Storage and Retrieval Codes in PIR Schemes with Colluding Servers

TL;DR

This paper investigates the design of star product PIR schemes with colluding servers, analyzing how the choice of storage and retrieval codes affects privacy and efficiency, especially in coded distributed storage systems.

Contribution

It provides new insights into selecting storage and retrieval codes, showing that matching algebraic code families is not always optimal for PIR schemes with colluding servers.

Findings

Nonzero retrieval rate schemes protect against small collusion groups.

Storage codes with special codewords limit collusion resistance.

Cyclic retrieval codes are not always optimal for cyclic storage codes.

Abstract

Private information retrieval (PIR) schemes (with or without colluding servers) have been proposed for realistic coded distributed data storage systems. Star product PIR schemes with colluding servers for general coded distributed storage system were constructed over general finite fields by R. Freij-Hollanti, O. W. Gnilke, C. Hollanti and A. Karpuk in 2017. These star product PIR schemes with colluding servers are suitable for the storage of files over small fields and can be constructed for coded distributed storage system with large number of servers. In this paper for an efficient storage code, the problem to find good retrieval codes is considered. In general if the storage code is a binary Reed-Muller code the retrieval code needs not to be a binary Reed-Muller code in general. It is proved that when the storage code contains some special codewords, nonzero retrieval rate star product PIR schemes with colluding servers can only protect against small number of colluding servers. We also give examples to show that when the storage code is a good cyclic code, the best choice of the retrieval code is not cyclic in general. Therefore in the design of star product PIR schemes with colluding servers, the scheme with the storage code and the retrieval code in the same family of algebraic codes is not always efficient.