The rapid development of the Internet has brought people close together, and on this basis, it has been extended and expanded into the Internet of Things (IoT), that is, the Internet of all things connected [1,2,3,4]. As an emerging technology, the Internet of Things has effectively promoted the intelligent development of industry, agriculture, transportation and other aspects. Applications in smart home, pollution monitoring, medical health, and other fields closely related to life have greatly improved people’s quality of life. Group-oriented technology shows its application potential in IoT. For example, the data collected jointly by using group-oriented technology can be used to analyze traffic conditions and realize multi-user interactive computing. As these group-oriented programs are applied on open and insecure networks, they face the need for security.
At the same time, cloud computing services are more popular for data storage and retrieval in the cloud environment. Because of the user’s data security, the encrypted data is persisted in the cloud to protect from permission denied users. The method considers cloud services provider (CSP) or trusted authority to take care the key management assurance like an “it confirmed that key cannot be compromised”. However, some entities may interrupt communications between users and CSP. Hence, it compels the CSP to release user secrecy. In cloud surroundings, data holders, store data on the clouds which are transmitted through the deniable encryption scheme. Therefore, secure and efficient multiparty communications in Untrusted Cloud Environments (UCE) attract widespread attentions [5,6,7,8,9].
With the development of these technologies, more and more advanced companies are adopting the IoT and tend to transform to the intelligent systems (IS) to achieve high performance with less risk. IS refers to the application of smart devices with perception technology, data processing technology, and network communication technology to all links of intelligent transportation devices, which are connected through the network to achieve efficient utilization of resources, improve product quality and reduce resource consumption, thus realizing transportation intelligence. The devices in the IS have the particularity of heterogeneity, and the IS communication is carried out in an untrusted cloud environment, which is an asynchronous network, that is, multiple users cannot transmit their messages simultaneously. In such an Untrusted Cloud Environment (UCE), ensuring secure communication between these heterogeneous transportation devices is a major challenge for multiparty communication applied in IS, which is shown in Fig. 1. Furthermore, in such an asynchronous environment, the asynchronous transmission can cause security problems in cryptographic functions. How to make implementation of rational secret sharing (RSS) in an asynchronous model for UCE has become a burning research topic.
The (t, n) secret sharing scheme (SS) refers to dividing the secret into n sub-secrets so that the secret can be recovered when any t or more sub-secrets are known, but when the number of known sub-secrets is less than t, no information about the secret can be obtained. The SS has become a building block in many current cryptographic applications. There are multiple technologies that can be used to achieve secret sharing. For instance, Shamir [10] designed a (k, n) threshold scheme using linear polynomial, Azimuth-Bloom [11] studied the secret sharing scheme based on the Chinese Remainder Theorem (CRT), and Blakely [12] introduced a secret sharing scheme using hyperplane geometry. Among them, Shamir’s SS has been received most attention. This is because his SS is flexible and efficient, and it is information-theoretic secure.
If t or more sub-secrets are known, Shamir’s SS can reconstruct the secret using Lagrange Interpolation formula, and this process is very simple. However, in 2004, Halpern and Teague [13] have proposed the concept of Rational SS (RSS). In an RSS, it assumes that the players in the game of secret reconstruction are rational. In other words, each player may be honest or malicious. The players in the secret reconstruction aim to maximize their own interests and take corresponding actions. The objective of an RSS is to find a solution that is composed of strategies to encourage players in the secret reconstruction to act honestly even players are rational to act for their own interest. If each player plays the game for the best response to the best response of other players, the game is in Nash equilibrium. Achieving the Nash equilibrium state corresponding to the global optima is the goal of an RSS. Halpern and Teague [13] have shown that Shamir’s SS is not an RSS. This can be easily understood. Suppose there are t players in the secret reconstruction, if one of them is malicious and he publishes a false sub-secret, at the same time, all other honest players publish valid sub-secrets, then only this malicious player can reconstruct the secret, while no other honest player has access to the secret. This is not an equilibrium state.
Many researchers have proposed RSS schemes [14,15,16]. In these schemes, players can only exchange information in a synchronized channel. Since the synchronous channel is difficult to be implemented in practice, the existing communication networks are asynchronous networks in which multiple users cannot transmit their messages simultaneously. The asynchronous transmission can cause security problems. Therefore, it is imminent to implement RSS in the asynchronous model. Up to now, there are very few RSS solutions designed for asynchronous channels, including those designed by Maleka et al. [17], designed by Fuchsbauer et al. [14], designed by Ong et al. [15], and designed by Moses et al. [18]. In [17], the game needs to be repeated and an interactive dealer is required [14]. requires the use of cryptographic primitives. In [15, 18], multiple rounds of the game are played and a certain number of players are assumed to be honest.
In 1995, Lin et al. [19] presented a fair secret reconstruction scheme suitable for asynchronous models. In this scheme, the secret s is hidden in the sequence, {d1, d2, …, dj − 1, dj, dj + 1, …, dk}, and it can be reconstructed as a whole in the asynchronous network, where dj = s, dj + 1 = D, D is public, and di, i ≠ j, j + 1, ∀ i, is random integer. Furthermore, Maleka et al. [20] first proposed the concept of repeated games in the RSS problem and proved that limited repeated games are impossible when the player knows the number of repetitions of the game. In [19], instead of requiring all players to release their sub-secrets at the same time, the secret reconstruction process restores one element at a time in the order of the sequence until the secret is derived. If the certificate and sub-secret submitted by the player pass the verification, the rebuild process continues down the line, otherwise it is forced to stop. Until D is restored, the players can be sure that the previous restoration is the secret. In addition, it should be noted that if the player correctly guesses the location of the secret, he can get exclusive access to the secret. Then the probability that this malicious player has exclusive access to the secret and other honest players cannot obtain the secret is \(\frac{1}{k}.\) In a recent paper [17], the scheme has been improved to include some other features.
In this paper, we propose an RSS protocol which uses VSS for secure group communications in UCE, one of the most popular cryptographic primitives, as a strategy to enforce all players in the secret reconstruction to act honestly to reach the Nash equilibrium state. Our proposed RSS is information-theoretic secure. And it has two phases. In the first phase, each player acts as the dealer in Peterson’s VSS [21] to divide his/her share into sub-shares for other players. In addition, the player needs to make Peterson’s commitments publicly known. Using these commitments, these sub-shares can be verified. After all sub-shares of players being verified successfully, the scheme is advanced into the second phase. In the second phase, players take turn to reveal their shares asynchronously. Each revealed share can be verified separately. If any share has been verified unsuccessfully, other players can work together to recover the share. Since each share has been shared by other players in the first phase, this feature encourages players to reveal their shares honestly in the second phase. In other words, if players release fake shares in the second phase, their “real” shares can still be recovered by other honest players. The main contributions of this paper are as follows.
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An information-theoretic secure RSS in asynchronous model is proposed.
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The RSS uses Peterson’s VSS as building block. It allows every player to divide his share into multiple pieces for other players. Then, shares can be revealed asynchronously. If any player acts maliciously, his share can be recovered by other players.
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The RSS is deterministic and simple. This unique feature can encourage players to act honestly since any malicious action (i.e., either revealing a fake share or refusing to release one) is useless.
The rest of paper is organized as follows. Petersen’s VSS is reviewed in Review of Petersen’s VSS section. Model section introduces the model of the presented protocol, including protocol description, type of attacks. In Proposed scheme section, our RSS scheme is proposed. Analysis section analyzes the security of RSS. We conclude this paper in Conclusion section.
