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.
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.
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.
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 problems with similar constraints.
Abstract A zero-knowledge proof protocol is a cryptographic protocol in which a prover, who knows the witness to a statement, can convince a verifier that the statement is true without revealing any information about the witness. Although zero-knowledge proof protocols are typically executed on electronic computers, there is a line of research to design zero-knowledge proof protocols based on physical objects (e.g., a deck of cards). This is called physical zero-knowledge proof. In this paper, we construct a physical zero-knowledge proof protocol for a logical puzzle called Sukoro. Sukoro has many cells on the puzzle board, like Sudoku, where each cell must be empty or filled with a number from one to four, and each number must match the number of adjacent filled cells, and the same numbers must not be adjacent to each other. In addition, it has a rule that all filled cells must be connected, which is called the connectivity condition. Although some existing protocols deal with the connectivity condition, all existing methods are interactive, which requires the prover’s knowledge to determine how the cards are manipulated during the execution of the protocols. In this paper, we give a new method for verifying the connectivity condition in the non-interactive setting, which means that the protocol can be executed without the prover’s knowledge, and construct a physical zero-knowledge proof protocol for Sukoro.
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 that he/she knows it, without revealing any information about the solution. The possible drawback is that the existing protocols have a soundness 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 for Sudoku that use a normal deck of playing cards and have no soundness error. Our protocols can be easily implemented by humans with a reasonable number of playing cards.
Ripple Effect is a logic puzzle with an objective to fill numbers into a rectangular grid divided into rooms. Each room must contain consecutive integers starting from 1 to its size. Also, if two cells in the same row or column have the same number x, the space separating the two cells must be at least x cells. In this paper, we propose a physical protocol of zero-knowledge proof for Ripple Effect puzzle using a deck of cards, which allows a prover to physically show that he/she knows a solution without revealing it. In particular, we develop a physical protocol that, given a secret number x and a list of numbers, verifies that x does not appear among the first x numbers in the list without revealing x or any number in the list.
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.