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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Floating-point Number Commitments in Card-based Cryptography and Their Applications

Shun Odaka, Yuichi Komano

Published Jan 1, 2026• 0 citations
Publisher

Abstract

Card-based cryptography enables the execution of cryptographic protocols such as secure multiparty computation, by using a deck of physical cards. To facilitate statistical analysis with card-based cryptography, we have previously presented protocols for integer-based arithmetic operations. These protocols utilize integer commitments, which are face-down card sequences encoding integers, and consist of simple operations. However, in statistical applications, they suffer from poor scalability, as the required number of cards increases significantly as the number of data points increases. To address this issue, this paper introduces a new representation of commitments, named floating-point number (FPN) commitments, which are face-down card sequences encoding floating-point numbers. These FPN commitments allow us to represent a wide range of numbers, including negative and arbitrary-precision values, with fewer cards. We then propose protocols for converting a conventional integer commitment to an FPN commitment and for performing arithmetic operations using FPN commitments. We also demonstrate that our new representation and protocols enhance the applicability of card-based cryptography to statistical processing.

DOI
10.1587/transfun.2026cip0019

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Card-based cryptography enables the execution of cryptographic protocols such as secure multiparty computation, by using a deck of physical cards. To facilitate statistical analysis with card-based cryptography, we have previously presented protocols for integer-based arithmetic operations. These protocols utilize integer commitments, which are face-down card sequences encoding integers, and consist of simple operations. However, in statistical applications, they suffer from poor scalability, as the required number of cards increases significantly as the number of data points increases. To address this issue, this paper introduces a new representation of commitments, named floating-point number (FPN) commitments, which are face-down card sequences encoding floating-point numbers. These FPN commitments allow us to represent a wide range of numbers, including negative and arbitrary-precision values, with fewer cards. We then propose protocols for converting a conventional integer commitment to an FPN commitment and for performing arithmetic operations using FPN commitments. We also demonstrate that our new representation and protocols enhance the applicability of card-based cryptography to statistical processing.
Card-based cryptography enables players to compute logical and arithmetic operations securely, such as bitwise AND and addition of integers. Several multiparty computation protocols and zero-knowledge proof protocols utilizing these secure computations have been developed as its applications. However, the realization of an efficient protocol for an arithmetic operation other than addition and subtraction remains an open problem. This paper proposes card-based protocols, based on integer commitment, for multiplication, division, and square root. Compared to general constructions for protocols for these operations based on binary integer commitment, the proposed protocols exhibit superior simplicity and efficiency. Furthermore, these protocols introduce novel applications for card-based cryptography to secure statistical data aggregation.
Card-based cryptography studies the problem of representing the functionality and security of cryptographic primitives visually by using physical cards, to demonstrate their security properties for those who are unfamiliar with cryptography. In this study, we propose efficient card-based protocols for secure computation of the four basic arithmetic operations (addition, subtraction, multiplication, and division). Existing protocols for securely performing these operations on ℓ-bit integers either require an exponential number of card manipulations in ℓ or need Ω (ℓ) additional cards. The proposed protocols simultaneously resolves these drawbacks for the first time by achieving a polynomial number of card manipulations in ℓ while requiring only a constant number of additional cards. The construction of our protocols is based on the long-hand algorithms for each of the four arithmetic operations. The technical novelty lies in demonstrating that the number of additional cards can be kept constant by freeing the cards representing intermediate values that are no longer needed during the process.
Research in secure multi-party computation using a deck of playing cards, often called card-based cryptography, dates back to 1989 when Den Boer introduced the five-card trick to compute the logical AND function. Since then, many protocols to compute different functions have been developed. In this paper, we propose a new encoding scheme using five cards to encode each integer in Z/6Z\mathbb{Z}/6\mathbb{Z}Z/6Z. Using this encoding scheme, we develop protocols that can copy a commitment with 13 cards, add two integers with 10 cards, and multiply two integers with 16 cards. All of our protocols are the currently best known protocols in terms of the required number of cards. Our encoding scheme can also be generalized to encode integers in Z/nZ\mathbb{Z}/n\mathbb{Z}Z/nZ for other values of nnn as well.
Abstract In the history of cryptography, many cryptographic protocols have relied on random coin tosses to prove their security. Although flipping coins is indispensable in this manner, the coins themselves have never been in the spotlight. Therefore, we would like to make physical coins rise to the level of cryptography, just as a deck of physical playing cards has been used to perform a secure multi-party computation. Such a card-based protocol is known to be helpful both to perform a secure computation without any black-box computers and to understand the principles of secure protocols. In this paper, we propose a new framework of secure multi-party computation using physical coins, named a coin-based protocol. One advantage of the use of coins is that they are more ubiquitous than cards. Whereas a face-down card can conceal the information about its face side, one side of a coin reveals the information of its other side. Hence, more careful design is required for a secure coin-based protocol than for a card-based one. We formalize a computational model of the coin-based protocol and explicitly give protocols for NOT, AND, COPY, OR, and XOR computations. We also discuss the composability of the extended protocols and how to implement them in practice.
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.
Card-based cryptography is an area involving cryptographic techniques that use a deck of physical cards to implement secure computation. In this paper, we study card-based cryptographic protocols for symmetric functions f: {0, 1}n → R, where R is an arbitrary finite set. Any symmetric Boolean function has the property that its output value is determined only by the sum of input bits, and the class of them includes many interesting functions such as the AND and equality functions. In 2020, Ruangwises and Itoh proposed a protocol which uses 2n + 2 cards to securely compute any symmetric function f: {0, 1}n → R; reducing the number of cards to less than 2n + 2 was left as an open problem. While the Ruangwises–Itoh protocol uses a two-colored deck of cards, this paper considers using three-colored and four-colored decks: to give a partial answer to the open problem, we propose three protocols for symmetric functions using 2n + 1 cards or less. Specifically, we first propose a protocol using 2n + 1 cards from a three-colored deck. We then propose two protocols using 2n cards from a four-colored deck; one is a simpler protocol and the other is a more shuffle-efficient protocol.