Interactive Physical ZKP for Connectivity: Applications to Nurikabe and Hitori

During the last years, several card-based Zero-Knowledge Proof (ZKP) protocols for Nikoli’s puzzles have been designed. Although there are relatively simple card-based ZKP protocols for a number of puzzles, such as Sudoku and Kakuro, some puzzles face difficulties in designing simple protocols. For example, Slitherlink requires novel and elaborate techniques to construct a protocol. In this study, we focus on three Nikoli puzzles: Nurikabe, Hitori, and Heyawake. To date, no card-based ZKP protocol for these puzzles has been developed, partially because they have a relatively tricky rule that colored cells should form a connected area (namely a polyomino); this rule, sometimes referred to as “Bundan-kin” (in Japanese), complicates the puzzles, as well as facilitating difficulties in designing card-based ZKP protocols. We address this challenging task and propose a method for verifying the connectivity of hidden colored cells in a ZKP manner, such that we construct card-based ZKP protocols for the three puzzles.

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“…An earlier version of this paper was presented and appeared as a conference paper [22]. The primary difference between the two is as follows.…”

Section: Contributionsmentioning

confidence: 99%

Card-Based ZKP for Connectivity: Applications to Nurikabe, Hitori, and Heyawake

et al. 2022

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“…Card-based ZKP protocols for many other popular pencil puzzles have been developed, including Sudoku [7,19,23], Nonogram [3,18], Akari [1], Takuzu [1,13], Kakuro [1,14], KenKen [1], Makaro [2], Norinori [4], Slitherlink [12], Juosan [13], Numberlink [20], Suguru [17], Ripple Effect [21], Nurikabe [16], Hitori [16], Cryptarithmetic [10], and Bridges [22].…”

Section: Related Workmentioning

confidence: 99%

“…In a recent work of Robert et al [16], the authors posed an open problem to extend the idea of their protocol to verify a solution of Shikaku or other puzzles that require to draw rectangles of certain sizes in a grid.…”

Section: Related Workmentioning

confidence: 99%

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How to Physically Verify a Rectangle in a Grid: A Physical ZKP for Shikaku

Ruangwises¹,

Itoh²

2022

Preprint

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Shikaku is a pencil puzzle consisting of a rectangular grid, with some cells containing a number. The player has to partition the grid into rectangles such that each rectangle contains exactly one number equal to the area of that rectangle. In this paper, we propose two physical zero-knowledge proof protocols for Shikaku using a deck of playing cards, which allow a prover to physically show that he/she knows a solution of the puzzle without revealing it. Most importantly, in our second protocol we develop a general technique to physically verify a rectangle-shaped area with a certain size in a rectangular grid, which can be used to verify other puzzles with similar constraints.

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“…After the discovery of the physical ZKP protocols for Sudoku, physical ZKP protocols for other popular logic puzzles have been proposed as well, including Nonogram [3], Akari [ Takuzu [1,14], Kakuro [1,15], KenKen [1], Makaro [2], Norinori [4], Slitherlink [12], Juosan [14], Numberlink [21], Suguru [20], Ripple Effect [22], Nurikabe [19], and Hitori [19].…”

Section: Related Workmentioning

confidence: 99%

Two Standard Decks of Playing Cards are Sufficient for a ZKP for Sudoku

Ruangwises

2021

Preprint

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Sudoku is a logic puzzle with an objective to fill a number between 1 and 9 in each empty cell of a 9 × 9 grid such that every number appears exactly once in each row, each column, and each 3 × 3 block. In 2020, Sasaki et al. proposed a physical zero-knowledge proof (ZKP) protocol for Sudoku using 90 cards, which allows a prover to physically show that he/she knows a solution without revealing it. However, their protocol requires nine identical copies of some cards, which cannot be found in a standard deck of playing cards. Therefore, nine decks of cards are actually required in order to perform that protocol. In this paper, we propose a new ZKP protocol for Sudoku that can be performed using only two standard decks of playing cards. In general, we develop the first ZKP protocol for an n × n Sudoku that can be performed using a deck of all different cards.

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Interactive Physical ZKP for Connectivity: Applications to Nurikabe and Hitori

Cited by 25 publications

References 22 publications

Card-Based ZKP for Connectivity: Applications to Nurikabe, Hitori, and Heyawake

Card-Based ZKP for Connectivity: Applications to Nurikabe, Hitori, and Heyawake

How to Physically Verify a Rectangle in a Grid: A Physical ZKP for Shikaku

Two Standard Decks of Playing Cards are Sufficient for a ZKP for Sudoku

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