Physical zero-knowledge proof for Ripple Effect

Abstract

Ripple Effect is a logic puzzle where the player has to fill numbers into empty cells in a rectangular grid. The grid is divided into rooms, and each room must contain consecutive integers starting from 1 to its size. Also, if two cells in the same row or column contain the same number x, there must be a space of at least x cells separating the two cells. In this paper, we develop a physical zero-knowledge proof for the Ripple Effect puzzle using a deck of cards, which allows a prover to convince a verifier that he/she knows a solution without revealing it. In particular, given a secret number x and a list of numbers, our protocol can physically verify that x does not appear among the first x numbers in the list without revealing x or any number in the list.