Physical Zero-Knowledge Proof Protocols for Topswops and Botdrops

Abstract Suppose that a sequence of n n cards, numbered 1 to n n, is placed face up in random order. Let k k be the number on the first card in the sequence. Then take the first k k cards from the sequence, rearrange that subsequence of k k cards in reverse order, and return them to the original sequence. Repeat this prefix reversal until the number on the first card in the sequence becomes 1. This is a one-player card game called Topswops. The computational complexity of Topswops has not been thoroughly investigated. For example, letting f (n) f (n) denote the maximum number of prefix reversals for Topswops with n n cards, values of f (n) f (n) for n 20 n ≥ 20 remain unknown. In general, there is no known efficient algorithm for finding an initial sequence of n n cards that requires exactly ℓ prefix reversals for any integers n n and ℓ. In this paper, using a deck of cards, we propose a physical zero-knowledge proof protocol that allows a prover to convince a verifier that the prover knows an initial sequence of n n cards that requires ℓ prefix reversals without leaking knowledge of that sequence. We also deal with Botdrops, a variant of Topswops.