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 $ε$-error MDS-TPIR scheme for $T\geq 3$ based on the disguise-and-squeeze framework.

翻译：本文研究从采用MDS编码且存在合谋服务器的数据库中检索私有信息的问题，即MDS-TPIR。在MDS-TPIR场景中，$M$个文件通过$(N,K)$-MDS编码独立存储于$N$个服务器上。用户希望在不向任意最多$T$个合谋服务器泄露目标文件索引的前提下，检索其中一个文件。研究PIR方案的核心问题在于最大化PIR速率，即目标文件大小与总下载量的比值。Freij-Hollanti等人曾提出MDS-TPIR容量（可达最大PIR速率）的猜想，但该猜想后来被Sun和Jafar通过$(M,N,T,K)=(2,4,2,2)$的反例证伪。本文提出一种基于伪装-压缩方法的新型MDS-TPIR方案，其特点包括：将Sun-Jafar反例推广至任意$(N,K)$-MDS编码系统中满足$N\geq K+2$的$(M,N,T,K)=(2,N,2,K)$参数组合，为Freij-Hollanti等人的猜想提供了更多反例；在$(M,N,T,K)=(2,N,2,K)$且采用GRS（广义Reed-Solomon码）编码的系统中，本方案实现了$\frac{N^2-N}{N^2+KN-2K}$的速率，超越了现有最优结果；进一步证明当$K=2$时该速率达到线性MDS-TPIR容量。本方案具有显著更小的实现所需域规模，并能自适应推广至多文件MDS-TPIR、抗循环相邻合谋服务器的MDS-PIR等广义PIR模型。最后，基于伪装-压缩框架提出了适用于$T\geq 3$情况的$ε$误差MDS-TPIR方案。