Efficient card-based zero-knowledge proof for Sudoku

Lafourcade

et al. 2021

Self Cite

During the last years, many Physical Zero-knowledge Proof (ZKP) protocols for Nikoli's puzzles ‡ have been designed. In this paper, we propose two ZKP protocols for the two Nikoli's puzzles called Nurikabe and Hitori. These two puzzles have some similarities, since in their rules at least one condition requires that some cells are connected to each other, horizontally or vertically. The novelty in this paper is to propose two techniques that allow us to prove such connectivity without leaking any information about a solution.

Section: Contributionsmentioning

confidence: 95%

“…Related Work: Efficient physical ZKP protocols for Nikoli puzzles have been proposed: Sudoku [8,17], Akari [2], Takuzu [2,13], Kakuro [2,14], Kenken [2], Makaro [3], Norinori [4], Slitherlink [12], Juosan [13], and Numberlink [16]. An important step in this line of research is to achieve no soundness error.…”

Section: Contributionsmentioning

confidence: 99%

Interactive Physical ZKP for Connectivity: Applications to Nurikabe and Hitori

Robert

Lafourcade

et al. 2021

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“…Card-based cryptography involves performing cryptographic tasks, such as secure multi-party computations, using a deck of physical cards; since den Boer [1] first proposed a protocol for a secure computation of the AND function with five cards, many elementary computations have been devised (e.g., [11,14]). For more complex tasks, millionaire protocols [9,12,13] that securely compare the properties of two players, a secure grouping protocol [5] that securely divides players into groups, and zero-knowledge proof protocols for pencil puzzles [3,10,15,16,18] were also proposed.…”

Section: Related Workmentioning

confidence: 99%

Efficient Generation of a Card-Based Uniformly Distributed Random Derangement

Murata

Mizuki

et al. 2021

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Consider a situation, known as Secret Santa, where n players wish to exchange gifts such that each player receives exactly one gift and no one receives a gift from oneself. Each player only wants to know in advance for whom he/she should purchase a gift. That is, the players want to generate a hidden uniformly distributed random derangement. (Note that a permutation without any fixed points is called a derangement.) To solve this problem, in 2015, Ishikawa et al. proposed a simple protocol with a deck of physical cards. In their protocol, players first prepare n piles of cards, each of which corresponds to a player, and shuffle the piles. Subsequently, the players verify whether the resulting piles have fixed points somehow: If there is no fixed point, the piles serve as a hidden random derangement; otherwise, the players restart the shuffle process. Such a restart occurs with a probability of approximately 0.6. In this study, we consider how to decrease the probability of the need to restart the shuffle based on the aforementioned protocol. Specifically, we prepare more piles of cards than the number n of players. This potentially helps us avoid repeating the shuffle, because we can remove fixed points even if they arise (as long as the number of remaining piles is at least n). Accordingly, we propose an efficient protocol that generates a uniformly distributed random derangement. The probability of the need to restart the shuffle can be reduced to approximately 0.1.

“…. Pile-scramble shuffle: A shuffle used in our protocol is a pile-scramble shuffle, which was first used by Ishikawa et al [10] and was used in other physical ZKP protocols for puzzles (e.g., Sudoku [18]). Consider that we have a sequence of piles of cards, each of which consists of the same number of face-down cards, denoted by (p 1 , p 2 , .…”

Section: Notationsmentioning

confidence: 99%

“…Related Work: In [18] the authors proposed an improved ZKP protocol for Sudoku that follows the pioneer work of [9]. In [5], the authors proposed a ZKP protocol for the Nikoli's puzzle Norinori.…”

Section: Introductionmentioning

confidence: 99%

Physical Zero-Knowledge Proof for Suguru Puzzle

Robert

Lafourcade

et al. 2020

Self Cite

Suguru is a paper and pencil puzzle invented by Naoki Inaba. The goal of the game is to fulfil a grid with numbers between 1 and 5 and to respect three simple constraints. In this paper we design a physical Zero-Knowledge Proof (ZKP) protocol for Suguru. A ZKP protocol allows a prover (P) to prove that he knows a solution of a Suguru grid to a verifier (V) without leaking any information on the solution. For constructing such a physical ZKP protocol, we only rely on a small number of physical cards and an adapted encoding. For a grid of Suguru with n cells, we only use 5n + 5 cards. Moreover, we prove the three classical security properties of a ZKP: completeness, extractability, and zero-knowledge.