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We propose three private information retrieval (PIR) protocols fordistributed storage systems (DSSs) where data is stored using an arbitrarylinear code. The first two protocols, named Protocol 1 and Protocol 2, achieveprivacy for the scenario with noncolluding nodes. Protocol 1 requires a filesize that is exponential in the number of files in the system, while Protocol 2requires a file size that is independent of the number of files and is hencesimpler. We prove that, for certain linear codes, Protocol 1 achieves themaximum distance separable (MDS) PIR capacity, i.e., the maximum PIR rate (theratio of the amount of retrieved stored data per unit of downloaded data) for aDSS that uses an MDS code to store any given (finite and infinite) number offiles, and Protocol 2 achieves the asymptotic MDS-PIR capacity (with infinitelylarge number of files in the DSS). In particular, we provide a necessary and asufficient condition for a code to achieve the MDS-PIR capacity with Protocols1 and 2 and prove that cyclic codes, Reed-Muller (RM) codes, and a class ofdistance-optimal local reconstruction codes achieve both the finite MDS-PIRcapacity (i.e., with any given number of files) and the asymptotic MDS-PIRcapacity with Protocols 1 and 2, respectively. Furthermore, we present a thirdprotocol, Protocol 3, for the scenario with multiple colluding nodes, which canbe seen as an improvement of a protocol recently introduced by Freij-Hollantiet al.. Similar to the noncolluding case, we provide a necessary and asufficient condition to achieve the maximum possible PIR rate of Protocol 3.Moreover, we provide a particular class of codes that is suitable for thisprotocol and show that RM codes achieve the maximum possible PIR rate for theprotocol. For all three protocols, we present an algorithm to optimize theirPIR rates.

Achieving Maximum Distance Separable Private Information Retrieval Capacity With Linear Codes