[Paper Review] On private information retrieval array codes

Yiwei Zhang, Xin Wang+2 more|arXiv (Cornell University)|Sep 29, 2016

Cryptography and Data Security7 references20 citations

TL;DR

This paper studies optimal private information retrieval (PIR) array codes for distributed storage systems, proposing new constructions and bounds for PIR rates when each server stores a fraction $1/s$ of the database. For $s > 2$, it derives a new upper bound on PIR rate and demonstrates that the Blackburn–Etzion construction achieves a higher rate than all prior constructions, supporting a conjecture on its optimality.

ABSTRACT

Given a database, the private information retrieval (PIR) protocol allows a user to make queries to several servers and retrieve a certain item of the database via the feedbacks, without revealing the privacy of the specific item to any single server. Classical models of PIR protocols require that each server stores a whole copy of the database. Recently new PIR models are proposed with coding techniques arising from distributed storage system. In these new models each server only stores a fraction $1/s$ of the whole database, where $s>1$ is a given rational number. PIR array codes are recently proposed by Fazeli, Vardy and Yaakobi to characterize the new models. Consider a PIR array code with $m$ servers and the $k$-PIR property (which indicates that these $m$ servers may emulate any efficient $k$-PIR protocol). The central problem is to design PIR array codes with optimal rate $k/m$. Our contribution to this problem is three-fold. First, for the case $12$, we derive a new upper bound on the rate of a PIR array code. Finally, for the case $s>2$, we analyze a new construction by Blackburn and Etzion and show that its rate is better than all the other existing constructions.

Motivation & Objective

• To determine the minimum number of servers required for optimal-rate PIR array codes when $1 < s ≤ 2$, particularly for $t > d^2 - d$.
• To derive a new upper bound on the rate of PIR array codes for $s > 2$, improving upon existing theoretical limits.
• To analyze and compare the Blackburn–Etzion construction with prior constructions, showing its rate superiority.
• To provide intuitive and structural evidence supporting the conjecture that the Blackburn–Etzion construction achieves optimal rate for $s > 2$.

Proposed method

• For $1 < s ≤ 2$, the authors determine the minimal number of servers that admit a PIR array code with optimal rate by analyzing parameter constraints and code structure.
• For $s > 2$, they derive a new theoretical upper bound on the PIR rate using combinatorial and linear algebraic techniques on the structure of array codes.
• They analyze the Blackburn–Etzion construction by expressing its rate as a weighted average of terms $\frac{t + p - i}{2p}$, with weights $\alpha_i$ corresponding to servers with $t - i$ singleton cells.
• They compare the rate of the Blackburn–Etzion construction to existing constructions using asymptotic and exact evaluations, showing it exceeds all prior rates.
• They use symmetry and permutation arguments to justify assuming uniform distribution of server types, supporting structural optimality reasoning.
• They perform inductive verification of key inequalities, such as $(4t+2)\binom{2t-1}{t-1} > 2^{2t}$, to validate rate superiority for $s = 3$.

Experimental results

Research questions

• RQ1 What is the minimum number of servers required to achieve optimal rate in PIR array codes when $1 < s ≤ 2$ for a given $t$?
• RQ2 Can a tighter upper bound on the PIR rate be derived for $s > 2$ than previously known?
• RQ3 Does the Blackburn–Etzion construction achieve a higher rate than all other known constructions for $s > 2$?
• RQ4 Is the Blackburn–Etzion construction optimal in rate for $s > 2$, and what structural properties support this conjecture?
• RQ5 How do the relative proportions of different server types (in terms of singleton and sum components) affect the overall PIR rate?

Key findings

• For $1 < s ≤ 2$, the paper determines the minimum number of servers that admit a PIR array code with optimal rate for the parameter range $t > d^2 - d$.
• For $s > 2$, the authors derive a new upper bound on the rate of PIR array codes, improving upon previous theoretical limits.
• The Blackburn–Etzion construction achieves a rate strictly greater than $\frac{4t+1}{6t+3}$ for $s = 3$, verified via inductive proof of the inequality $(4t+2)\binom{2t-1}{t-1} > 2^{2t}$.
• The rate of the Blackburn–Etzion construction exceeds all other known constructions for $s > 2$, as shown through direct comparison and rate expression analysis.
• The construction's rate is a weighted average of terms $\frac{t + p - i}{2p}$, and the inclusion of servers with fewer than $t-1$ singleton cells is shown to likely reduce the overall rate.
• The paper provides strong intuitive evidence supporting Conjecture 18: the Blackburn–Etzion construction achieves optimal rate for $s > 2$, based on optimal partner selection and server type distribution.