This paper proposes new card-based cryptographic protocols to calculate logic functions with the minimum number of cards using private operations under the semi-honest model. Though various card-based cryptographic protocols were shown, the minimum number of cards used in the protocol has not been achieved yet for many problems. Operations executed by a player where the other players cannot see are called private operations. Private operations have been introduced in some protocols to solve a particular problem or to input private values. However, the effectiveness of introducing private operations to the calculation of general logic functions has not been considered. This paper introduces three new private operations: private random bisection cuts, private reverse cuts, and private reveals. With these three new operations, we show that all of AND, XOR, and copy protocols are achieved with the minimum number of cards by simple three-round protocols. This paper then shows a protocol to calculate any logical functions using these private operations. Next, we consider protocols with malicious players.

中文翻译：

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使用私有操作的基于卡的密码逻辑计算

本文提出了新的基于卡片的密码协议，在半诚实模型下使用私有操作以最少的卡片数量计算逻辑函数。尽管显示了各种基于卡的加密协议，但由于许多问题，协议中使用的卡的最少数量尚未达到。由玩家执行的其他玩家无法看到的操作称为私人操作。一些协议中引入了私有操作来解决特定问题或输入私有值。但是，没有考虑将私有操作引入到通用逻辑函数的计算中的有效性。本文介绍了三种新的私有操作：私有随机二分切割、私有反向剪切和私有揭示。通过这三个新操作，我们证明了所有的 AND，XOR 和复制协议是通过简单的三轮协议以最少数量的卡实现的。然后，本文展示了使用这些私有操作计算任何逻辑函数的协议。接下来，我们考虑与恶意玩家的协议。