The Role of Reusable and Single-Use Side Information in Private Information Retrieval

This paper introduces the problem of Private Information Retrieval with Reusable and Single-use Side Information (PIR-RSSI). In this problem, one or more remote servers store identical copies of a set of $K$ messages, and there is a user that initially knows $M$ of these messages, and wants to privately retrieve one other message from the set of $K$ messages. The objective is to design a retrieval scheme in which the user downloads the minimum amount of information from the server(s) while the identity of the message wanted by the user and the identities of an $M_1$-subset of the $M$ messages known by the user (referred to as reusable side information) are protected, but the identities of the remaining $M_2=M-M_1$ messages known by the user (referred to as single-use side information) do not need to be protected. The PIR-RSSI problem reduces to the classical Private Information Retrieval (PIR) problem when ${M_1=M_2=0}$, and reduces to the problem of PIR with Private Side Information or PIR with Side Information when ${M_1\geq 1,M_2=0}$ or ${M_1=0,M_2\geq 1}$, respectively. In this work, we focus on the single-server setting of the PIR-RSSI problem. We characterize the capacity of this setting for the cases of ${M_1=1,M_2\geq 1}$ and ${M_1\geq 1,M_2=1}$, where the capacity is defined as the maximum achievable download rate over all PIR-RSSI schemes. Our results show that for sufficiently small values of $K$, the single-use side information messages can help in reducing the download cost only if they are kept private; and for larger values of $K$, the reusable side information messages cannot help in reducing the download cost.