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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Card-Based Arithmetic Operations Using Integer Commitments and Their Application to Statistical Data Aggregation

Shun Odaka, Yuichi Komano

Published Jul 24, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1587/transfun.2025cip0006

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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 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 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.
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 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.
We construct statistical zero-knowledge authentication protocols for smart cards based on general assumptions. The main protocol is only secure against active attacks, but we present a modication based on trapdoor commitments that can resist concurrent attacks as well. Both protocols are instantiated using lattice-based primitives, which are conjectured to be secure against quantum attacks. We illustrate the practicality of our main protocol on smart cards in terms of storage, computation, communication, and round complexities. Furthermore, we compare it to other lattice-based authentication protocols, which are either zero-knowledge or have a similar structure. The comparison shows that our protocol improves the best previous protocol.