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Register a new userA new Capacity-Achieving Private Information Retrieval Scheme with (Almost) Optimal File Length for Coded Servers
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Added by
Jinbao Zhu
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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In a distributed storage system, private information retrieval (PIR) guarantees that a user retrieves one file from the system without revealing any information about the identity of its interested file to any individual server. In this paper, we investigate $(N,K,M)$ coded sever model of PIR, where each of $M$ files is distributed to the $N$ servers in the form of $(N,K)$ maximum distance separable (MDS) code for some $N>K$ and $M>1$. As a result, we propose a new capacity-achieving $(N,K,M)$ coded linear PIR scheme such that it can be implemented with file length $frac{K(N-K)}{gcd(N,K)}$, which is much smaller than the previous best result $Kbig(frac{N}{gcd(N,K)}big)^{M-1}$. Notably, among all the capacity-achieving coded linear PIR schemes, we show that the file length is optimal if $M>biglfloor frac{K}{gcd(N,K)}-frac{K}{N-K}bigrfloor+1$, and within a multiplicative gap $frac{K}{gcd(N,K)}$ of a lower bound on the minimum file length otherwise.
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This paper investigates reducing sub-packetization of capacity-achieving schemes for uncoded Storage Constrained Private Information Retrieval (SC-PIR) systems. In the SC-PIR system, a user aims to retrieve one out of $K$ files from $N$ servers while revealing nothing about its identity to any individual server, in which the $K$ files are stored at the $N$ servers in an uncoded form and each server can store up to $mu K$ equivalent files, where $mu$ is the normalized storage capacity of each server. We first prove that there exists a capacity-achieving SC-PIR scheme for a given storage design if and only if all the packets are stored exactly at $Mtriangleq mu N$ servers for $mu$ such that $M=mu Nin{2,3,ldots,N}$. Then, the optimal sub-packetization for capacity-achieving linear SC-PIR schemes is characterized as the solution to an optimization problem, which is typically hard to solve because of involving indicator functions. Moreover, a new notion of array called Storage Design Array (SDA) is introduced for the SC-PIR system. With any given SDA, an associated capacity-achieving SC-PIR scheme is constructed. Next, the SC-PIR schemes that have equal-size packets are investigated. Furthermore, the optimal equal-size sub-packetization among all capacity-achieving linear SC-PIR schemes characterized by Woolsey et al. is proved to be $frac{N(M-1)}{gcd(N,M)}$. Finally, by allowing unequal size of packets, a greedy SDA construction is proposed, where the sub-packetization of the associated SC-PIR scheme is upper bounded by $frac{N(M-1)}{gcd(N,M)}$. Among all capacity-achieving linear SC-PIR schemes, the sub-packetization is optimal when $min{M,N-M}|N$ or $M=N$, and within a multiplicative gap $frac{min{M,N-M}}{gcd(N,M)}$ of the optimal one otherwise. In particular, for the case $N=dcdot Mpm1$ where $dgeq 2$, another SDA is constructed to obtain lower sub-packetization.
We consider constructing capacity-achieving linear codes with minimum message size for private information retrieval (PIR) from $N$ non-colluding databases, where each message is coded using maximum distance separable (MDS) codes, such that it can be recovered from accessing the contents of any $T$ databases. It is shown that the minimum message size (sometimes also referred to as the sub-packetization factor) is significantly, in fact exponentially, lower than previously believed. More precisely, when $K>T/textbf{gcd}(N,T)$ where $K$ is the total number of messages in the system and $textbf{gcd}(cdot,cdot)$ means the greatest common divisor, we establish, by providing both novel code constructions and a matching converse, the minimum message size as $textbf{lcm}(N-T,T)$, where $textbf{lcm}(cdot,cdot)$ means the least common multiple. On the other hand, when $K$ is small, we show that it is in fact possible to design codes with a message size even smaller than $textbf{lcm}(N-T,T)$.
In the classical private information retrieval (PIR) setup, a user wants to retrieve a file from a database or a distributed storage system (DSS) without revealing the file identity to the servers holding the data. In the quantum PIR (QPIR) setting, a user privately retrieves a classical file by receiving quantum information from the servers. The QPIR problem has been treated by Song emph{et al.} in the case of replicated servers, both without collusion and with all but one servers colluding. In this paper, the QPIR setting is extended to account for maximum distance separable (MDS) coded servers. The proposed protocol works for any $[n,k]$-MDS code and $t$-collusion with $t=n-k$. Similarly to the previous cases, the rates achieved are better than those known or conjectured in the classical counterparts. Further, it is demonstrated how the protocol can adapted to achieve significantly higher retrieval rates from DSSs encoded with a locally repairable code (LRC) with disjoint repair groups, each of which is an MDS code.
In quantum private information retrieval (QPIR), a user retrieves a classical file from multiple servers by downloading quantum systems without revealing the identity of the file. The QPIR capacity is the maximal achievable ratio of the retrieved file size to the total download size. In this paper, the capacity of QPIR from MDS-coded and colluding servers is studied. Two classes of QPIR, called stabilizer QPIR and dimension squared QPIR induced from classical strongly linear PIR are defined, and the related QPIR capacities are derived. For the non-colluding case, the general QPIR capacity is derived when the number of files goes to infinity. The capacities of symmetric and non-symmetric QPIR with coded and colluding servers are proved to coincide, being double to their classical counterparts. A general statement on the converse bound for QPIR with coded and colluding servers is derived showing that the capacities of stabilizer QPIR and dimension squared QPIR induced from any class of PIR are upper bounded by twice the classical capacity of the respective PIR class. The proposed capacity-achieving scheme combines the star-product scheme by Freij-Hollanti et al. and the stabilizer QPIR scheme by Song et al. by employing (weakly) self-dual Reed--Solomon codes.
This work investigates a system where each user aims to retrieve a scalar linear function of the files of a library, which are Maximum Distance Separable coded and stored at multiple distributed servers. The system needs to guarantee robust decoding in the sense that each user must decode its demanded function with signals received from any subset of servers whose cardinality exceeds a threshold. In addition, (a) the content of the library must be kept secure from a wiretapper who obtains all the signals from the servers;(b) any subset of users together can not obtain any information about the demands of the remaining users; and (c) the users demands must be kept private against all the servers even if they collude. Achievable schemes are derived by modifying existing Placement Delivery Array (PDA) constructions, originally proposed for single-server single-file retrieval coded caching systems without any privacy or security or robustness constraints. It is shown that the PDAs describing the original Maddah-Ali and Niesens coded caching scheme result in a load-memory tradeoff that is optimal to within a constant multiplicative gap, except for the small memory regime when the number of file is smaller than the number of users. As by-products, improved order optimality results are derived for three less restrictive systems in all parameter regimes.
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