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New Generation Computing

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

Samuel Hand, Alexander Koch, Pascal Lafourcade, Daiki Miyahara, Léo Robert

Published Aug 19, 2024• 7 citations
Publisher

Abstract

Abstract 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.

DOI
10.1007/s00354-024-00274-1

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Abstract 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.
Card-based zero-knowledge proof (ZKP) protocols allow a prover to convince a verifier that it knows a witness of a given statement, without revealing any information, using a physical deck of playing cards. Previous studies have focused on puzzles with a specific connected component, such as a simple cycle and a polyomino. In this study, we propose a unified approach to handle a family of connected components, including a tree, path, cycle, and polyomino. This approach achieves this verification in O(mn) steps relative to a given grid size m × n. Using this approach, we construct a card-based ZKP protocol for Nurimeizu, where the goal is to find the shortest path on a given grid.
A Zero-Knowledge Proof (ZKP) protocol allows a participant to prove the knowledge of some secret without revealing any information about it. While such protocols are typically executed by computers, there exists a line of research proposing physical instances of ZKP protocols. Up to now, many card-based ZKP protocols for pen-and-pencil puzzles, like Sudoku, have been designed. Those games, mostly edited by Nikoli, have simple rules, yet designing them in card-based ZKP protocols is non-trivial. In this work, we propose a card-based ZKP protocol for Usowan, a Nikoli game. In Usowan, for each room of a puzzle instance, there is exactly one piece of false information. The goal of the game is to detect this wrong data amongst the correct data and also to satisfy the other rules. Designing a card-based ZKP protocol to deal with the property of detecting a liar has never been done. In some sense, we propose a physical ZKP for hiding of a liar. This work extends a previous paper appearing in Ref. [1]. In this extension, we propose two other protocols, for Herugolf and Five Cells. The puzzles are specifically chosen because each of those three puzzles shares a common constraint, connectivity. However, showing the connected configuration cannot be done with generic approach and brings new construction to the existing connectivity ZKP protocol. Indeed, in Herugolf, the connectivity is handled with a given length of cell which is decremental (i.e., the length of each connected cell decreases by one at each step). For Five Cells, there is an additional step in the setup allowing to encode all the information needed to ensure a valid ZKP protocol.
A zero-knowledge proof (ZKP) is an interactive proof between a prover and a verifier, with the prover's goal to convince the verifier that a solution of a computation problem exists without revealing any information about the solution itself. A card-based physical ZKP is an implementation of a ZKP using physical cards. Several metrics are used to measure the efficiency of card-based ZKPs, such as the numbers of required cards and shuffles. In this paper, we propose the first card-based ZKP for a puzzle Sudoku that uses only one shuffle. To achieve this, we use a technique of applying private operations interactively to prepare inputs for the protocol.
Kakuro is a popular logic puzzle, in which a player fills in all empty squares with digits from 1 to 9 so that the sum of digits in each (horizontal or vertical) line is equal to a given number, called a clue, and digits in each line are all different. In 2016, Bultel, Dreier, Dumas, and Lafourcade proposed a physical zero-knowledge proof protocol for Kakuro using a deck of cards; their proposed protocol enables a prover to convince a verifier that the prover knows the solution of a Kakuro puzzle without revealing any information about the solution. One possible drawback of their protocol would be that the protocol is not perfectly extractable, implying that a prover who does not know the solution can convince a verifier with a small probability; therefore, one has to repeat the protocol to make such an error become negligible. In this paper, to overcome this, we design zero-knowledge proof protocols for Kakuro having perfect extractability property. Our improvement relies on the ideas behind the copy protocols in the field of card-based cryptography. By executing our protocols with a real deck of physical playing cards, humans can practically perform an efficient zero-knowledge proof of knowledge for Kakuro.
Abstract Decomposition puzzles are pencil-and-paper logic puzzles that involve partitioning a rectangular grid into several regions to satisfy certain rules. In this paper, we construct a generic card-based protocol called printing protocol, which can be used to physically verify solutions of decompositon puzzles. We apply the printing protocol to develop card-based zero-knowledge proof protocols for two such puzzles: Five Cells and Meadows. These protocols allow a prover to physically show that he/she knows solutions of the puzzles without revealing them.
In 2009, Gradwohl, Naor, Pinkas, and Rothblum proposed physical zero-knowledge proof protocols for Sudoku. That is, for a puzzle instance of Sudoku, their excellent protocols allow a prover to convince a verifier that there is a solution to the Sudoku puzzle and the prover knows it, without revealing any information about the solution. The possible drawback is that the existing protocols have an extractability error with a non-zero probability, or need special cards (such as scratch-off cards). Thus, in this study, we propose new protocols to perform zero-knowledge proof of knowledge for Sudoku using a normal deck of playing cards with no extractability error. Our protocols can be easily implemented by humans with a reasonable number of playing cards.