A classification proof for commutative three-element semigroups with local AND structure and its application to card-based protocols

The matching situation with AND operations between two parties is a common application in card-based protocols, 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 logic by Bochvar and Lukasiewicz respectively. We also discuss some examples applied to the card-based protocols that have been devised. The examples are based on the same pattern cards with indistinguishability on the backside, which is another card protocol different from the general encoding rules.