On an Equivalence Between Single-Server PIR with Side Information and Locally Recoverable Codes
This work addresses a theoretical problem for researchers in information theory and coding theory by linking two previously separate areas, though it appears incremental as it builds on existing concepts to derive new bounds.
The paper tackles the problem of establishing a relationship between single-server Private Information Retrieval (PIR) with side information and Locally Recoverable Codes (LRCs), showing that PIR schemes for single-message retrieval are equivalent to classical LRCs and for multiple-message retrieval to cooperative LRCs, leading to recovered upper bounds on download rates for PIR-SI and a novel rate upper bound on cooperative LRCs.
Private Information Retrieval (PIR) problem has recently attracted a significant interest in the information-theory community. In this problem, a user wants to privately download one or more messages belonging to a database with copies stored on a single or multiple remote servers. In the single server scenario, the user must have prior side information, i.e., a subset of messages unknown to the server, to be able to privately retrieve the required messages in an efficient way. In the last decade, there has also been a significant interest in Locally Recoverable Codes (LRC), a class of storage codes in which each symbol can be recovered from a limited number of other symbols. More recently, there is an interest in 'cooperative' locally recoverable codes, i.e., codes in which multiple symbols can be recovered from a small set of other code symbols. In this paper, we establish a relationship between coding schemes for the single-server PIR problem and LRCs. In particular, we show the following results: (i) PIR schemes designed for retrieving a single message are equivalent to classical LRCs; and (ii) PIR schemes for retrieving multiple messages are equivalent to cooperative LRCs. These equivalence results allow us to recover upper bounds on the download rate for PIR-SI schemes, and to obtain a novel rate upper bound on cooperative LRCs. We show results for both linear and non-linear codes.