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Theoretical Computer Science

Card-based protocols for secure ranking computations

Ken Takashima, Yuta Abe, Tatsuya Sasaki, Daiki Miyahara, Kazumasa Shinagawa, Takaaki Mizuki, Hideaki Sone

Published Sep 11, 2020• 28 citations
Publisher

Abstract

Consider a group of people who want to know the “rich list” among them, namely the ranking in terms of their total assets, without revealing any information about the actual value of their assets. This can be achieved by a “secure ranking computation,” which was first considered by Jiang and Gong (2006) [2]; they constructed a secure ranking computation protocol based on a public-key cryptosystem. In this paper, instead of using a public-key cryptosystem, we use a deck of physical cards to provide secure ranking computation protocols. Therefore, our card-based protocols do not rely on computers, and they are simple and easy for humans to implement. Specifically, we design four protocols considering tradeoffs between the number of cards and the number of shuffles required to execute the protocols. We also present a guide to choose an appropriate protocol according to the number of people participating in the protocol and the size of the input range. To be precise, whereas our protocols make all players know the rich list, the Jiang–Gong scheme makes each player know his/her rank only; to achieve the same task (as the Jiang–Gong scheme) using a deck of cards is an intriguing open problem.

DOI
10.1016/j.tcs.2020.09.008

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Consider a group of people who want to know the “rich list” among them, namely the ranking in terms of their total assets, without revealing any information about the actual value of their assets. This can be achieved by a “secure ranking computation,” which was first considered by Jiang and Gong (2006) [2]; they constructed a secure ranking computation protocol based on a public-key cryptosystem. In this paper, instead of using a public-key cryptosystem, we use a deck of physical cards to provide secure ranking computation protocols. Therefore, our card-based protocols do not rely on computers, and they are simple and easy for humans to implement. Specifically, we design four protocols considering tradeoffs between the number of cards and the number of shuffles required to execute the protocols. We also present a guide to choose an appropriate protocol according to the number of people participating in the protocol and the size of the input range. To be precise, whereas our protocols make all players know the rich list, the Jiang–Gong scheme makes each player know his/her rank only; to achieve the same task (as the Jiang–Gong scheme) using a deck of cards is an intriguing open problem.
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.
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.
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.
Secure computation enables a number of players each holding a secret input value to compute a function of the inputs without revealing the inputs. It is known that secure computation is possible physically when the inputs are given as a sequence of physical cards. This research area is called card-based cryptography. One of the important problems in card-based cryptography is to minimize the number of cards and shuffles, where a shuffle is the most important (and somewhat heavy) operation in card-based protocols. In this paper, we determine the minimum number of shuffles for achieving general secure computation. Somewhat surprisingly, the answer is just one, i.e., we design a protocol which securely computes any Boolean circuit with only a single shuffle. The number of cards required for our protocol is proportional to the size of the circuit to be computed.
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.
We consider a problem, which we call secure grouping, of dividing a number of parties into some subsets (groups) in the following manner: Each party has to know the other members of his/her group, while he/she may not know anything about how the remaining parties are divided (except for certain public predetermined constraints, such as the number of parties in each group). In this paper, we construct an information-theoretically secure protocol using a deck of physical cards to solve the problem, which is jointly executable by the parties themselves without a trusted third party. Despite the non-triviality and the potential usefulness of the secure grouping, our proposed protocol is fairly simple to describe and execute. Our protocol is based on algebraic properties of conjugate permutations. A key ingredient of our protocol is our new techniques to apply multiplication and inverse operations to hidden permutations (i.e., those encoded by using face-down cards), which would be of independent interest and would have various potential applications.