12 research outputs found
Physical Zero-Knowledge Proof for Numberlink
Numberlink is a logic puzzle for which the player has to connect all pairs of cells with the same numbers by non-crossing paths in a rectangular grid. In this paper, we propose a physical protocol of zero-knowledge proof for Numberlink using a deck of cards, which allows a player to physically show that he/she knows a solution without revealing it. In particular, we develop a physical protocol to count the number of elements in a list that are equal to a given secret value without revealing that value, the positions of elements in the list that are equal to it, or the value of any other element in the list. Our protocol can also be applied to verify the existence of vertex-disjoint paths connecting all given pairs of endpoints in any undirected graph
Physical Zero-Knowledge Proof for Ball Sort Puzzle
Ball sort puzzle is a popular logic puzzle consisting of several bins
containing balls of multiple colors. Each bin works like a stack; a ball has to
follow the last-in first-out order. The player has to sort the balls by color
such that each bin contains only balls of a single color. In this paper, we
propose a physical zero-knowledge proof protocol for the ball sort puzzle using
a deck of playing cards, which enables a prover to physically show that he/she
knows a solution with moves of the ball sort puzzle without revealing it.
Our protocol is the first zero-knowledge proof protocol for an interactive
puzzle involving moving objects.Comment: arXiv admin note: text overlap with arXiv:2302.0123
Two Standard Decks of Playing Cards are Sufficient for a ZKP for Sudoku
Sudoku is a logic puzzle with an objective to fill a number between 1 and 9
in each empty cell of a grid such that every number appears
exactly once in each row, each column, and each 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 Sudoku that can
be performed using a deck of all different cards.Comment: A shortened version of this paper has appeared at COCOON 202
An Improved Physical ZKP for Nonogram
Nonogram is a logic puzzle consisting of a rectangular grid with an objective
to color every cell black or white such that the lengths of blocks of
consecutive black cells in each row and column are equal to the given numbers.
In 2010, Chien and Hon developed the first physical zero-knowledge proof for
Nonogram, which allows a prover to physically show that he/she knows a solution
of the puzzle without revealing it. However, their protocol requires special
tools such as scratch-off cards and a machine to seal the cards, which are
difficult to find in everyday life. Their protocol also has a nonzero soundness
error. In this paper, we propose a more practical physical zero-knowledge proof
for Nonogram that uses only a deck of regular paper cards and also has perfect
soundness.Comment: This paper has appeared at COCOA 202
Physical Zero-Knowledge Proof for Five Cells
Five Cells is a pencil puzzle consisting of a rectangular grid, with some
cells containg a number. The player has to partition the grid into blocks, each
consisting of five cells, such that the number in each cell must be equal to
the number of edges of that cell that are borders of blocks. In this paper, we
propose a physical zero-knowledge proof protocol for Shikaku using a deck of
playing cards, which allows a prover to physically show that he/she knows a
solution of the puzzle without revealing it. More importantly, in the
optimization we develop a technique to verify a graph coloring that no two
adjacent vertices have the same color without revealing any information about
the coloring. This technique reduces the number of required cards in our
protocol from quadratic to linear in the number of cells, and can also be used
in other protocols related to graph coloring.Comment: arXiv admin note: text overlap with arXiv:2202.0978
Card-Based ZKP Protocols for Takuzu and Juosan
International audienc
Check Alternating Patterns: A Physical Zero-Knowledge Proof for Moon-or-Sun
A zero-knowledge proof (ZKP) allows a party to prove to another party that it knows some secret, such as the solution to a difficult puzzle, without revealing any information about it. We propose a physical zero-knowledge proof using only a deck of playing cards for solutions to a pencil puzzle called \emph{Moon-or-Sun}. In this puzzle, one is given a grid of cells on which rooms, marked by thick black lines surrounding a connected set of cells, may contain a number of cells with a moon or a sun symbol. The goal is to find a loop passing through all rooms exactly once, and in each room either passes through all cells with a moon, or all cells with a sun symbol.
Finally, whenever the loop passes from one room to another, it must go through all cells with a moon if in the previous room it passed through all cells with a sun, and visa-versa. This last rule constitutes the main challenge for finding a physical zero-knowledge proof for this puzzle, as this must be verified without giving away through which borders the loop enters or leaves a given room. We design a card-based zero-knowledge proof of knowledge protocol for Moon-or-Sun solutions, together with an analysis of their properties. Our technique of verifying the alternation of a pattern along a non-disclosed path might be of independent interest for similar puzzles