In symmetric private information retrieval (SPIR), a user communicates with multiple servers to retrieve from them a message in a database, while not revealing the message index to any individual server (user privacy), and learning no additional information about the database (database privacy). We study the problem of SPIR on graph-replicated database systems, where each node of the graph represents a server and each link represents a message. Each message is replicated at exactly two servers; those at which the link representing the message is incident. To ensure database privacy, the servers share a set of common randomness, independent of the database and the user's desired message index. We study two cases of common randomness distribution to the servers: i) graph-replicated common randomness, and ii) fully-replicated common randomness. Given a graph-replicated database system, in i), we assign one randomness variable independently to every pair of servers sharing a message, while in ii), we assign an identical set of randomness variable to all servers, irrespective of the underlying graph. In both settings, our goal is to characterize the SPIR capacity, i.e., the maximum number of desired message symbols retrieved per downloaded symbol, and quantify the minimum amount of common randomness required to achieve the capacity. To this goal, in setting i), we derive a general lower bound on the SPIR capacity, and show it to be tight for path and regular graphs through a matching converse. Moreover, we establish that the minimum size of common randomness required for SPIR is equal to the message size. In setting ii), the SPIR capacity improves over the first, more restrictive setting. We show this through capacity lower bounds for a class of graphs, by constructing SPIR schemes from PIR schemes.

翻译：在对称私有信息检索（SPIR）中，用户与多个服务器通信以从数据库中检索消息，同时不向任何单个服务器透露消息索引（用户隐私），并且不获取数据库的额外信息（数据库隐私）。我们研究了基于图复制数据库系统的SPIR问题，其中图的每个节点代表一个服务器，每条边代表一条消息。每条消息恰好复制在两个服务器上；即代表该消息的边所连接的两个服务器。为确保数据库隐私，服务器共享一组与数据库及用户所需消息索引无关的公共随机性。我们研究了两种向服务器分发公共随机性的情况：i) 图复制公共随机性，以及 ii) 完全复制公共随机性。给定一个图复制数据库系统，在情况 i) 中，我们为每对共享消息的服务器独立分配一个随机变量；而在情况 ii) 中，我们向所有服务器分配一组相同的随机变量，与底层图结构无关。在这两种设置下，我们的目标是表征SPIR容量（即每下载一个符号所能检索到的期望消息符号的最大数量），并量化实现该容量所需的最小公共随机性量。为此，在设置 i) 中，我们推导了SPIR容量的一般下界，并通过匹配的逆证明该下界对于路径图和正则图是紧的。此外，我们证明了SPIR所需的最小公共随机性大小等于消息大小。在设置 ii) 中，SPIR容量相较于第一种更受限的设置有所提升。我们通过为一类图构造从PIR方案导出的SPIR方案，给出了容量下界以证明这一点。