Private information retrieval (PIR) allows a user to retrieve one out of $ M $ messages from $ N $ servers without revealing the identity of the desired message. Every message consists of $ L $ symbols (packets) from an additive group and the length $ L $ is called the sub-packetization. A PIR scheme with download cost (DC) $ D/L $ is implemented by querying $ D $ sums of the symbols to servers. We assume that each uncoded server can store up to $ tLM/N $ symbols, $ t\in\{1,2,\cdots,N\} $. The minimum DC of storage constrained PIR was determined by Attia et al. in 2018 to be $ DC_{min} = 1+1/t+1/t^{2}+\cdots+1/t^{M-1} $. The capacity of storage constrained PIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired symbols that can be privately retrieved per bit of downloaded symbols. Tandon et al. designed a capacity-achieving PIR scheme with sub-packetization $ L' = {N\choose t}t^{M} $ of each message. In this paper, we design a PIR scheme with $ t $ times smaller sub-packetization $ L'/t $ and with the minimum DC for any parameters $ N, M, t $. We also prove that $ t^{M-1} $ is a factor of sub-packetization $ L $ for any capacity-achieving PIR scheme from storage constrained servers.

中文翻译：

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具有较小子分组的可实现容量的私人信息检索方案

专用信息检索（PIR）允许用户从$ N $服务器中检索$ M $消息中的一条，而无需透露所需消息的身份。每个消息都包含来自加性组的$ L $符号（数据包），长度$ L $称为子数据包化。通过向服务器查询符号的总和来实现具有下载成本（DC）$ D / L $的PIR方案。我们假设每个未编码的服务器最多可以存储$ tLM / N $个符号$ t \ in \ {1,2，\ cdots，N \} $。由存储限制的PIR的最小DC由Attia等确定。在2018年为$ DC_ {min} = 1 + 1 / t + 1 / t ^ {2} + \ cdots + 1 / t ^ {M-1} $。受存储限制的PIR的容量（等效于最小下载成本的倒数）是所下载符号的每位可私下检索的所需符号的最大位数。Tandon等。设计了一种容量达到的PIR方案，其中每个消息的子分组$ L'= {N \选择t} t ^ {M} $。在本文中，我们设计了一种PIR方案，其中$ t $乘以较小的子分组化$ L'/ t $，并且对于任何参数$ N，M，t $具有最小DC。我们还证明，对于来自存储受限服务器的任何实现容量的PIR方案，$ t ^ {M-1} $都是子分组化$ L $的一个因素。