Cross Subspace Alignment and the Asymptotic Capacity of $X$ -Secure $T$ -Private Information Retrieval

Jia, Zhonghua; Sun, Hua; Jafar, Syed A.

doi:10.1109/tit.2019.2916079

Cited by 104 publications

(89 citation statements)

References 53 publications

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“…which is strictly worse than that of [11] when E > 0. This essentially stems from the fact that it is not clear how to align the noise terms arising from the function-privacy randomness and data-privacy randomness as in [11] for non-linear functions. Thus our PC scheme leaves room for improvement in the case of non-trivial data privacy.…”

Section: E Comparison With Previous Workmentioning

confidence: 79%

“…Additional interesting and practically relevant variants on PIR appear Parts of this work were presented at the International Symposium on Information Theory (ISIT), Vail, CO, USA, 2018. when one accounts for unreliability of the servers, whether they be unresponsive, malicious, or curious as to the contents of the data they are storing. Variants along these lines have been studied and capacity expressions have been derived in [3,11,24].…”

Section: A Private Information Retrievalmentioning

confidence: 99%

“…The capacity for the case E > 0 is subtler. The case of K = 1, G = 1, P = A = 0 is that of the recent work [11], which studies the PIR problem under the constraints of T -function privacy and E-data privacy, for storage systems in which every server stores an amount of data which is comparable to the entire dataset. In [11], the authors show that…”

Section: E Comparison With Previous Workmentioning

confidence: 99%

“…In the first round, the polynomial r (1) (z) has degree ≤ N − 1 = 10, hence its evaluation vector lives in a Reed-Solomon code with parameters [14,11,4], allowing the user to correct the erasure and error from the straggler and adversary. The user then decodes γ 10 , .…”

Section: Examplementioning

confidence: 99%

“…In all successive rounds, it is clear from the above expressions that all coefficients of negative powers of z are decoded in previous rounds, and can thus be subtracted from the response vector. After this subtraction, what is left is in a Reed-Solomon code with parameters [14,11,4], which can correct P = 1 erasure and A = 1 error. The user performs this erasure/error correction, and then decodes the coefficients of 1, z, z 2 , z 3 in every round.…”

Section: Examplementioning

confidence: 99%

See 4 more Smart Citations

Private Polynomial Computation From Lagrange Encoding

Raviv

Karpuk

2020

IEEE Trans.Inform.Forensic Secur.

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Private computation is a generalization of private information retrieval, in which a user is able to compute a function on a distributed dataset without revealing the identity of that function to the servers. In this paper it is shown that Lagrange encoding, a powerful technique for encoding Reed-Solomon codes, enables private computation in many cases of interest. In particular, we present a scheme that enables private computation of polynomials of any degree on Lagrange encoded data, while being robust to Byzantine and straggling servers, and to servers colluding to attempt to deduce the identities of the functions to be evaluated. Moreover, incorporating ideas from the well-known Shamir secret sharing scheme allows the data itself to be concealed from the servers as well. Our results extend private computation to high degree polynomials and to data-privacy, and reveal a tight connection between private computation and coded computation.

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Section: E Comparison With Previous Workmentioning

confidence: 79%

Section: A Private Information Retrievalmentioning

confidence: 99%

Section: E Comparison With Previous Workmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

Section: Examplementioning

confidence: 99%

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Private Polynomial Computation From Lagrange Encoding

Raviv

Karpuk

2020

IEEE Trans.Inform.Forensic Secur.

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On the Capacity and Straggler-Robustness of Distributed Secure Matrix Multiplication

2019

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Computationally efficient matrix multiplication is a fundamental requirement in various fields, including and particularly in data analytics. To do so, the computation task of large-scale matrix multiplication is typically outsourced to multiple servers. However, due to data misusage at the servers, security is typical of concern. In this paper, we first study the two-sided secure matrix multiplication problem, where a user is interested in the matrix product AB of two finite field private matrices A and B from an information-theoretic perspective. In this problem, the user exploits the computational resources of N servers to compute the matrix product but simultaneously tries to conceal the private matrices from the servers. Our goal is twofold: (i) to maximize the downlink communication rate, and (ii) to minimize the effective number of server observations needed to determine AB, while preserving security, where we allow for up to ≤ N servers to collude. To this end, we propose two schemes -an aligned secret sharing scheme (A3S) and a secure cross subspace alignment (SCSA) scheme. For A3S, we optimize the partitioning of matrices A and B in order to either optimize objective (i) or (ii) as a function of the system parameters (e.g., N and ). A proposed inductive approach gives us analytical, close-to-optimal solutions for both (i) and (ii). The SCSA, on the other hand, is shown to be (rate) capacity-optimal for the general J -sided distributed secure matrix multiplication problem J j=1 M j . We show this by developing a recursive information-theoretic upper bound (converse) on the downlink rate for the J -sided secure matrix multiplication problem. With respect to (i), both A3S and SCSA, significantly outperform the state-of-the-art in terms of (a) communication rate, (b) maximum tolerable number of colluding servers, and (c) computational complexity. Overall SCSA (A3S) is the preferred choice when the focus is on the downlink (uplink).

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The Capacity of Private Information Retrieval with Eavesdroppers

Wang

Sun

Skoglund

2018

2018 52nd Asilomar Conference on Signals, Systems, and Computers

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We consider the problem of private information retrieval (PIR) with colluding servers and eavesdroppers (abbreviated as ETPIR). The ETPIR problem is comprised of K messages, N servers where each server stores all K messages, a user who wants to retrieve one of the K messages without revealing the desired message index to any set of T colluding servers, and an eavesdropper who can listen to the queries and answers of any E servers but is prevented from learning any information about the messages. The information theoretic capacity of ETPIR is defined to be the maximum number of desired message symbols retrieved privately per information symbol downloaded. We show that the capacity of ETPIR is

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Cross Subspace Alignment and the Asymptotic Capacity of $X$ -Secure $T$ -Private Information Retrieval

Cited by 104 publications

References 53 publications

Private Polynomial Computation From Lagrange Encoding

Private Polynomial Computation From Lagrange Encoding

On the Capacity and Straggler-Robustness of Distributed Secure Matrix Multiplication

The Capacity of Private Information Retrieval with Eavesdroppers

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