Security Considerations for the Fourth Data Over Non-Committed 3-Valued 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 allows the input of a third value “indefinite”, which is neither 0 nor 1, by extending the normal AND operation with two input choices, 0 and 1. A previous protocol that allows the input of 0, 1, and
$\theta$
with almost the same operation as the Five Card Trick has been actually proposed. The protocols are designed to have an algebraic structure such that it is a 3-valued logic satisfying the condition of the semigroups with 3 elements, and the transition rule. Due to satisfying the transition rule, we can change uncommitted protocols to committed protocols. In this paper, we consider the case where the fourth undefined value (illegal value) is input to the extended protocol using cards that can be distinguishable as top or bottom. As an example of a method that can maintain the indistinguishability of the reverse side by shuffling the cards up and down, business cards printed on one side can be used, which is a practical method because it is easy to prepare and shuffle.