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

## TL;DR

This paper investigates the minimal download cost for private information retrieval when the user has private coded side information, providing lower bounds and proposing protocols that achieve these bounds.

## Contribution

It establishes fundamental lower bounds for PIR with private coded side information and introduces optimal protocols matching these bounds.

## Key findings

- Lower bounds on download cost for PIR with coded side information.
- Proposed PIR protocols that achieve these lower bounds.
- Analysis of models where demand is or isn't part of side information.

## Abstract

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 their indices and coefficients are unknown to the server a priori. The user wants to retrieve a message from the server (with minimum download cost), while protecting the identities of both the demand and 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 protocols; and for the other 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.