Linear Symmetric Private Information Retrieval for MDS Coded Distributed Storage with Colluding Servers

TL;DR

This paper derives the maximum efficiency of linear symmetric private information retrieval from MDS-coded distributed storage with colluding servers, establishing capacity limits based on server collusion and shared randomness.

Contribution

It provides the first explicit capacity characterization for linear schemes in MDS-TSPIR with colluding servers and shared randomness requirements.

Findings

Capacity equals $1-rac{M+T-1}{N}$ with sufficient shared randomness.

Capacity drops to zero without shared randomness.

Conjecture that the capacity result applies to all schemes, not just linear ones.

Abstract

The problem of symmetric private information retrieval (SPIR) from a coded database which is distributively stored among colluding servers is studied. Specifically, the database comprises $K$ files, which are stored among $N$ servers using an $(N,M)$-MDS storage code. A user wants to retrieve one file from the database by communicating with the $N$ servers, without revealing the identity of the desired file to any server. Furthermore, the user shall learn nothing about the other $K-1$ files in the database. In the $T$-colluding SPIR problem (hence called TSPIR), any $T$ out of $N$ servers may collude, that is, they may communicate their interactions with the user to guess the identity of the requested file. We show that for linear schemes, the information-theoretic capacity of the MDS-TSPIR problem, defined as the maximum number of information bits of the desired file retrieved per downloaded bit, equals $1-\frac{M+T-1}{N}$, if the servers share common randomness (unavailable at the user) with amount at least $\frac{M+T-1}{N-M-T+1}$ times the file size. Otherwise, the capacity equals zero. We conjecture that our capacity holds also for general MDS-TSPIR schemes.