How to implement non-committed card protocols to realize AND operations satisfying the three-valued logics
Yuji Suga · 2022
The matching situation with AND operations between two parties is a common application in card-based proto-cols, and it is known to provide a non-embarrassing confession of love. This means that the other party does not know whether the input was 0 or 1, which is said to avoid embarrassment. In this paper, we consider an extended AND protocol that extends the usual AND operations with two input choices, 0 and 1, to allow the input of a third value, “indefinite,” which is neither 0 nor 1. Considering the continuity of the protocols, it is better to design them to satisfy the transitivity law, and in practice, we need to deal with the classification of three-element semigroups. We attempted a complete classification with restrictions to have a local AND structure, and found that except for trivial examples and considerations of the algebraic isomorphism, we could reduce the classification to only seven cases. These include cases of three-valued logics by Bochvar and Lukasiewicz respectively. We also discuss how to implement card-based protocols to realize these three-element semigroups, but the usual encoding rule of the previous works is not applied to select one value from three candidates. So we try to use the same pattern cards with indistinguishability on the backside, which is another card protocol different from the general encoding rule. Finally, we propose actual implementations with 2 types of card-shuffling; ordinary random cut and top-to-bottom swapping.