Private Information Retrieval (PIR) is a mechanism for efficiently downloading messages while keeping the index secret. Here, PIRs in which servers do not communicate with each other are called standard PIRs, and PIRs in which some servers communicate with each other are called colluding PIRs. The information-theoretic upper bound on efficiency has been given in previous studies. However, the conditions for PIRs to keep privacy, to decode the desired message, and to achieve that upper bound have not been clarified. In this paper, we prove the necessary and sufficient conditions for achieving the capacity in standard PIRs and colluding PIRs.

翻译：私有信息检索（PIR）是一种在保密索引的同时高效下载消息的机制。其中，服务器之间不相互通信的PIR称为标准PIR，而部分服务器相互通信的PIR称为共谋PIR。先前研究已给出了基于信息论效率的上界，但关于PIR在保持隐私性、正确解码目标消息以及达到该上界所需的条件尚未明确。本文证明了标准PIR和共谋PIR中达到容量最优的充要条件。