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Numberlink is a logic puzzle with an objective to connect all pairs of cellswith the same number by non-crossing paths in a rectangular grid. In thispaper, we propose a physical protocol of zero-knowledge proof for Numberlinkusing a deck of cards, which allows a prover to convince a verifier that he/sheknows a solution without revealing it. In particular, the protocol shows how tophysically count the number of elements in a list that are equal to a givensecret value without revealing that value, the positions of elements in thelist 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 thewell-known $k$ vertex-disjoint paths problem, both the undirected and directedsettings.

Physical Zero-Knowledge Proof for Numberlink Puzzle and $k$ Vertex-Disjoint Paths Problem