Abstract In card-based cryptography, designing AND protocols in committed format is a major research topic. The state-of-the-art AND protocol proposed by Koch, Walzer, and Härtel in ASIACRYPT 2015 uses only four cards, which is the minimum permissible number. The minimality of their protocol relies on somewhat complicated shuffles having non-uniform probabilities of possible outcomes. Restricting the allowed shuffles to uniform closed ones entails that, to the best of our knowledge, six cards are sufficient: the six-card AND protocol proposed by Mizuki and Sone in 2009 utilizes the random bisection cut, which is a uniform and cyclic (and hence, closed) shuffle. Thus, a question has arisen: “Can we improve upon this six-card protocol using only uniform closed shuffles?” In other words, the existence or otherwise of a five-card AND protocol in committed format using only uniform closed shuffles has been one of the most important open questions in this field. In this paper, we answer the question affirmatively by designing five-card committed-format AND protocols using only uniform cyclic shuffles. The shuffles that our protocols use are the random cut and random bisection cut, both of which are uniform cyclic shuffles and can be easily implemented by humans.
Abstract In card-based cryptography, designing AND protocols in committed format is a major research topic. The state-of-the-art AND protocol proposed by Koch, Walzer, and Härtel in ASIACRYPT 2015 uses only four cards, which is the minimum permissible number. The minimality of their protocol relies on somewhat complicated shuffles having non-uniform probabilities of possible outcomes. Restricting the allowed shuffles to uniform closed ones entails that, to the best of our knowledge, six cards are sufficient: the six-card AND protocol proposed by Mizuki and Sone in 2009 utilizes the random bisection cut, which is a uniform and cyclic (and hence, closed) shuffle. Thus, a question has arisen: “Can we improve upon this six-card protocol using only uniform closed shuffles?” In other words, the existence or otherwise of a five-card AND protocol in committed format using only uniform closed shuffles has been one of the most important open questions in this field. In this paper, we answer the question affirmatively by designing five-card committed-format AND protocols using only uniform cyclic shuffles. The shuffles that our protocols use are the random cut and random bisection cut, both of which are uniform cyclic shuffles and can be easily implemented by humans.
In card-based cryptography, designing AND protocols in committed format is a major topic of research. The state-of-the-art AND protocol proposed by Koch, Walzer, and Härtel in ASIACRYPT 2015 uses only four cards, which is the minimum permissible number. Their protocol»s minimality relies on somewhat complicated shuffles having non-uniform probabilities of possible outcomes. Restricting the allowed shuffles to "practical»» ones, namely uniform closed shuffles, to our knowledge, six cards are sufficient: The six-card AND protocol proposed by Mizuki and Sone in 2009 utilizes the random bisection cut, which is a uniform and cyclic (and hence, closed) shuffle. Thus, a question has arisen: Can we improve upon this six-card protocol using only practical shuffles? In other words, whether there exists a five-card AND protocol in committed format using only uniform closed shuffles has been one of the most important open questions in this field. In this paper, we answer the question affirmatively by designing a five-card committed-format AND protocol using only practical shuffles. The shuffles that our protocol uses are random cut and random bisection cut, both of which are uniform cyclic shuffles and can be easily implemented by humans.
Card-based cryptographic protocols provide secure multi-party computations using a deck of physical cards. The most important primitive of those protocols is the shuffling operation, and most of the existing protocols rely on uniform cyclic shuffles (such as the random cut and random bisection cut) in which each possible outcome is equally likely and all possible outcomes constitute a cyclic subgroup. However, a couple of protocols with non-uniform and/or non-cyclic shuffles were proposed by Koch, Walzer, and Härtel at Asiacrypt 2015. Compared to the previous protocols, their protocols require fewer cards to securely produce a hidden AND value, although to implement of such unconventional shuffles appearing in their protocols remains an open problem. This paper introduces “pile-shifting scramble,” which can be a secure implementation of those shuffles. To implement such unconventional shuffles, we utilize physical cases that can store piles of cards, such as boxes and envelopes. Therefore, humans are able to perform the shuffles using these everyday objects. Furthermore, we show that a certain class of non-uniform and/or non-cyclic shuffles having two possible outcomes can be implemented by the pile-shifting scramble. This also implies that we can improve upon the known COPY protocol using three card cases so that the number of cases required can be reduced to two.
