One extra bit of download ensures perfectly private information retrieval

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Abstract
Private information retrieval (PIR) systems allow a user to retrieve a record from a public database without revealing to the server which record is being retrieved. The literature on PIR considers only replication-based systems, wherein each storage node stores a copy of the entire data. However, systems based on erasure codes are gaining increasing popularity due to a variety of reasons. This paper initiates an investigation into PIR in erasure-coded systems by establishing its capacity and designing explicit codes and algorithms. The notion of privacy considered here is information-theoretic, and the metric optimized is the amount of data downloaded by the user during PIR. In this paper, we present four main results. First, we design an explicit erasure code and PIR algorithm that requires only one extra bit of download to provide perfect privacy. In contrast, all existing PIR algorithms require a download of at least twice the size of the requisite data. Second, we derive lower bounds proving the necessity of downloading at least one additional bit. This establishes the precise capacity of PIR with respect to the metric of download. These results are also applicable to PIR in replication-based systems, which are a special case of erasure codes. Our third contribution is a negative result showing that capacity-achieving codes necessitate super-linear storage overheads. This motivates the fourth contribution of this paper: an erasure code and PIR algorithm that requires a linear storage overhead, provides high reliability to the data, and is a small factor away from the capacity.