Checking equivalence among distributed sets

• I have elements from $\{0, 1\}^{n}$ (range of a hash function)
• The master $A$ can have any subset of this range.
• There are clients that each have a subset from the space, too.
• I want to make sure that the union of the clients' sets is equal to the master set
• The communication should be as least as possible.
• The elements are secret. (this requirement can be relaxed with a solution that could potentially leak)

thanks to @kelalaka for refining the question.

A potential option would be to use Bloom filters to identify potential duplicates probabilistically, and then check if they really are duplicates by sending the few potentials. If you use large enough bloom filters this would be sufficient for any size.

Alternatively, which may not be a great fit after all, you could use miniscketch.

From the readme,

Sketches, as produced by this library, can be seen as "set checksums" with two peculiar properties:

1. Sketches have a predetermined capacity, and when the number of elements in the set is not higher than the capacity, libminisketch will always recover the entire set from the sketch. A sketch of b-bit elements with capacity c can be stored in bc bits.
2. The sketches of two sets can be combined by adding them (XOR) to obtain a sketch of the symmetric difference between the two sets (i.e., all elements that occur in one but not both input sets).

So assuming you use a large enough capacity, you can just XOR the sketches to identify the unique elements and remove the rest.