Physical Zero-Knowledge Proof for Five Cells

Five Cells is a logic 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 Five Cells 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.