Akari, Takuzu, Kakuro and KenKen are logic games similar to Sudoku. In Akari, a labyrinth on a grid has to be lit by placing lanterns, respecting various constraints. In Takuzu a grid has to be filled with 0's and 1's, while respecting certain constraints. In Kakuro a grid has to be filled with numbers such that the sums per row and column match given values; similarly in KenKen a grid has to be filled with numbers such that in given areas the product, sum, difference or quotient equals a given value. We give physical algorithms to realize zero-knowledge proofs for these games which allow a player to show that he knows a solution without revealing it. These interactive proofs can be realized with simple office material as they only rely on cards and envelopes. Moreover, we formalize our algorithms and prove their security.We also prove the security of our constructions.Related Work: Sudoku, introduced under this name in 1986 by the Japanese puzzle company Nikoli, and similar games such as Akari, Takuzu, Kakuro and Ken-Ken have gained immense popularity in recent years. Following the success of Sudoku, generalizations such as Mojidoku which uses letters instead of digits, and other similar logic puzzles like Hitori, Masyu, Futoshiki, Hashiwokakero, or Nurikabe were developed. Many of them have been proved to be NP-complete [12,5].Interactive ZKPs were introduced by Goldwasser et al. [8], and it was then shown that for any NP-complete problem there exists an interactive ZKP protocol [7]. An extension by Ben-Or et al. [1] showed that every provable statement can be proved in zero-knowledge. They also showed that physical protocols using envelopes exist, yet their construction is -due to its generality -rather involved an often impractical for real problem instances. Proofs can also be non-interactive in the sense that the prover and verifier do not need to interact during the protocol [3]. For more background on ZKPs see for example [15].As ZKPs have always been difficult to explain, there are works on how to explain the concepts to non experts, partly using physical protocols as illustrations. For example, in their famous paper [18], Quisquater et al. propose "Ali Bababa's cave" as a tool to explain Zero-Knowledge Proof to kids. In [20], ZKP's are illustrated using a magician that can count the number of sand grains in a heap of sand. Naor et al. used the well-known "Where's Waldo?" cartoons to explain the concept of ZKP to kids, and also proposed an efficient physical protocol for the problem in [17].In 2007, the same authors proposed a ZKP for Sudoku using cards [9], which partly inspired Cobra's solution in our paper. This was later extended for Hanjie [4].As in [9], we here also assume an abstract shuffle functionality which is essentially an indistinguishable shuffle of a set of sealed envelopes or of face down cards. This functionality is necessary to prevent information leakage, and cannot be realized neither by the verifier nor the prover. The verifier cannot perform this action, as otherwise he could be pe...

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Norinori is a logic game similar to Sudoku. In Norinori, a grid of cells has to be filled with either black or white cells so that the given areas contain exactly two black cells, and every black cell shares an edge with exactly one other black cell. We propose a secure interactive physical algorithm, relying only on cards, to realize a zero-knowledge proof of knowledge for Norinori. It allows a player to show that he or she knows a solution without revealing it. For this, we show in particular that it is possible to physically prove that a particular element is present in a list, without revealing any other value in the list, and without revealing the actual position of that element in the list.

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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.