Card-based cryptography is a research field for realizing secure computation using physical cards. In card-based protocols, the efficiency of a protocol is mainly measured by the number of cards and the number of shuffles. Shinagawa and Nuida proposed a batching technique, which is a technique for reducing the number of pile-scramble shuffles (PSS) by combining multiple PSSs into a single PSS. In this paper, we propose three new methods for the batching technique. The first method uses a new shuffle called a generalized PSS; unlike a PSS, this allows the number of cards in each pile to be uneven. The second method uses a shuffle called a branching PSS, which is a PSS for a sequence including face-up cards. The third method uses a shuffle called a branching generalized PSS, which is a generalized PSS for a sequence including face-up cards. Our methods are efficient in terms of the number of cards and shuffles compared to the existing batching technique. As applications, we use our methods to reduce the number of shuffles for AND protocols and XOR protocols.
Card-based cryptography is a research field for realizing secure computation using physical cards. In card-based protocols, the efficiency of a protocol is mainly measured by the number of cards and the number of shuffles. Shinagawa and Nuida proposed a batching technique, which is a technique for reducing the number of pile-scramble shuffles (PSS) by combining multiple PSSs into a single PSS. In this paper, we propose three new methods for the batching technique. The first method uses a new shuffle called a generalized PSS; unlike a PSS, this allows the number of cards in each pile to be uneven. The second method uses a shuffle called a branching PSS, which is a PSS for a sequence including face-up cards. The third method uses a shuffle called a branching generalized PSS, which is a generalized PSS for a sequence including face-up cards. Our methods are efficient in terms of the number of cards and shuffles compared to the existing batching technique. As applications, we use our methods to reduce the number of shuffles for AND protocols and XOR protocols.
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.
Abstract Card-based cryptography allows us to securely compute arbitrary functions using a deck of physical cards. Its performance is mainly measured by the number of used cards and shuffles, and there is a line of work that aims to reduce either of them. One seminal work is the card-based garbled circuit technique by Shinagawa and Nuida (Discret Appl Math 289:248–261, 2021, https://doi.org/10.1016/j.dam.2020.10.013 ), which allows the construction of a card-based protocol for any Boolean function with a single shuffle. Their construction requires 2n+24g 2 n + 24 g cards for an n-input Boolean function that is represented by g logical gates. In this paper, we reduce the number of cards to 2n+8g 2 n + 8 g for arbitrary functions while keeping it working with only one shuffle. In addition, we propose two types of extensions to support numerical encoding and multi-input gates. In the extended scheme, the free-ADD technique, obtained by generalizing the free-XOR technique by Manabe and Shinagawa (Deng J, Kolesnikov V, Schwarzmann AA (eds) CANS 2023, LNCS, vol 14342. Springer, Singapore, pp 232–248, 2023, https://doi.org/10.1007/978-981-99-7563-1-11 ), is available. The free-ADD technique allows our scheme to evaluate any n-input symmetric Boolean function using 2n2+6n+2 2 n 2 + 6 n + 2 cards.
In card-based cryptography, the numbers of cards and shuffles are the complexity measures of protocols for secure computations, and the smaller these values are, the better. As the state-of-the-art study to minimize the latter measure, Shinagawa and Nuida showed a surprising result that any n-input logical function can be securely computed with only one shuffle, based on the idea of Yao's garbled circuit. When executing their protocol, the number of required cards is 2n+24q, where the n-input logical function to be computed is represented by q gates. For example, when applied to the n-input AND and XOR functions, the number of gates is n-1, and hence, 26n-24 cards are required. In this paper, we show that the number of required cards can be reduced by focusing on these two specific functions. Specifically, we construct a single-shuffle protocol for the n-input AND function using 4n-2 cards, and construct a single-shuffle protocol for the n-input XOR function using 2n cards.
Abstract Card-based protocols perform secure computation using physical playing cards, and their efficiency is primarily evaluated by two metrics: the number of cards and the number of shuffles. In this paper, we present three new card-based protocols that reduce the number of cards. The first protocol supports any Boolean circuit and decreases the number of cards. The second protocol extends the first to a committed-format version, while the third is specialized for Boolean circuits without a branched input, achieving an even smaller number of cards. All the proposed protocols are based on Yao’s garbled-circuit framework, similar to previous single-shuffle card-based protocols. Our main contribution is a novel encoding method that represents the truth table of each Boolean gate with fewer cards than existing schemes. Consequently, our protocols require only six cards per gate in a single-shuffle execution, providing a significant improvement in efficiency over prior work. Furthermore, by incorporating the card-based free-XOR introduced by Manabe–Shinagawa, we can further reduce the number of cards in certain configurations.
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.
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.