The Capacity of $T$-Private Information Retrieval with Private Side Information

TL;DR

This paper determines the capacity of T-private information retrieval with private side information, providing exact formulas for both standard and symmetric variants, and resolves an open problem in the field.

Contribution

It derives the capacity formula for TPIR-PSI and establishes the capacity of STPIR-PSI with and without common randomness, solving previously open problems.

Findings

Capacity of TPIR-PSI is +rac{T}{N}+\u2026+ig(rac{T}{N}ig)^{K-M-1}"] ,

Capacity of STPIR-PSI is 1-rac{T}{N} with sufficient common randomness, otherwise zero.

It settles the capacity of PIR-PSI for T=1, an open problem noted by Kadhe et al.

Abstract

We consider the problem of $T$-Private Information Retrieval with private side information (TPIR-PSI). In this problem, $N$ replicated databases store $K$ independent messages, and a user, equipped with a local cache that holds $M$ messages as side information, wishes to retrieve one of the other $K-M$ messages. The desired message index and the side information must remain jointly private even if any $T$ of the $N$ databases collude. We show that the capacity of TPIR-PSI is $\left(1+\frac{T}{N}+\cdots+\left(\frac{T}{N}\right)^{K-M-1}\right)^{-1}$. As a special case obtained by setting $T=1$, this result settles the capacity of PIR-PSI, an open problem previously noted by Kadhe et al. We also consider the problem of symmetric-TPIR with private side information (STPIR-PSI), where the answers from all $N$ databases reveal no information about any other message besides the desired message. We show that the capacity of STPIR-PSI is $1-\frac{T}{N}$ if the databases have access to common randomness (not available to the user) that is independent of the messages, in an amount that is at least $\frac{T}{N-T}$ bits per desired message bit. Otherwise, the capacity of STPIR-PSI is zero.