Several caveats should be carefully considered before using PSI.

1. Server set size
2. Client set size (Note that Each of these can be turned into upper bounds by adding dummy elements.)

There are two configurations for instantiating a new client/server pair by passing in a boolean switch into their respective constructors.

1. One that reveals only the size (cardinality) of the intersection to the client.
2. One that reveals the actual intersecion to the client.

In the case of #1, coordinated clients could get the actual intersection. However, server set items not in any of the client sets will never be uncovered. Situations where it’s feasible for clients to send one request per element in the domain - there is a possbility that coordinated clients could uncover server set.

Presence of new client set members or absence of former client set members can be detected by server/eavesdroppers if client secret is reused.

In the absence of any rate limiting and assuming the client and server have enough computing power and bandwidth, small domains may be brute-forceable. However, a query needs to be performed for each brute-force attempt. An example for this situation would be suppose you were trying to limit sending antibody tests to people based on whether they’d been in an infected location, so that people would have to share their location history to prove they’d been somewhere infected, and you were using PSI so people wouldn’t have to share their location history without good reason. If your health authority only covers 10 possible geohashes, people could sidestep the PSI step entirely and submit location histories which unlock tests by brute force.

A potential limitation with the PSI approach is the communication complexity, which scales linearly with the size of the larger set. This is of particular concern when performing PSI between a constrained device (cellphone) holding a small set, and a large service provider (e.g. WhatsApp), such as in the Private Contact Discovery application. Assuming a bloom filter is used, the Client set size affects the algorithmic complexity in linear time O(n), with a constant number of lookups. The bloom filter has linear size in the server's set, hence the algorithmic complexity of our protocol is O(n). However, a bloom filter requires a large number of lookups on each query, if the false positive rate is low. An alternative is the Golomb Compressed Set, which requires O(n log n) time due to sorting operations, but in practice takes around 25-30% less space than a bloom filter.