Abstract 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.
Abstract 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.
Private operations (private permutations) were independently introduced by Nakai et al. and Marcedone et al. for implementing card-based cryptographic protocols efficiently. Recently, 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 four-card majority voting protocol with three inputs. In this paper, it is shown that only three cards are sufficient to construct the majority voting protocol with three inputs. Specifically, we propose two constructions of three-input majority voting protocols. First, assuming that players are allowed to announce their outputs, we show that one card can be reduced from Nakai et al.'s protocol without any additional private operations and communications. Our second construction requires two more private operations and communications, whereas it removes the assumption on announcement from the first construction.
Card-based cryptography realizes secure multiparty computation using physical cards. In 2018, Watanabe et al. proposed a card-based 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.
Abstract 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, 2ncards are necessary for every protocol withn-bit input since at least two cards are required to express a bit. In this paper, we proposen-bit input protocols with fewer than 2ncards 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.
Abstract Card-based cryptography is a secure computation protocol realized by using physical cards. There are two models on card-based cryptography: public and private models. We adopt private one that allows players to handle cards privately. While much of the existing works for card-based cryptography use two-colored cards, it is also a vital task to construct an efficient protocol with playing cards. In the public model, 2n cards are necessary for any n-bit input protocol since at least two cards are required to express a Boolean value. It holds true for both two-colored and playing-card settings. On the other hand, the private model enables us to construct a protocol with fewer than 2n cards. However, all existing protocols that achieve such properties are only in the two-colored setting. This paper shows that the private model enables us to construct a protocol with fewer than 2n cards using the playing cards. We first show two-bit input protocols with fewer than four cards for logical operations, AND, OR, and XOR. Furthermore, we show a three-input majority voting protocol using only three cards, which is constructed by combining our AND and OR protocols. Notably, our proposed protocols require no randomness. All operations are deterministic and depend only on players’ private inputs.
Card-based protocol is a multi-party computation using cards. The card-based protocol using operations called private operation has an advantage that the number of cards and the number of times of communication are smaller than the card-based protocol using operations called shuffle. However, there is a disadvantage that private operation allows dishonest players to perform malicious behaviors. Although the method to detect malicious behaviors in private operations was proposed, the method was available only in committed-format protocols, where inputs and outputs are represented by a pair of cards called commitment. In this paper, we show how to detect malicious behaviors in non-committed-format protocol with an example of a three-input majority voting protocol using private operations. Our majority voting protocol requires a smaller number of cards than the minimum number of cards required for committed-format protocols.
This paper shows new card-based cryptographic protocols with the minimum number of rounds, using private operations under the semi-honest model. Physical cards are used in card-based cryptographic protocols instead of computers to achieve secure multiparty computation. Operations that a player executes in a place where the other players cannot see are called private operations. Using three private operations—private random bisection cuts, private reverse cuts, and private reveals—the calculations of two variable Boolean functions and copy operations were realized with the minimum number of cards. Though the number of cards has been discussed, the efficiency of these protocols has not been discussed. This paper defines the number of rounds to evaluate the efficiency of the protocols, using private operations. Most of the meaningful calculations using private operations need at least two rounds. This paper presents a new two-round committed-input, committed-output logical XOR protocol, using four cards. Then, we show new two-round committed-input, committed-output logical AND and copy protocols, using six cards. Even if private reveal operations are not used, logical XOR, logical AND, and copy operations can be executed with the minimum number of rounds. Protocols for general n-variable Boolean functions and protocols that preserve an input are also shown. Lastly, protocols with asymmetric cards are shown.