Storage-Rate Trade-off in A-XPIR

2026-01-20

AI Summary

This study addresses the trade-off between average server storage and download cost in asymmetric X-secure private information retrieval (A-XPIR). For arbitrary communication structures, it proposes a storage-and-retrieval scheme tailored to non-overlapping communication groups, combining information-theoretic analysis with code constructions to characterize the achievable storage–download trade-off region. The key contributions include showing that under asymmetric security, the original three constraints reduce to two; designing a novel protocol that achieves the replication-optimal rate without message replication; and establishing, for the first time, the exact capacity under specific asymmetric security and collusion patterns. Notably, the capacity is fully characterized as 1/3 for the case with N=4 servers, K=2 messages, and communication groups {1,2} and {3,4}, and this result is extended to a broader class of settings.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

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.

Problem

• private information retrieval
• storage-rate trade-off
• asymmetric security
• download cost
• server storage

Innovation

• asymmetric X-secure PIR
• storage-rate trade-off
• non-replicated coded storage
• private information retrieval capacity
• asymmetric collusion