Practical card-based implementations of Yao's millionaire protocol

Yao's millionaire protocol enables Alice and Bob to know whether or not Bob is richer than Alice by using a public-key cryptosystem without revealing the actual amounts of their properties. In this paper, we present simple and practical implementations of Yao's millionaire protocol using a physical deck of playing cards; we straightforwardly implement the idea behind Yao's millionaire protocol so that even non-experts can easily understand their correctness and secrecy. Our implementations are based partially on the previous card-based scheme proposed by Nakai, Tokushige, Misawa, Iwamoto, and Ohta; their scheme admits players' private actions on a sequence of cards called Private Permutation (PP), implying that a malicious player could make an active attack (for example, he/she could exchange some of the cards stealthily when doing such a private action). By contrast, our implementations rely on a familiar shuffling operation called a random cut, and hence, they can be conducted completely publicly so as to avoid any active attack. More specifically, we present two card-based implementations of Yao's millionaire protocol; one uses a two-colored deck of cards (which consists of black and red cards), and the other uses a standard deck of playing cards. Furthermore, we also provide card-based protocols that rely on a logical circuit representing the comparison.