Relationship between AND extension and XOR extension of 3-valued input with 2-party 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. Extended AND protocols that extend the normal AND operation with two input choices of 0, 1 to allow a third "indeterminate" value that is neither 0 nor 1 have been proposed in ICCE-TW2022. Similar extensions to the XOR operation were also proposed at ICCE-TW2023, each of which is given commutative semigroup classifications.By finding algebraic isomorphism in the commutative semi-groups of AND and XOR extensions discussed above, this paper presents two examples of compatible implementations of the card-based protocols. We also show that it is possible to construct card protocols with different boolean tables by changing only the encoding rules while keeping the same procedures exactly.