[Paper Review] Achieving Maximum Distance Separable Private Information Retrieval Capacity With Linear Codes

Siddhartha Kumar, Hsuan-Yin Lin+2 more|arXiv (Cornell University)|Dec 11, 2017

Advanced Data Storage Technologies4 citations

TL;DR

This paper proposes three private information retrieval (PIR) protocols for distributed storage systems using arbitrary linear codes. Protocol 1 achieves the finite maximum distance separable (MDS) PIR capacity with exponential file size, while Protocol 2 achieves the asymptotic MDS-PIR capacity with polynomial file size. The key contribution is identifying necessary and sufficient conditions for codes to achieve optimal PIR rates, with cyclic codes, Reed-Muller codes, and distance-optimal local reconstruction codes shown to attain the MDS-PIR capacity under both protocols.

ABSTRACT

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.

Motivation & Objective

• To design efficient PIR protocols for distributed storage systems (DSSs) that ensure user query privacy while minimizing download cost.
• To achieve the maximum possible PIR rate—defined as the ratio of retrieved file size to downloaded data—using linear codes.
• To establish necessary and sufficient conditions under which linear codes can achieve the finite and asymptotic MDS-PIR capacity.
• To extend PIR protocols to handle colluding servers, improving upon prior schemes.
• To provide a general algorithm to optimize PIR rates across all proposed protocols.

Proposed method

• Proposes Protocol 1 and Protocol 2 for noncolluding servers, using query design based on code automorphisms and generator matrix structure to ensure privacy and optimize download rate.
• Employs generalized Hamming weights and code decomposition to analyze the rank of the query matrix and derive bounds on achievable PIR rates.
• Introduces Protocol 3 for colluding servers, based on a modified query construction that accounts for collusion constraints and maximizes the PIR rate under this threat model.
• Derives necessary and sufficient conditions for a linear code to achieve the maximum PIR rate in each protocol, based on structural properties such as code automorphisms and subcode decomposition.
• Develops a general optimization algorithm to compute the optimal PIR rate for any given linear code by analyzing the rank of the query matrix and exploiting code symmetry.
• Uses Gaussian elimination on the query matrix to simplify the rank computation and derive upper bounds on the dimension of the code spanned by the queries.

Experimental results

Research questions

• RQ1 What conditions on a linear code ensure that it can achieve the finite MDS-PIR capacity in a noncolluding DSS?
• RQ2 Can a PIR protocol be designed with file size independent of the number of files while still achieving the asymptotic MDS-PIR capacity?
• RQ3 What structural properties of a linear code allow it to achieve the maximum PIR rate under colluding server models?
• RQ4 How can the PIR rate be optimized for arbitrary linear codes in both noncolluding and colluding server scenarios?
• RQ5 Which families of codes—such as cyclic codes, Reed-Muller codes, or local reconstruction codes—achieve the MDS-PIR capacity under the proposed protocols?

Key findings

• Protocol 1 achieves the finite MDS-PIR capacity for certain linear codes, with file size exponential in the number of files.
• Protocol 2 achieves the asymptotic MDS-PIR capacity with file size independent of the number of files, making it more practical.
• Cyclic codes, Reed-Muller codes, and a class of distance-optimal local reconstruction codes achieve both the finite and asymptotic MDS-PIR capacity under Protocols 1 and 2, respectively.
• For colluding servers, Protocol 3 achieves the maximum possible PIR rate when the code satisfies a specific structural condition based on automorphisms and subcode decomposition.
• Reed-Muller codes achieve the maximum PIR rate under Protocol 3, demonstrating their robustness in colluding server settings.
• An optimization algorithm is provided that computes the optimal PIR rate for any linear code by analyzing the rank of the query matrix and exploiting code symmetry.