In the classical model for (information theoretically secure) Private Information Retrieval (PIR) due to Chor, Goldreich, Kushilevitz and Sudan, a user wishes to retrieve one bit of a database that is stored on a set of ${n}$ servers, in such a way that no individual server gains information about which bit the user is interested in. The aim is to design schemes that minimise the total communication between the user and the servers. More recently, there have been moves to consider more realistic models where the total storage of the set of servers, or the per server storage, should be minimised (possibly using techniques from distributed storage), and where the database is divided into ${R}$ -bit records with ${R}>1$ , and the user wishes to retrieve one record rather than one bit. When ${R}$ is large, downloads from the servers to the user dominate the communication complexity and so the aim is to minimise the total number of downloaded bits. Work of Shah, Rashmi and Ramchandran shows that at least ${R}+1$ bits must be downloaded from servers in the worst case, and provides PIR schemes meeting this bound. Sun and Jafar have considered the download cost of a scheme, defined as the ratio of the message length ${R}$ and the total number of bits downloaded. They determine the best asymptotic download cost of a PIR scheme (as ${R}\rightarrow \infty $ ) when a database of ${k}$ messages is stored by ${n}$ servers. This paper provides various bounds on the download complexity of a PIR scheme, generalising those of Shah et al. to the case when the number ${n}$ of servers is bounded, and providing links with classical techniques due to Chor et al. The paper also provides a range of constructions for PIR schemes that are either simpler or perform better than previously known schemes. These constructions include explicit schemes that achieve the best asymptotic download complexity of Sun and Jafar with significantly lower upload complexity, and general techniques for constructing a scheme with good worst case download complexity from a scheme with good download complexity on average.

中文翻译：

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下载复杂度小、存储要求低的PIR方案

在 Chor、Goldreich、Kushilevitz 和 Sudan 的（信息理论上安全的）私人信息检索 (PIR) 的经典模型中，用户希望检索存储在一组 ${n}$ 服务器，以这样的方式，没有单个服务器获得有关用户感兴趣的位的信息。目的是设计方案，最大限度地减少用户和服务器之间的总通信。最近，人们开始考虑更现实的模型，其中应该最小化一组服务器的总存储量或每台服务器的存储量（可能使用分布式存储技术），并将数据库划分为 ${R}$ -位记录 ${R}>1$ ，并且用户希望检索一个记录而不是一位。什么时候 ${R}$ 大，从服务器到用户的下载在通信复杂性中占主导地位，因此目标是最小化下载的总位数。Shah、Rashmi 和 Ramchandran 的工作表明，至少 ${R}+1$ 在最坏的情况下必须从服务器下载位，并提供满足此限制的 PIR 方案。Sun 和 Jafar 考虑了一个方案的下载成本，定义为消息长度的比率 ${R}$ 以及下载的总位数。他们确定 PIR 方案的最佳渐近下载成本（如 ${R}\rightarrow \infty $ ) 当一个数据库 ${k}$ 消息由 ${n}$ 服务器。本文对 PIR 方案的下载复杂性提供了各种界限，概括了 Shah 等人的界限。数的情况下 ${n}$ 服务器的数量是有界的，并且由于 Chor 等人提供了与经典技术的链接。该论文还为 PIR 方案提供了一系列结构，这些结构比以前已知的方案更简单或性能更好。这些构造包括显式方案，以显着降低上传复杂度实现 Sun 和 Jafar 的最佳渐近下载复杂度，以及用于从平均下载复杂度良好的方案构建具有良好最坏情况下载复杂度的方案的通用技术。