Card-based cryptographic protocols provide secure multi-party computation using a deck of physical cards. The most important primitive of those protocols is the shuffling operation, and most known protocols rely on uniform shuffles (such as the random cut and random bisection cut) in which each possible outcome is equally likely. However, several protocols with non-uniform shuffles have recently been proposed by Koch et al. Compared to other protocols, their protocols require fewer cards to securely produce a hidden AND value, although implementation of the non-uniform shuffle appearing in their protocols remains an open problem. This paper presents a secure implementation of their non-uniform shuffle. To implement the shuffle, we utilize physical cases that can store piles of cards, such as boxes and envelopes. Therefore, humans are able to perform the non-uniform shuffle using these everyday objects.
Card-based protocols are cryptographic protocols that use a deck of physical cards. In this paper, we deal with card-based protocols using a standard deck of playing cards, which are commonly and commercially available. For finite-runtime committed-format protocols, Mizuki (CANS 2016) proposed an eight-card AND protocol with four random bisection cuts and a six-card COPY protocol with one random bisection cut. In this paper, we propose a partial-open action, which reveals any position of the face of cards, by generalizing the half-open action introduced by Miyahara and Mizuki (IJTCS-FAW 2022). Using the partial-open action, we propose four-card AND protocols with three random cuts, a four-card COPY protocol with three random cuts, and a four-card base conversion protocol with a random cut. We note that, without partial-open actions, these AND and COPY protocols with random cuts only are known to be impossible to construct using four cards. Therefore, the partial-open actions are inherently necessary to obtain our results.
In card-based cryptography, a commitment to a Boolean value is usually represented by two face-down cards of different colors or numbers, whose order specifies the one-bit value (namely, 0 or 1). One of the most important primitives in card-based cryptography is a "copy protocol," which is supposed to make two identical copies of a given commitment. In the literature, there are several copy protocols, which can be categorized by kinds of shuffles they use; this paper focuses on those using only the so-called random cut, which is the simplest shuffle, and we propose two copy protocols that are more efficient than the existing ones. Specifically, we first work on a standard deck of cards and design a six-card copy protocol using three random cuts (on average). Since the previous protocol needs 5.5 random cuts, our protocol improves upon it. Next, we shift our attention to the case of a two-colored deck of cards, and construct a six-card copy protocol using three random cuts (on average). Because the previous protocol requires eight cards, our protocol uses two cards fewer than the previous one (although it uses one more shuffle). In addition, going back to the standard-deck setting, we provide a four-card XOR protocol using only one random cut for the first time.
Abstract Card-based cryptography started with the “five-card trick” designed by Den Boer (EUROCRYPT 1989); it enables Alice and Bob to securely evaluate the AND value of their private bits using a physical deck of five cards. It was then shown that the same task can be done with only four cards, i.e., Mizuki et al. proposed a four-card AND protocol (ASIACRYPT 2012). These two AND protocols are simple and easy even for non-experts, such as high school students, to execute. Their only common drawback is the need to prepare a customized deck consisting of red and black cards such that all cards of the same color must be identical. Fortunately, several existing protocols are based on a standard deck of playing cards (commercially available). Among them, the state-of-the-art AND protocol was constructed by Koch et al. (ASIACRYPT 2019); it uses four playing cards (such as ‘A, J, Q, K’) to securely evaluate the AND value. The protocol is elaborate, while its possible drawback is the need to repeat a shuffling operation six times (in expectation), which makes it less practical. This paper aims to provide the first practical protocol working on a standard deck of playing cards. We present an extremely simple AND protocol that terminates after only one shuffle using only four cards; our proposed protocol relies on a new operation, called the “half-open” action, whereby players can check only the suit of a face-down card without revealing the number on it. We believe that this new operation is easy-to-implement, and hence, our four-card AND protocol working on a standard deck is practical. We formalize the half-open action to present a formal description of our proposed protocol. Moreover, we discuss what is theoretically implied by introducing the half-open action and show that it can be applied to efficiently solving Yao’s Millionaires’ problem with a standard deck of cards.