Physical Zero-Knowledge Proof Protocols for Topswops and Botdrops

Abstract

AbstractSuppose that a sequence of $${\varvec{n}}$$ n cards, numbered 1 to $${\varvec{n}}$$ n , is placed face up in random order. Let $${\varvec{k}}$$ k be the number on the first card in the sequence. Then take the first $${\varvec{k}}$$ k cards from the sequence, rearrange that subsequence of $${\varvec{k}}$$ 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 $${\varvec{f}}({\varvec{n}})$$ f ( n ) denote the maximum number of prefix reversals for Topswops with $${\varvec{n}}$$ n cards, values of $${\varvec{f}}({\varvec{n}})$$ f ( n ) for $${\varvec{n}}\ge 20$$ n ≥ 20 remain unknown. In general, there is no known efficient algorithm for finding an initial sequence of $${\varvec{n}}$$ n cards that requires exactly $$\ell $$ ℓ prefix reversals for any integers $${\varvec{n}}$$ n and $${\varvec{\ell }}$$ 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 $${\varvec{n}}$$ n cards that requires $${\varvec{\ell }}$$ ℓ prefix reversals without leaking knowledge of that sequence. We also deal with Botdrops, a variant of Topswops.