t-Private Information Retrieval Schemes Using Transitive Codes

TL;DR

This paper develops private information retrieval schemes for coded storage systems using transitive codes, achieving optimal rates even with colluding servers and focusing on binary fields with Reed-Muller codes.

Contribution

It introduces PIR schemes applicable to general linear codes, especially transitive codes, and explicitly calculates their rates, matching known capacities for MDS codes.

Findings

Transitive codes yield the highest PIR rates in the proposed scheme.

The PIR rate matches the asymptotic capacity for MDS-coded storage without collusion.

Binary fields with Reed-Muller codes are effectively utilized in the scheme.

Abstract

This paper presents private information retrieval (PIR) schemes for coded storage with colluding servers, which are not restricted to maximum distance separable (MDS) codes. PIR schemes for general linear codes are constructed and the resulting PIR rate is calculated explicitly. It is shown that codes with transitive automorphism groups yield the highest possible rates obtainable with the proposed scheme. This rate coincides with the known asymptotic PIR capacity for MDS-coded storage systems without collusion. While many PIR schemes in the literature require field sizes that grow with the number of servers and files in the system, we focus especially on the case of a binary base field, for which Reed- Muller codes serve as an important and explicit class of examples.