[Paper Review] On Sub-Packetization of Capacity-Achieving PIR Schemes for MDS Coded Databases.

Jingke Xu, Zhifang Zhang|arXiv (Cornell University)|Dec 7, 2017

Cryptography and Data Security3 citations

TL;DR

This paper establishes the optimal sub-packetization level for linear capacity-achieving private information retrieval (PIR) schemes in MDS-coded databases. It proves that the minimal sub-packetization is $ Kn^{M-1} $, where $ n = N / ext{gcd}(N,K) $, and presents a scheme achieving this bound, thus resolving the optimality of sub-packetization in nontrivial coded-PIR settings.

ABSTRACT

Consider the problem of private information retrieval (PIR) over a distributed storage system where $M$ records are stored across $N$ servers by using an $[N,K]$ MDS code. For simplicity, this problem is usually referred as the coded-PIR problem. The capacity of coded-PIR with privacy against any individual server was determined by Banawan and Ulukus in 2016, i.e., $\mathcal{C}_{ iny C-PIR}=(1+\frac{K}{N}+\dots+\frac{K^{M-1}}{N^{M-1}})^{-1}$. They also presented a linear capacity-achieving scheme with sub-packetization $KN^{M}$. In this paper we focus on minimizing the sub-packetization for linear capacity-achieving coded-PIR schemes. We prove that the sub-packetization for all linear capacity-achieving coded-PIR schemes in the nontrivial cases (i.e. $N>K\geq 1$ and $M>1$) must be no less than $Kn^{M-1}$, where $n=N/{ m gcd}(N,K)$. Moreover, we design a linear capacity-achieving coded-PIR scheme with sub-packetization $Kn^{M-1}$ for all $N>K\geq 1$ and $M>1$. Therefore, $Kn^{M-1}$ is the optimal sub-packetization for linear capacity-achieving coded-PIR schemes.

Motivation & Objective

• To determine the minimal possible sub-packetization for linear capacity-achieving coded-PIR schemes in MDS-coded databases.
• To close the gap between known upper bounds and theoretical lower bounds on sub-packetization in nontrivial coded-PIR scenarios.
• To provide a constructive scheme that achieves the proven lower bound on sub-packetization.
• To establish that $ Kn^{M-1} $ is the optimal sub-packetization level for all $ N > K o 1 $ and $ M > 1 $.
• To generalize the understanding of sub-packetization trade-offs in linear coded-PIR constructions.

Proposed method

• Derives a theoretical lower bound on sub-packetization for linear capacity-achieving coded-PIR schemes using algebraic and combinatorial arguments.
• Introduces a new construction based on structured coding and symmetric access patterns to achieve the lower bound.
• Defines $ n = N / ext{gcd}(N,K) $ as a key parameter to scale the sub-packetization in the scheme.
• Constructs a linear coded-PIR scheme with sub-packetization $ Kn^{M-1} $, matching the proven lower bound.
• Uses MDS code properties and symmetry in server access patterns to ensure privacy and capacity optimality.
• Employs linear algebra and code structure to ensure that all queries are linear combinations of stored symbols, preserving linearity.

Experimental results

Research questions

• RQ1 What is the minimal possible sub-packetization for linear capacity-achieving coded-PIR schemes in nontrivial settings?
• RQ2 Can a linear coded-PIR scheme be constructed with sub-packetization matching the theoretical lower bound?
• RQ3 How does the greatest common divisor of $ N $ and $ K $ affect the minimal sub-packetization in coded-PIR?
• RQ4 Is $ Kn^{M-1} $, with $ n = N / ext{gcd}(N,K) $, the optimal sub-packetization across all $ N > K o 1 $ and $ M > 1 $?
• RQ5 Can the lower bound on sub-packetization be achieved constructively while maintaining capacity optimality and privacy?

Key findings

• The minimal sub-packetization for linear capacity-achieving coded-PIR schemes in nontrivial cases ($ N > K o 1 $, $ M > 1 $) is $ Kn^{M-1} $, where $ n = N / ext{gcd}(N,K) $.
• This lower bound is tight, as a linear capacity-achieving scheme is explicitly constructed with exactly $ Kn^{M-1} $ sub-packetization.
• The sub-packetization bound is strictly smaller than the previously known $ KN^M $, representing a significant reduction.
• The optimality holds for all $ N > K o 1 $ and $ M > 1 $, covering all nontrivial coded-PIR configurations.
• The construction achieves both capacity optimality and privacy while minimizing sub-packetization, confirming the theoretical bound as achievable.
• The result resolves the open problem of sub-packetization minimization in linear coded-PIR, establishing $ Kn^{M-1} $ as the optimal value.