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

In many practical settings, the user needs to retrieve information messages from a server in a periodic manner, over multiple rounds of communication. The messages are retrieved one at a time and the identity of future requests are not known to the server. In this paper, we focus on the private information retrieval protocols that ensure that the identities of all the messages retrieved from the server are protected. This scenario can occur in practical settings such as periodic content download from text and multimedia repositories. We refer to this problem of minimizing the rate of data download as online private information retrieval problem.Following the previous line of work by Kadhe et al. we assume that the user knows a subset of M messages in the database as side information. The identities of these M messages are initially unknown to the server. Focusing on scalar-linear settings, we characterize the per-round capacity, i.e., the maximum achievable download rate at each round. In particular, we show that for the setting with K messages stored at the server, the per-round capacity of the scalar-linear setting is C₁ = (M + 1)/K for round i = 1 and C_i = (2^{i -1}(M + 1))/KM for round i ≥ 2, provided that K/(M + 1) is a power of 2. The key idea≥of our achievability scheme is to combine the data downloaded during the current round and the previous rounds with the original side information messages and use the resulting data as side information for the subsequent rounds.