Achieving Maximum Distance Separable Private Information Retrieval Capacity With Linear Codes
Journal article, 2019

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.

linear codes

distributed storage

Reed-Muller codes

private information retrieval

Code automorphisms

colluding servers

generalized Hamming weight

local reconstruction codes


Siddhartha Kumar

Simula UiB

Hsuan-Yin Lin

Simula UiB

Eirik Rosnes

Simula UiB

Alexandre Graell i Amat

Chalmers, Electrical Engineering, Communication and Antenna Systems, Communication Systems

IEEE Transactions on Information Theory

0018-9448 (ISSN)

Vol. 65 7 4243-4273

Rethinking Distributed Storage for Data Storage and Wireless Content Delivery

Swedish Research Council (VR), 2016-01-01 -- 2019-12-31.

Subject Categories

Computer Engineering


Communication Systems

Areas of Advance

Information and Communication Technology



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