Weakly Private Information Retrieval from Heterogeneously Trusted Servers

We study the problem of weakly private information retrieval (PIR) when there is heterogeneity in servers' trustworthiness under the maximal leakage (Max-L) metric and mutual information (MI) metric. A user wishes to retrieve a desired message from N non-colluding servers efficiently, such that the identity of the desired message is not leaked in a significant manner; however, some servers can be more trustworthy than others. We propose a code construction for this setting and optimize the probability distribution for this construction. For the Max-L metric, it is shown that the optimal probability allocation for the proposed scheme essentially separates the delivery patterns into two parts: a completely private part that has the same download overhead as the capacity-achieving PIR code, and a non-private part that allows complete privacy leakage but has no download overhead by downloading only from the most trustful server. The optimal solution is established through a sophisticated analysis of the underlying convex optimization problem and a reduction between the homogeneous setting and the heterogeneous setting. For the MI metric, the homogeneous case is studied first for which the code can be optimized with an explicit probability assignment, while a closed-form solution becomes intractable for the heterogeneous case. Numerical results are provided for both cases to corroborate the theoretical analysis.