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.”

中文翻译：

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如何用两种卡解决百万富翁问题

den Boer 提出的基于卡的密码学旨在通过使用物理卡实现多方计算（MPC）。我们为百万富翁的问题提出了几种有效的基于卡片的协议，通过引入一种称为私有排列 (PP) 的新操作，而不是大多数现有的基于卡片的密码学中使用的随机排列。Shuffle 是一种利用洗牌特性的有用随机化技术，但从算术 MPC 的角度来看，它需要一个强有力的假设，因为 shuffle 假设公共随机化是可能的。另一方面，PP 中可以使用私有随机性，这使我们能够设计基于卡的协议，同时考虑算术 MPC 的思想。实际上，我们展示了姚氏百万富翁协议可以通过使用 PP 轻松转换为基于卡片的协议，使用 shuffle 并不简单，因为 Yao 的协议使用私有随机性。此外，我们通过安全地更新两个数字之间的按位比较，提出了基于 PP 的全新且高效的基于卡片的百万富翁协议，这揭示了 PP 的力量。作为这些协议的另一个兴趣点，我们指出它们与著名的逻辑难题“道路分叉”有着深刻的联系。