Private information retrieval (PIR) is a mechanism for efficiently downloading messages while keeping the index of the desired message secret from the servers. PIR schemes have been extended to various scenarios with adversarial servers: PIR schemes where some servers are unresponsive or return noisy responses are called robust PIR and Byzantine PIR, respectively; PIR schemes where some servers collude to reveal the index are called colluding PIR. The information-theoretic upper bound on the download efficiency of these PIR schemes has been proved in previous studies. However, systematic ways to construct PIR schemes that achieve the upper bound are not known. In order to construct a capacity-achieving PIR schemes systematically, it is necessary to clarify the conditions that the queries should satisfy. This paper proves the necessary and sufficient conditions for capacity-achieving PIR schemes.

翻译：私有信息检索（PIR）是一种高效下载消息的机制，同时确保目标消息的索引对服务器保密。PIR方案已扩展至多种对抗性服务器场景：部分服务器无响应或返回噪声响应的方案分别称为鲁棒PIR与拜占庭PIR；部分服务器共谋以揭示索引的方案称为共谋PIR。已有研究证明了这些PIR方案下载效率的信息理论上界，但系统化构建达到该上界的PIR方案的方法尚未明确。为系统化构建容量最优PIR方案，必须明确查询应满足的条件。本文证明了容量最优PIR方案的充要条件。