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Cryptography

Minimum Round Card-Based Cryptographic Protocols Using Private Operations

Hibiki Ono, Yoshifumi Manabe

Published Jul 13, 2021• 8 citations
Publisher

Abstract

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.

DOI
10.3390/cryptography5030017

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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.
Abstract 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.
Abstract This paper shows new card-based cryptographic protocols using private operations that are secure against malicious players. Physical cards are used in card-based cryptographic protocols instead of computers. Operations that a player executes in a place where the other players cannot see are called private operations. Using several private operations, calculations of two variable Boolean functions and copy operations were realized with the minimum number of cards. Though private operations are very powerful in card-based cryptographic protocols, there is a problem that it is very hard to prevent malicious actions during private operations. Though most card-based protocols are discussed in the semi-honest model, there might be cases when the semi-honest model is not enough. Thus, this paper shows new protocols that are secure against malicious players. We show logical XOR, logical AND, n-variable Boolean function, and copy protocols. We can execute any logical computations with a combination of these protocols. We use envelopes as an additional tool that can be easily prepared and used by people.
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.
This paper proposes new efficient card-based cryptographic protocols for the millionaires' problem using private input operations. The millionaires' problem is one of the fundamental problems in cryptography. Two players, Alice and Bob, want to know which of them is richer without revealing their actual amount of asset. Many cryptographic protocols were proposed to solve the problem. Card-based cryptographic protocols were proposed to execute cryptographic protocols using physical cards instead of computers. Though some card-based cryptographic protocols for the millionaires' problem were proposed, most of them use many cards whose number depends on the size of the amount of asset. Though Nakai et al. implicitly proposed a new protocol that uses a constant number of cards using private input operations, their protocol is not efficient since the number of rounds is 2n+1, where n is the maximum number of bits of the asset. This paper shows new card-based protocols whose number of rounds is n+1. Another important feature of the proposed protocols is no-open property. No cards are opened until the end of the protocol.
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+1n+1n+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 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.