We consider the storage problem in an asymmetric $X$-secure private information retrieval (A-XPIR) setting. The A-XPIR setting considers the $X$-secure PIR problem (XPIR) when a given arbitrary set of servers is communicating. We focus on the trade-off region between the average storage at the servers and the average download cost. In the case of $N=4$ servers and two non-overlapping sets of communicating servers with $K=2$ messages, we characterize the achievable region and show that the three main inequalities compared to the no-security case collapse to two inequalities in the asymmetric security case. In the general case, we derive bounds that need to be satisfied for the general achievable region for an arbitrary number of servers and messages. In addition, we provide the storage and retrieval scheme for the case of $N=4$ servers with $K=2$ messages and two non-overlapping sets of communicating servers, such that the messages are not replicated (in the sense of a coded version of each symbol) and at the same time achieve the optimal achievable rate for the case of replication. Finally, we derive the exact capacity for the case of asymmetric security and asymmetric collusion for $N=4$ servers, with the communication links $\{1,2\}$ and $\{3,4\}$, which splits the servers into two groups, i.e., $g=2$, and with the collusion links $\{1,3\}$, $\{2,4\}$, as $C=\frac{1}{3}$. More generally, we derive a capacity result for a certain family of asymmetric collusion and asymmetric security cases.

翻译：本文研究非对称$X$安全私有信息检索（A-XPIR）框架下的存储问题。A-XPIR框架考察当给定任意服务器集合进行通信时的$X$安全PIR问题（XPIR）。我们重点关注服务器平均存储量与平均下载成本之间的权衡区域。针对$N=4$台服务器、存在两组非重叠通信服务器集合且消息数$K=2$的场景，我们刻画了可达区域，并证明相较于无安全情形下的三个主要不等式，在非对称安全情形中这些不等式将缩减为两个。对于一般情况，我们推导了任意服务器数量与消息数量下通用可达区域所需满足的边界条件。此外，我们针对$N=4$台服务器、$K=2$条消息且存在两组非重叠通信服务器集合的场景，提出了无需复制消息（即每个符号的编码版本）的存储与检索方案，同时该方案在复制情形下能达到最优可达速率。最后，我们推导出$N=4$台服务器在非对称安全与非对称共谋场景下的精确容量：通信链路为$\{1,2\}$与$\{3,4\}$（即将服务器划分为两组，即$g=2$），共谋链路为$\{1,3\}$、$\{2,4\}$时，容量为$C=\frac{1}{3}$。更一般地，我们针对特定类型的非对称共谋与非对称安全场景推导了容量结果。