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.

中文翻译：

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涟漪效应的物理零知识证明

Ripple Effect 是一个逻辑谜题，其目标是将数字填入划分为多个房间的矩形网格。每个房间必须包含从 1 到其大小的连续整数。此外，如果同一行或同一列中的两个单元格具有相同的编号 $x$，则分隔两个单元格的空间必须至少为 $x$ 个单元格。在本文中，我们提出了一种使用一副纸牌的 Ripple Effect 难题的零知识证明物理协议，它允许证明者在不透露解决方案的情况下物理表明他/她知道解决方案。特别是，我们开发了一个物理协议，给定一个秘密数字 $x$ 和一个数字列表，验证 $x$ 没有出现在列表中的第一个 $x$ 数字中，而不泄露 $x$ 或任何数字列表。