Capacity of Single-Server Single-Message Private Information Retrieval with Private Coded Side Information

We study the problem of single-server single-message Private Information Retrieval with Private Coded Side Information (PIR-PCSI). In this problem, there is a server that stores a database, and a user who knows a random linear combination of a random subset of messages in the database. The number of messages contributing to the user's side information is known to the server a priori, whereas the indices and the coefficients of these messages are unknown to the server a priori. The user wants to retrieve a message from the server, while protecting the identities of both the demand message and the side information messages. Depending on whether the demand is part of the coded side information or not, we consider two different models for the problem. For the model in which the demand does not contribute to the side information, we prove a lower bound on the minimum download cost for all (linear and non-linear) PIR schemes; and for the model wherein the demand is one of the messages contributing to the side information, we prove a lower bound for all scalar-linear PIR protocols. In addition, we propose novel PIR protocols that achieve these lower bounds.