Efficient Card-Based ZKP for Single Loop Condition and Its Application to Moon-or-Sun

A zero-knowledge proof (ZKP) allows a prover to prove to a verifier that it knows some secret, such as a solution to a difficult puzzle, without revealing any information about it. In recent years, ZKP protocols using only a deck of playing cards for solutions to various pencil puzzles have been proposed. The previous work of Lafourcade et al. deals with a famous puzzle called Slitherlink. Their proposed protocol can verify that a solution forms a single loop without revealing anything about the solution, except this fact. Their protocol guarantees that the solution satisfies the single-loop condition, by interactively constructing a solution starting from a state that holds a simple single loop, and proceeding via steps that preserve the invariant of encoding a single loop, until the proper solution is reached. A drawback of their protocol is that it requires additional verifications to guarantee a single loop. In this study, we propose a more efficient ZKP protocol for such a puzzle with fewer additional verifications. For this, we employ the previous work of Robert et al., which addressed the connectivity property in a puzzle. That is, we verify that a solution is connected but not split, to be a single loop. Applying our proposal, we construct a card-based ZKP protocol for Moon-or-Sun, which has its specific rule of alternating pattern in addition to the single-loop condition.