PIR Array Codes with Optimal Virtual Server Rate

There has been much recent interest in Private information Retrieval (PIR) in models where a database is stored across several servers using coding techniques from distributed storage, rather than being simply replicated. In particular, a recent breakthrough result of Fazelli, Vardy and Yaakobi introduces the notion of a PIR code and a PIR array code, and uses this notion to produce efficient PIR protocols. In this paper we are interested in designing PIR array codes. We consider the case when we have $m$ servers, with each server storing a fraction $(1/s)$ of the bits of the database; here $s$ is a fixed rational number with $s > 1$. A PIR array code with the $k$-PIR property enables a $k$-server PIR protocol (with $k\leq m$) to be emulated on $m$ servers, with the overall storage requirements of the protocol being reduced. The communication complexity of a PIR protocol reduces as $k$ grows, so the virtual server rate, defined to be $k/m$, is an important parameter. We study the maximum virtual server rate of a PIR array code with the $k$-PIR property. We present upper bounds on the achievable virtual server rate, some constructions, and ideas how to obtain PIR array codes with the highest possible virtual server rate. In particular, we present constructions that asymptotically meet our upper bounds, and the exact largest virtual server rate is obtained when $1 < s \leq 2$. A $k$-PIR code (and similarly a $k$-PIR array code) is also a locally repairable code with symbol availability $k-1$. Such a code ensures $k$ parallel reads for each information symbol. So the virtual server rate is very closely related to the symbol availability of the code when used as a locally repairable code. The results of this paper are discussed also in this context, where subspace codes also have an important role

Achieving Maximum Distance Separable Private Information Retrieval Capacity With Linear Codes

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 noncolluding nodes. Protocol 1 requires a file size that is exponential in the number of files in the system, while Protocol 2 requires a file size that is independent of the number of files and is hence simpler. We prove that, for certain linear codes, Protocol 1 achieves the maximum distance separable (MDS) PIR capacity, i.e., the maximum PIR rate (the ratio of the amount of retrieved stored data per unit of downloaded data) for a DSS that uses an MDS code to store any given (finite and infinite) number of files, and Protocol 2 achieves the asymptotic MDS-PIR capacity (with infinitely large number of files in the DSS). In particular, we provide a necessary and a sufficient condition for a code to achieve the MDS-PIR capacity with Protocols 1 and 2 and prove that cyclic codes, Reed-Muller (RM) codes, and a class of distance-optimal local reconstruction codes achieve both the finite MDS-PIR capacity (i.e., with any given number of files) and the asymptotic MDS-PIR capacity with Protocols 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.. Similar to the noncolluding case, we provide a necessary and a sufficient condition to achieve the maximum possible PIR rate of Protocol 3. Moreover, 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. For all three protocols, we present an algorithm to optimize their PIR rates.Comment: This work is the extension of the work done in arXiv:1612.07084v2. The current version introduces further refinement to the manuscript. Current version will appear in the IEEE Transactions on Information Theor

Robust Private Information Retrieval on Coded Data

We consider the problem of designing PIR scheme on coded data when certain nodes are unresponsive. We provide the construction of $\nu$-robust PIR schemes that can tolerate up to $\nu$ unresponsive nodes. These schemes are adaptive and universally optimal in the sense of achieving (asymptotically) optimal download cost for any number of unresponsive nodes up to $\nu$

Asymmetry Helps: Improved Private Information Retrieval Protocols for Distributed Storage

We consider private information retrieval (PIR) for distributed storage systems (DSSs) with noncolluding nodes where data is stored using a non maximum distance separable (MDS) linear code. It was recently shown that if data is stored using a particular class of non-MDS linear codes, the MDS-PIR capacity, i.e., the maximum possible PIR rate for MDS-coded DSSs, can be achieved. For this class of codes, we prove that the PIR capacity is indeed equal to the MDS-PIR capacity, giving the first family of non-MDS codes for which the PIR capacity is known. For other codes, we provide asymmetric PIR protocols that achieve a strictly larger PIR rate compared to existing symmetric PIR protocols.Comment: To be presented at 2018 IEEE Information Theory Workshop (ITW'18). See arXiv:1808.09018 for its extended versio

Multiround private information retrieval: Capacity and storage overhead

Private information retrieval (PIR) is the problem of retrieving one message out of $K$ messages from $N$ non-communicating replicated databases, where each database stores all $K$ messages, in such a way that each database learns no information about which message is being retrieved. The capacity of PIR is the maximum number of bits of desired information per bit of downloaded information among all PIR schemes. The capacity has recently been characterized for PIR as well as several of its variants. In every case it is assumed that all the queries are generated by the user simultaneously. Here we consider multiround PIR, where the queries in each round are allowed to depend on the answers received in previous rounds. We show that the capacity of multiround PIR is the same as the capacity of single-round PIR. The result is generalized to also include $T$ -privacy constraints. Combined with previous results, this shows that there is no capacity advantage from multiround over single-round schemes, non-linear over linear schemes or from $\epsilon$ -error over zero-error schemes. However, we show through an example that there is an advantage in terms of storage overhead. We provide an example of a multiround, non-linear, $\epsilon$ -error PIR scheme that requires a strictly smaller storage overhead than the best possible with single-round, linear, zero-error PIR schemes