Single-Server Single-Message Online Private Information Retrieval with Side Information

In many practical settings, the user needs to retrieve information from a server in a periodic manner, over multiple rounds of communication. In this paper, we discuss the setting in which this information needs to be retrieved privately, such that the identity of all the information retrieved until the current round is protected. This setting can occur in practical situations in which the user needs to retrieve items from the server or a periodic basis, such that the privacy needs to be guaranteed for all the items been retrieved until the current round. We refer to this setting as an \emph{online private information retrieval} as the user does not know the identities of the future items that need to be retrieved from the server. Following the previous line of work by Kadhe \emph{et al.}~we assume that the user knows a random subset of $M$ messages in the database as a side information which are unknown to the server. Focusing on scalar-linear settings, we characterize the \emph{per-round capacity}, i.e., the maximum achievable download rate at each round, and present a coding scheme that achieves this capacity. The key idea of our scheme is to utilize the data downloaded during the current round as a side information for the subsequent rounds. We show for the setting with $K$ messages stored at the server, the per-round capacity of the scalar-linear setting is $C_1= ({M+1})/{K}$ for round $i=1$ and ${C_i= {(2^{i-1}(M+1))}/{KM}}$ for round $i\geq2$, provided that ${K}/({M+1})$ is a power of $2$.