Nearly optimal constructions of PIR and batch codes

Asi, Hilal; Yaakobi, Eitan

doi:10.1109/isit.2017.8006508

Cited by 24 publications

(51 citation statements)

References 14 publications

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“…, m} be disjoint sets of servers, chosen so that the cells in each subset of servers span a subspace containing x i . We choose these subsets so that k i is as large as possible subject to this condition; so k = min{k 1…”

Section: Upper Boundsmentioning

confidence: 99%

“…The classical model of PIR assumes that each server stores a copy of an n-bit database, so the storage overhead, namely the ratio between the total number of bits stored by all servers and the size of the database, is k. However, recent work combines PIR protocols with techniques from distributed storage (where each server stores only some of the database) to reduce the storage overhead. This approach was first considered in [33], and several papers have developed this direction further: [1,2,3,4,5,13,14,15,32,36,37,38,39,43,45]. Our discussion will follow the breakthrough approach presented by Fazeli, Vardy, and Yaakobi [13,14].…”

Section: Introductionmentioning

confidence: 99%

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PIR Array Codes With Optimal Virtual Server Rate

Blackburn

Etzion

2019

IEEE Trans. Inform. Theory

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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 ≤ 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 ≤ 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.

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Section: Upper Boundsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

PIR Array Codes With Optimal Virtual Server Rate

Blackburn

Etzion

2019

IEEE Trans. Inform. Theory

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“…Those constructions were based on unbalanced expanders, on recursive application of trivial batch codes, on smooth and Reed-Muller codes, and others. Many other constructions proposed later in [2]- [4] improve the redundancy of batch codes. In particular, a systematic linear code, defined by the generator matrix G = [I n |E], is shown [3] to be a kbatch code, where k is the minimal number of ones in rows of E and the bipartite graph, whose biadjacency matrix is E, has no cycle of length at most 6.…”

Section: B Related Workmentioning

confidence: 99%

Constructions of Batch Codes via Finite Geometry

Polyanskii

Vorobyev

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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“…Server S 1 receives q 1 , and so (since W ⊆ T ) replies with c * 1 . But there are at most 2 d possible replies c ℓ other from the remaining servers by (1), and so there are at most 2 d responses (c * 1 , c ℓ other ) to the query (q 1 , q ℓ other ). Since |W | = 2 d + 1, there are two databases X, Y ∈ W such that the servers respond identically.…”

Section: Lower Bounds On the Download Complexitymentioning

confidence: 99%

“…A new subspace approach for such codes was given recently in [36,37]. Another family of related codes with similar properties are batch codes, which were first defined by Ishai, Kushilevitz, Ostrovsky, and Sahai [26] and were recently studied by many others, for example [1,2,35]. It is important to note that all these codes are very important in the theory of distributed storage codes.…”

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confidence: 99%

PIR schemes with small download complexity and low storage requirements

Blackburn

Etzion

Paterson

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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In the classical model for (information theoretically secure) Private Information Retrieval (PIR) due to Chor, Goldreich, Kushilevitz and Sudan, a user wishes to retrieve one bit of a database that is stored on a set of n servers, in such a way that no individual server gains information about which bit the user is interested in. The aim is to design schemes that minimise the total communication between the user and the servers. More recently, there have been moves to consider more realistic models where the total storage of the set of servers, or the per server storage, should be minimised (possibly using techniques from distributed storage), and where the database is divided into R-bit records with R > 1, and the user wishes to retrieve one record rather than one bit. When R is large, downloads from the servers to the user dominate the communication complexity and so the aim is to minimise the total number of downloaded bits. Work of Shah, Rashmi and Ramchandran shows that at least R + 1 bits must be downloaded from servers in the worst case, and provides PIR schemes meeting this bound. Sun and Jafar have considered the download cost of a scheme, defined as the ratio of the message length R and the total number of bits downloaded. They determine the best asymptotic download cost of a PIR scheme (as R → ∞) when a database of k messages is stored by n servers.This paper provides various bounds on the download complexity of a PIR scheme, generalising those of Shah et al. to the case when the number n of servers is bounded, and providing links with classical techniques due to Chor et al. The paper also provides a range of constructions for PIR schemes that are either simpler or perform better than previously known schemes. These constructions include explicit schemes that achieve the best asymptotic download complexity of Sun and Jafar with significantly lower upload complexity, and general techniques for constructing a scheme with good worst case download complexity from a scheme with good download complexity on average. * Parts of this paper were presented at the International Symposium on Information Theory, The PIR ModelIn the classical model for private information retrieval (PIR) due to Chor, Goldreich, Kushilevitz and Sudan [14], a database X is replicated across n servers S 1 , S 2 , . . . , S n . A user wishes to retrieve one bit of the database, so sends a query to each server and downloads their reply. The user should be able to deduce the bit from the servers' replies. Moreover, no single server should gain any information on which bit the user wishes to retrieve (without collusion). The resulting protocol is known as an (information-theoretic) PIR scheme; there are also computational variants of the security model [30]. The goal of PIR is to minimise the total communication between the user and the servers.In practice, the assumption that the user only wishes to retrieve one bit of the database, and the assumption that there is no shortage of server storage seem unrealistic. Because of this, many recen...

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Nearly optimal constructions of PIR and batch codes

Cited by 24 publications

References 14 publications

PIR Array Codes With Optimal Virtual Server Rate

PIR Array Codes With Optimal Virtual Server Rate

Constructions of Batch Codes via Finite Geometry

PIR schemes with small download complexity and low storage requirements

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