A Disguise-and-Squeeze PIR Scheme for the MDS-TPIR Setting and Beyond

TL;DR

This paper introduces a novel disguise-and-squeeze PIR scheme for MDS-TPIR systems, improving capacity results, providing more counterexamples to existing conjectures, and offering practical advantages like smaller field sizes and adaptability to various models.

Contribution

It proposes a new PIR scheme that generalizes previous counterexamples, achieves higher rates for certain parameters, and extends to multi-file and cyclic collusion models.

Findings

The scheme generalizes Sun-Jafar counterexamples to broader parameters.

Achieves higher PIR rates for GRS-coded systems with specific parameters.

Provides an $ extit{ extbf{epsilon}}$-error scheme for T ≥ 3.

Abstract

We consider the problem of private information retrieval (PIR) from MDS coded databases with colluding servers, i.e., MDS-TPIR. In the MDS-TPIR setting, $M$ files are stored across $N$ servers, where each file is stored independently using an $(N,K)$-MDS code. A user wants to retrieve one file without disclosing the index of the desired file to any set of up to $T$ colluding servers. The general problem in studying PIR schemes is to maximize the PIR rate, defined as the ratio of the size of the desired file to the size of the total download. Freij-Hollanti et al. proposed a conjecture of the MDS-TPIR capacity (the maximum achievable PIR rate), which was later disproved by Sun and Jafar by a counterexample with $(M,N,T,K)=(2,4,2,2)$. In this paper, we propose a new MDS-TPIR scheme based on a disguise-and-squeeze approach. The features of our scheme include the following. Our scheme generalizes the Sun-Jafar counterexample to $(M,N,T,K)=(2,N,2,K)$ with $N\geq K+2$ for an arbitrary $(N,K)$-MDS coded system, providing more counterexamples to the conjecture by Freij-Hollanti et al. For $(M,N,T,K)=(2,N,2,K)$ and a GRS (generalized Reed-Solomon codes) coded system, our scheme has rate $\frac{N^2-N}{N^2+KN-2K}$, beating the state-of-the-art results. We further show that this rate achieves the linear MDS-TPIR capacity when $K=2$. Our scheme features a significantly smaller field size for implementation and the adaptiveness to generalized PIR models such as multi-file MDS-TPIR and MDS-PIR against cyclically adjacent colluding servers. Lastly, we provide an $\epsilon$-error MDS-TPIR scheme for $T\geq 3$ based on the disguise-and-squeeze framework.