The Capacity of Symmetric Private Information Retrieval under Arbitrary Collusion and Eavesdropping Patterns

We study the symmetric private information retrieval (SPIR) problem under arbitrary collusion and eavesdropping patterns for replicated databases. We find its capacity, which is the same as the capacity of the original SPIR problem with the number of databases $N$ replaced by a number $F^*$. The number $F^*$ is the optimal solution to a linear programming problem that is a function of the joint pattern, which is the union of the collusion and eavesdropping pattern. This is the first result that shows how two arbitrary patterns collectively affect the capacity of the PIR problem. We draw the conclusion that for SPIR problems, the collusion and eavesdropping constraints are interchangeable in terms of capacity. As special cases of our result, the capacity of the SPIR problem under arbitrary collusion patterns and the capacity of the PIR problem under arbitrary eavesdropping patterns are also found.