Private Polynomial Computation for Noncolluding Coded Databases

Obead, Sarah A.; Lin, Hsuan-Yin; Rosnes, Eirik; Kliewer, Joerg

doi:10.1109/isit.2019.8849825

Cited by 9 publications

(19 citation statements)

References 29 publications

(220 reference statements)

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“…upload constraints [46], arbitrary collusion patterns [21], [47], single server PIR with user side information [48]- [54], latent-variable single server PIR [55], as well as applications of PIR to private computation [56]- [59], private search [60], private set intersection [45], coded computing [61], locally decodable codes [62], etc.…”

Section: Introductionmentioning

confidence: 99%

Double Blind T-Private Information Retrieval

Jia

Jafar

2021

IEEE J. Sel. Areas Inf. Theory

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Double blind T -private information retrieval (DB-TPIR) enables two users, each of whom specifies an index (θ 1 , θ 2 , resp.), to efficiently retrieve a message W (θ 1 , θ 2 ) labeled by the two indices, from a set of N servers that store all messagesthat the two users' indices are kept private from any set of up to T 1 , T 2 colluding servers, respectively, as well as from each other. A DB-TPIR scheme based on cross-subspace alignment is proposed in this paper, and shown to be capacity-achieving in the asymptotic setting of large number of messages and bounded latency. The scheme is then extended to M -way blind X-secure T -private information retrieval (MB-XS-TPIR) with multiple (M ) indices, each belonging to a different user, arbitrary privacy levels for each index (T 1 , T 2 , • • • , T M ), and arbitrary level of security (X) of data storage, so that the message W (θ 1 , θ 2 , • • • , θ M ) can be efficiently retrieved while the stored data is held secure against collusion among up to X colluding servers, the m th user's index is private against collusion among up to T m servers, and each user's index θ m is private from all other users. The general scheme relies on a tensor-product based extension of cross-subspace alignment and retrieves 1 − (X + T 1 + • • • + T M )/N bits of desired message per bit of download.

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Section: Introductionmentioning

confidence: 99%

Double Blind T-Private Information Retrieval

Jia

Jafar

2021

IEEE J. Sel. Areas Inf. Theory

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“…This problem is referred to as Private Computation (PC), which seeks efficient solutions for the user to compute a function of files stored at distributed servers, without disclosing the identity of the desired function to the servers. The PC problem was firstly introduced in [12], [20] and has attracted remarkable attention in the past few years within information-theoretic community [9], [13], [15], [16]. In the classical PC setup, the user wishes to compute one out of any ξ candidate functions over M files from N non-colluding servers, each of which stores all the M files, while preventing any individual server from obtaining information about which function is being computed.…”

Section: Introductionmentioning

confidence: 99%

“…−1 . Moreover, they [15] further constructed PLC schemes on arbitrary linear storage codes and showed that the capacity of MDS-PLC can be achieved for a large class of linear codes.…”

Section: Introductionmentioning

confidence: 99%

“…Particularly in [15], [9], [16], the problem of PC was focused on the setup that the candidate functions are polynomials with maximum degree G in M variables (files) over a finite field F p , called Private Polynomial Computation (PPC). Very recently, Obeady et al [15] presented two novel non-colluding PPC schemes from systematic and nonsystematic Reed-Solomon coded servers for arbitrary number of candidate polynomial functions, also referred to as systematic MDS-PPC and nonsystematic MDS-PPC, respectively. In [9], [16], the ξ candidate functions were restricted to be a finite-dimensional vector space (or sub-space) of polynomials over F p .…”

Section: Introductionmentioning

confidence: 99%

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Symmetric Private Polynomial Computation From Lagrange Encoding

Zhu¹,

Yan²,

Tang³

2020

Preprint

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The problem of X-secure T -colluding symmetric Private Polynomial Computation (PPC) from coded storage system with B Byzantine and U unresponsive servers is studied in this paper. Specifically, a dataset consisting of M files are stored across N distributed servers according to (N, K + X) Maximum Distance Separable (MDS) codes such that any group of up to X colluding servers can not learn anything about the data files. A user wishes to privately evaluate one out of a set of candidate polynomial functions over the M files from the system, while guaranteeing that any T colluding servers can not learn anything about the identity of the desired function and the user can not learn anything about the M data files more than the desired polynomial function, in the presence of B Byzantine servers that can send arbitrary responses maliciously to confuse the user and U unresponsive servers that will not respond any information at all. Two novel symmetric PPC schemes using Lagrange encoding are proposed. Both the two schemes achieve the same PPC rate 1 − G(K+X−1)+T +2B N−U, secrecy rate G(K+X−1)+T N−(G(K+X−1)+T +2B+U ) , finite field size and decoding complexity, where G is the maximum degree over all the candidate polynomial functions. Particularly, the first scheme focuses on the general case that the candidate functions are consisted of arbitrary polynomials, and the second scheme restricts the candidate functions to be a finite-dimensional vector space (or sub-space) of polynomials over Fp but requires less upload cost, query complexity and server computation complexity. Remarkably, the PPC setup studied in this paper generalizes all the previous MDS coded PPC setups and the two degraded schemes strictly outperform the best known schemes in terms of (asymptotical) PPC rate, which is the main concern of the PPC schemes.

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“…In [10] the capacity for private linear computation in MDS coded databases is studied. Recently the new problem of retrieving a polynomial function of files from some servers has been introduced and discussed in [11] and [12] by using Lagrange encoding in coded databases.…”

Section: Introductionmentioning

confidence: 99%

Private Inner Product Retrieval for Distributed Machine Learning

Mousavi

Maddah-Ali

Mirmohseni

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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In this paper, we argue that in many basic algorithms for machine learning, including support vector machine (SVM) for classification, principal component analysis (PCA) for dimensionality reduction, and regression for dependency estimation, we need the inner products of the data samples, rather than the data samples themselves.Motivated by the above observation, we introduce the problem of private inner product retrieval for distributed machine learning, where we have a system including a database of some files, duplicated across some non-colluding servers. A user intends to retrieve a subset of specific size of the inner products of the data files with minimum communication load, without revealing any information about the identity of the requested subset. For achievability, we use the algorithms for multi-message private information retrieval. For converse, we establish that as the length of the files becomes large, the set of all inner products converges to independent random variables with uniform distribution, and derive the rate of convergence. To prove that, we construct special dependencies among sequences of the sets of all inner products with different length, which forms a time-homogeneous irreducible Markov chain, without affecting the marginal distribution. We show that this Markov chain has a uniform distribution as its unique stationary distribution, with rate of convergence dominated by the second largest eigenvalue of the transition probability matrix. This allows us to develop a converse, which converges to a tight bound in some cases, as the size of the files becomes large. While this converse is based on the one in multi-message private information retrieval due to the nature of retrieving inner products instead of data itself some changes are made to reach the desired result.

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Private Polynomial Computation for Noncolluding Coded Databases

Cited by 9 publications

References 29 publications

Double Blind T-Private Information Retrieval

Double Blind T-Private Information Retrieval

Symmetric Private Polynomial Computation From Lagrange Encoding

Private Inner Product Retrieval for Distributed Machine Learning

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