[Paper Review] New Results on the Storage-Retrieval Tradeoff in Private Information Retrieval Systems

Tao Guo, Ruida Zhou+1 more|arXiv (Cornell University)|Aug 3, 2020

Cryptography and Data Security40 references4 citations

TL;DR

This paper advances the understanding of the storage-retrieval tradeoff in private information retrieval (PIR) systems by introducing a regime-wise 2-approximate characterization, a cyclic permutation lemma for code construction, and a relaxed entropic linear program (LP) for tighter lower bounds. The authors derive new upper and lower bounds that significantly improve upon prior art, even though they do not achieve a tighter approximate characterization in general.

ABSTRACT

In a private information retrieval (PIR) system, the user needs to retrieve one of the possible messages from a set of storage servers, but wishes to keep the identity of requested message private from any given server. Existing efforts in this area have made it clear that the efficiency of the retrieval will be impacted significantly by the amount of the storage space allowed at the servers. In this work, we consider the tradeoff between the storage cost and the retrieval cost. We first present three fundamental results: 1) a regime-wise 2-approximate characterization of the optimal tradeoff, 2) a cyclic permutation lemma that can produce more sophisticated codes from simpler ones, and 3) a relaxed entropic linear program (LP) lower bound that has a polynomial complexity. Equipped with the cyclic permutation lemma, we then propose two novel code constructions, and by applying the lemma, obtain new storage-retrieval points. Furthermore, we derive more explicit lower bounds by utilizing only a subset of the constraints in the relaxed entropic LP in a systematic manner. Though the new upper bound and lower bound do not lead to a more precise approximate characterization in general, they are significantly tighter than the existing art.

Motivation & Objective

• To characterize the fundamental tradeoff between storage cost and retrieval cost in information-theoretic PIR systems without structural storage constraints.
• To develop a general technique—cyclic permutation lemma—that generates more sophisticated PIR codes from simpler ones.
• To design a relaxed entropic LP framework with polynomial complexity to compute tighter lower bounds on the retrieval cost.
• To propose novel code constructions that yield new storage-retrieval tradeoff points.
• To derive explicit, close-form lower bounds by systematically utilizing a subset of constraints in the relaxed LP.

Proposed method

• Proposes a regime-wise 2-approximate characterization of the optimal storage-retrieval tradeoff, partitioning the curve into two regimes where either storage or retrieval cost is approximated within a factor of 2.
• Introduces a cyclic permutation lemma that enables the transformation of simpler PIR codes into more complex ones, recovering known codes like uncoded and MDS-PIR codes as special cases.
• Develops a relaxed entropic LP by selecting a subset of inequalities from the full entropic LP framework, reducing complexity from exponential to polynomial while preserving tightness.
• Applies the cyclic permutation lemma to two novel code constructions to generate new storage-retrieval tradeoff points.
• Derives a family of explicit lower bounds parametrized by real values by systematically applying constraints from the relaxed LP.
• Uses submodularity and privacy constraints in information-theoretic inequalities to derive bounds on conditional entropies in the PIR system.

Experimental results

Research questions

• RQ1 What is the fundamental tradeoff between storage cost and retrieval cost in PIR systems when no structural constraints are imposed on storage codes?
• RQ2 Can a general transformation technique be developed to generate more sophisticated PIR codes from simpler ones?
• RQ3 How can the exponential complexity of the full entropic LP be reduced while maintaining useful lower bounds for the PIR problem?
• RQ4 Can tighter upper and lower bounds be derived for the storage-retrieval tradeoff than previously known?
• RQ5 What are the explicit, closed-form lower bounds achievable by selectively applying constraints in the relaxed entropic LP?

Key findings

• The paper establishes a regime-wise 2-approximate characterization of the optimal storage-retrieval tradeoff, providing a coarse but general approximation across the entire curve.
• The cyclic permutation lemma successfully recovers known PIR codes such as the uncoded storage PIR and MDS-PIR codes from simpler base codes, demonstrating its generality and utility.
• The relaxed entropic LP achieves polynomial complexity and provides significantly tighter lower bounds than previous methods, despite not being the full entropic LP.
• New code constructions, enhanced via the cyclic permutation lemma, yield previously unknown storage-retrieval tradeoff points, expanding the known region of feasible solutions.
• Explicit lower bounds are derived by systematically applying a subset of constraints from the relaxed LP, resulting in a parametrized family of bounds with improved tightness.
• Although the new upper and lower bounds do not yield a more precise approximate characterization than existing work, they are substantially tighter than prior art, improving the practical understanding of the tradeoff.