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I was wondering if there exists an algorithm, paper, etc. for the following problem:

Assume we have a public list of numbers, let's say {1, 2, 3, 4, 5}. Alice and Bob both pick any subset of those numbers in secret. Is there a way for Alice and Bob to exchange their selections in such a way that neither Alice nor Bob know what the other person has picked, however they still see which numbers they both picked?

For instance: Alice picks {1, 2, 5} and Bob picks {2, 3, 4}. At the end Alice and Bob should know that they have {2} in common, however without knowing the other person's selection.

I don't know where to start looking for a solution.

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The technique to which you refer is known as private set intersection.

These slides give an introduction to the problem.

This paper describes one solution using fully homomorphic encryption.

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