only be zero.…”

“…, m |L| } is the vector of the number of side information messages used in each coding subspace which satisfies Eqn. (16) and…”

“…By exploiting the advantages of replications of the database in non-colluding servers, private information retrieval can be achieved without downloading all messages and the capacity of this problem is characterized in [4]. Ensuing work has studied many variations of this theme, including databases coded by erasure codes [6], [7], [8], [9], [10], [11], partial colluding servers [5], [12], [13], side information messages available at users [14], [15], [16], [17], [18] and multiple messages [19], [20],…”

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.…”