Private Information Retrieval from MDS Coded Data in Distributed Storage Systems

The problem of providing privacy, in the private information retrieval (PIR) sense, to users requesting data from a distributed storage system (DSS), is considered. The DSS is coded by an $(n,k,d)$ Maximum Distance Separable (MDS) code to store the data reliably on unreliable storage nodes. Some of these nodes can be spies which report to a third party, such as an oppressive regime, which data is being requested by the user. An information theoretic PIR scheme ensures that a user can satisfy its request while revealing, to the spy nodes, no information on which data is being requested. A user can trivially achieve PIR by downloading all the data in the DSS. However, this is not a feasible solution due to its high communication cost. We construct PIR schemes with low download communication cost. When there is $b=1$ spy node in the DSS, we construct PIR schemes with download cost $\frac{1}{1-R}$ per unit of requested data ($R=k/n$ is the code rate), achieving the information theoretic limit for linear schemes. The proposed schemes are universal since they depend on the code rate, but not on the generator matrix of the code. Also, when $b\leq n-δk$, for some $δ\in \mathbb{N^+}$, we construct linear PIR schemes with $cPoP = \frac{b+δk}δ$.