Private information retrieval (PIR) allows a user to retrieve a desired message from a set of databases without revealing the identity of the desired message. The replicated database scenario, where $N$ databases store each of the $K$ messages was considered by Sun and Jafar, and the optimal download cost was characterized as $\left ({1+ \frac {1}{N}+ \frac {1}{N^{2}}+ \cdots + \frac {1}{N^{K-1}}}\right)$ . In this work, we consider the problem of PIR from uncoded storage constrained databases. Each database has a storage capacity of $\mu KL$ bits, where $L$ is the size of each message in bits, and $\mu \in [{1/N, 1}]$ is the normalized storage. The novel aspect of this work is to characterize the optimum download cost of PIR from uncoded storage constrained databases for any “normalized storage” value in the range $\mu \in [{1/N, 1}]$ . In particular, for any $(N,K)$ , we show that the optimal trade-off between normalized storage, $\mu $ , and the download cost, $D(\mu)$ , is a piece-wise linear function given by the lower convex hull of the $N$ pairs $\left ({\frac {t}{N}, \left ({1+ \frac {1}{t}+ \frac {1}{t^{2}}+ \cdots + \frac {1}{t^{K-1}}}\right)}\right)$ for $t=1,2,\ldots, N$ . To prove this result, we first present a storage constrained PIR scheme for any $(N,K)$ . Next, we obtain a general lower bound on the download cost for PIR, which is valid for any arbitrary storage architecture. The uncoded storage assumption is then applied which allows us to express the lower bound as a linear program (LP). Finally, we solve the LP to obtain tight lower bounds on the download cost for different regimes of storage, which match the proposed storage constrained PIR scheme.

中文翻译：

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从非编码存储约束数据库中检索私人信息的能力

私人信息检索 (PIR) 允许用户从一组数据库中检索所需的消息，而无需透露所需消息的身份。复制数据库场景，其中 $N$ 数据库存储每个 $K$ 消息被 Sun 和 Jafar 考虑，最优下载成本被表征为 $\left ({1+ \frac {1}{N}+ \frac {1}{N^{2}}+ \cdots + \frac {1}{N^{K-1}}}\right) $ . 在这项工作中，我们从未编码存储受限数据库。每个数据库的存储容量为 $\亩KL$ 位，其中 $L$ 是每条消息的大小（以位为单位），并且 $\mu \in [{1/N, 1}]$ 是归一化存储。这项工作的新颖之处在于，针对范围内的任何“标准化存储”值，从未编码的存储受限数据库中表征 PIR 的最佳下载成本 $\mu \in [{1/N, 1}]$ . 特别地，对于任何 $(N,K)$ ，我们展示了标准化存储之间的最佳权衡， $\亩 $ ，以及下载成本， $D(\mu)$ , 是由下凸包给出的分段线性函数 $N$ 对 $\left ({\frac {t}{N}, \left ({1+ \frac {1}{t}+ \frac {1}{t^{2}}+ \cdots + \frac {1} {t^{K-1}}}\right)}\right)$ 为了 $t=1,2,\ldots, N$ . 为了证明这个结果，我们首先提出了一个存储约束的 PIR 方案，用于任何 $(N,K)$ . 接下来，我们获得了 PIR 下载成本的一般下限，这适用于任何任意存储架构。然后应用未编码的存储假设，这允许我们将下限表示为线性程序 (LP)。最后，我们求解 LP 以获得不同存储机制的下载成本的严格下限，这与所提出的存储约束 PIR 方案相匹配。