Achieving Private Information Retrieval Capacity in Distributed Storage Using an Arbitrary Linear Code

We propose three private information retrieval (PIR) protocols for distributed storage systems (DSSs) where data is stored using an arbitrary linear code. The first two protocols, named Protocol 1 and Protocol 2, achieve privacy for the scenario with non-colluding nodes. Protocol 1 requires a file size that is exponential in the number of files in the system, while the file size required for Protocol 2 is independent of the number of files and is hence simpler. We prove that, for certain linear codes, Protocol 1 achieves the PIR capacity, i.e., its PIR rate (the ratio of the amount of retrieved stored data per unit of downloaded data) is the maximum possible for any given (finite and infinite) number of files, and Protocol 2 achieves the asymptotic PIR capacity (with infinitely large number of files in the DSS). In particular, we provide a sufficient and a necessary condition for a code to be PIR capacity achieving and prove that cyclic codes, Reed-Muller (RM) codes, and optimal information locality local reconstruction codes achieve both the finite PIR capacity (i.e., with any given number of files) and the asymptotic PIR capacity with Protocol 1 and 2, respectively. Furthermore, we present a third protocol, Protocol 3, for the scenario with multiple colluding nodes, which can be seen as an improvement of a protocol recently introduced by Freij-Hollanti et al.. We also present an algorithm to optimize the PIR rate of the proposed protocol. Finally, we provide a particular class of codes that is suitable for this protocol and show that RM codes achieve the maximum possible PIR rate for the protocol.