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We consider a special case of $X$-secure $T$-private information retrieval(XSTPIR), where the security requirement is \emph{asymmetric} due to possiblemissing communication links between the $N$ databases considered in the system.We define the problem with a communication matrix that indicates all possiblecommunications among the databases, and propose a database grouping mechanismthat collects subsets of databases in an optimal manner, followed by agroup-based PIR scheme to perform asymmetric XSTPIR with the goal of maximizingthe communication rate (minimizing the download cost). We provide an upperbound on the general achievable rate of asymmetric XSTPIR, and show that theproposed scheme achieves this upper bound in some cases. The proposed approachoutperforms classical XSTPIR under certain conditions, and the results of thiswork show that unlike in the symmetric case, some databases with certainproperties can be dropped to achieve higher rates, concluding that moredatabases is not always better.

Asymmetric $X$-Secure $T$-Private Information Retrieval: More Databases is Not Always Better