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Card-based cryptography, introduced by den Boer aims to realize multiparty computation (MPC) by using physical cards. We propose several efficient card-based protocols for the millionaires’ problem by introducing a new operation called Private Permutation (PP) instead of the shuffle used in most of existing card-based cryptography. Shuffle is a useful randomization technique by exploiting the property of card shuffling, but it requires a strong assumption from the viewpoint of arithmetic MPC because shuffle assumes that public randomization is possible. On the other hand, private randomness can be used in PPs, which enables us to design card-based protocols taking ideas of arithmetic MPCs into account. Actually, we show that Yao’s millionaires’ protocol can be easily transformed into a card-based protocol by using PPs, which is not straightforward by using shuffles because Yao’s protocol uses private randomness. Furthermore, we propose entirely novel and efficient card-based millionaire protocols based on PPs by securely updating bitwise comparisons between two numbers, which unveil a power of PPs. As another interest of these protocols, we point out they have a deep connection to the well-known logical puzzle known as “The fork in the road.”

Card-based cryptography is a variant of multi-party computation using physical cards like playing cards. There are two models on card-based cryptography, called public and private models. The public model assumes that all operations are executed publicly, while the private model allows the players private operations called private permutations (PP, for short). Much of the existing card-based protocols were developed under the public model. Under the public model, 2n cards are necessary for every protocol with n-bit input since at least two cards are required to express a bit. In this paper, we propose n-bit input protocols with fewer than 2n cards by utilizing PP, which shows the power of PP. In particular, we show that a protocol for (n-bit input) threshold function can be realized with only $$n+1$$ n + 1 cards by reducing the threshold function to the majority voting. Toward this end, we first offer that two-bit input protocols for logic gates can be realized with fewer than four cards. Furthermore, we construct a new protocol for three-input majority voting with only four cards by observing the relationship between AND/OR operations. This protocol can be easily extended to more participants, and to the protocol for threshold functions.

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