Numberlink is a logic puzzle with an objective to connect all pairs of cells with the same number 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 prover to convince a verifier that he/she knows a solution without revealing it. In particular, the protocol shows how to physically 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. Finally, we show that our protocol can be modified to verify a solution of the well-known k vertex-disjoint paths problem, both the undirected and directed settings.

中文翻译：

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Numberlink Puzzle 和 k 顶点不相交路径问题的物理零知识证明

Numberlink 是一个逻辑谜题，其目标是通过矩形网格中的非交叉路径连接所有具有相同数字的单元格对。在本文中，我们提出了一种使用一副纸牌为 Numberlink 进行零知识证明的物理协议，它允许证明者说服验证者他/她知道一个解决方案而无需透露它。特别是，该协议展示了如何在不透露该值的情况下物理计算列表中等于给定秘密值的元素数量、列表中与其相等的元素的位置或任何其他元素的值在列表中。最后，我们表明可以修改我们的协议以验证众所周知的 k 顶点不相交路径问题的解决方案，包括无向和有向设置。