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 matrix form. In this paper, we prove the necessary and sufficient conditions for the properties of standard PIR and colluding PIR. Further, we represent the properties in matrix form.

翻译：私有信息检索（PIR）是一种在高效下载消息的同时保持索引机密性的机制。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称为合谋PIR。已有研究给出了信息论意义上的效率上界。然而，PIR在保持隐私性、解码目标消息以及达到该上界所需的条件尚未以矩阵形式阐明。本文证明了标准PIR与合谋PIR属性成立的充要条件，并进一步以矩阵形式表示了这些属性。