# Private Information Retrieval with Private Coded Side Information: The Multi-Server Case

## TL;DR

This paper characterizes the maximum download rate for multi-server private information retrieval with private coded side information, considering whether the demand message is part of the side information or not.

## Contribution

It derives the capacity bounds for multi-server PIR with private coded side information, extending previous single-server results and introducing new achievable schemes.

## Key findings

- Capacity for side information excluding demand message: (1+1/N+...+1/N^{K-M-1})^{-1}
- Lower bound on capacity when demand message is part of side information: (1+1/N+...+1/N^{K-M})^{-1}
- Utilizes techniques from single-server PIR-PCSI and multi-server private computation literature.

## Abstract

In this paper, we consider the multi-server setting of Private Information Retrieval with Private Coded Side Information (PIR-PCSI) problem. In this problem, there is a database of $K$ messages whose copies are replicated across $N$ servers, and there is a user who knows a random linear combination of a random subset of $M$ messages in the database as side information. The user wishes to download one message from the servers, while protecting the identities of both the demand message and the messages forming the side information. We assume that the servers know the number of messages forming the user's side information in advance, whereas the indices of these messages and their coefficients in the side information are not known to any of the servers a priori. Our goal is to characterize (or derive a lower bound on) the capacity, i.e., the maximum achievable download rate, for the following two settings. In the first setting, the set of messages forming the linear combination available to the user as side information, does not include the user's demanded message. For this setting, we show that the capacity is equal to $\left(1+{1}/{N}+\dots+{1}/{N^{K-M-1}}\right)^{-1}$. In the second setting, the demand message contributes to the linear combination available to the user as side information, i.e., the demand message is one of the messages that form the user's side information. For this setting, we show that the capacity is lower-bounded by $\left(1+{1}/{N}+\dots+{1}/{N^{K-M}}\right)^{-1}$. The proposed achievability schemes and proof techniques leverage ideas from both our recent methods proposed for the single-server PIR-PCSI problem as well as the techniques proposed by Sun and Jafar for multi-server private computation problem.