Card-Based Protocols for Private Set Intersection and Union

Card-based cryptography aims to realize secure multiparty computation with physical cards. This paper is the first to address Private Set Intersection (PSI) and Private Set Union (PSU) in card-based cryptography. PSI and PSU are well-studied secure computation protocols to compute the set intersection and the set union, respectively. We show two-party PSI and PSU protocols in each of the two operation models: one is the shuffle-based model in which parties perform all operations publicly, and the other is the private-permutation-based model that allows parties to perform some operations privately. In the shuffle-based model, we show PSI and PSU protocols can be realized with existing secure AND and OR protocols, respectively. However, these protocols have an issue of increasing the number of shuffles depending on the size of the universal set. To resolve the issue, we further propose PSI and PSU protocols with only one shuffle at the cost of increasing the number of cards. In the private-permutation-based model, we show PSI and PSU protocols can be achieved with existing secure AND and OR protocols, respectively, as in the shuffle-based protocols. These protocols have an advantage of requiring only one private permutation and one communication. We further show that the number of cards of these protocols can be reduced at the cost of increasing the number of private permutations and communications.