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Card-based cryptography is a variety of secure multiparty computation (MPC). Recently, a new technique called private operations was introduced because the protocol can be implemented with fewer cards than that by using the conventional technique called the shuffle. For example, Nakai et al. showed that if the private operations are available, secure computations of AND and OR operations for two inputs can be realized simultaneously by using four cards, and the protocol is applied to a four-card majority voting protocol with three inputs. This paper shows that only three cards are sufficient to construct a majority voting protocol with three inputs. Specifically, we propose two constructions of three-input majority voting protocols. One is a protocol assuming that players can announce their output, and the other is not allowed. Compared to Nakai et al.’s protocol, the protocol with the announcement is realized without any additional private operations and communications. On the other hand, the second construction requires two more private operations and communications because it removes the assumption on the announcement from the first construction. More importantly, the idea of the second protocol can be extended to an n-input majority voting protocol with n cards, which is the main result of this paper.

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Card-based cryptography realizes secure multiparty computation using physical cards. In 2018, Watanabe et al. proposed a cardbased three-input majority voting protocol using three cards. In a card-based cryptographic protocol with 𝑛-bit inputs, it is known that a protocol using shuffles requires at least 2𝑛 cards. In contrast, as Watanabe et al.'s protocol, a protocol using private permutations can be constructed with fewer cards than the lower bounds above. Moreover, an 𝑛-input protocol using private permutations would not even require 𝑛 cards in principle since a private permutation depending on an input can represent the input without using additional cards. However, there are only a few protocols with fewer than 𝑛 cards. Recently, Abe et al. extended Watanabe et al.'s protocol and proposed an 𝑛-input majority voting protocol with 𝑛 cards and 𝑛 + ⌊𝑛/2⌋ + 1 private permutations. This paper proposes an 𝑛-input majority voting protocol with ⌈𝑛/2⌉ + 1 cards and 2𝑛 − 1 private permutations, which is also obtained by extending Watanabe et al.'s protocol. Compared with Abe et al.'s protocol, although the number of private permutations increases by about 𝑛/2, the number of cards is reduced by about 𝑛/2. In addition, unlike Abe et al.'s protocol, our protocol includes Watanabe et al.'s protocol as a special case where 𝑛 = 3.