Rational quantum secret sharing

Scientific Reports, Jul 2018

The traditional quantum secret sharing does not succeed in the presence of rational participants. A rational participant’s motivation is to maximize his utility, and will try to get the secret alone. Therefore, in the reconstruction, no rational participant will send his share to others. To tackle with this problem, we propose a rational quantum secret sharing scheme in this paper. We adopt the game theory to analyze the behavior of rational participants and design a protocol to prevent them from deviating from the protocol. As proved, the rational participants can gain their maximal utilities when they perform the protocol faithfully, and the Nash equilibrium of the protocol is achieved. Compared to the traditional quantum secret sharing schemes, our scheme is fairer and more robust in practice.

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Rational quantum secret sharing

Abstract The traditional quantum secret sharing does not succeed in the presence of rational participants. A rational participant’s motivation is to maximize his utility, and will try to get the secret alone. Therefore, in the reconstruction, no rational participant will send his share to others. To tackle with this problem, we propose a rational quantum secret sharing scheme in this paper. We adopt the game theory to analyze the behavior of rational participants and design a protocol to prevent them from deviating from the protocol. As proved, the rational participants can gain their maximal utilities when they perform the protocol faithfully, and the Nash equilibrium of the protocol is achieved. Compared to the traditional quantum secret sharing schemes, our scheme is fairer and more robust in practice. Introduction “Secret sharing” (SS) was first proposed by Shamir1, suggesting a secure way to distribute information (secret) to a set of participants. SS splits the secret into several parts and distributes them to different participants, so that only qualified participants can recover the original secret. In Shamir’s scheme, a participant is classified as “good” or “bad”. A good participant always performs the protocol faithfully, while the bad one would try his best to break it. However, this kind of classification may not reflect practical situations. Indeed, a participant can be neither good nor bad, but rational and always try to maximize his utility. Hence, each rational participant aims to get the secret, but at the same time, prevents others to get it. The involvement of rational participants leads to a major problem in SS. In SS, a participant can recover the secret alone even not sending his share to others, if others have sent out theirs. On the other hand, if participants did not send their shares, none can recover the secret. Therefore, from the viewpoint of a rational participant, not sending his share weakly dominates sending his share. This implies the Nash equilibrium corresponds to the case that nobody sends his share to others, resulting in a failure of Shamir’s scheme in the presence of rational participants. To mitigate this problem, Halpern et al.2 introduced the concept of “rational secret sharing” (RSS), and it has become an active area of research in recent years3,4,5. In classical RSS, signed share is used to prevent cheating of participants, while another approach is to use verifiable secret sharing6. On the other hand, Hillery et al.7 have proposed “Quantum secret sharing” (QSS), which can be considered as an extension of Shamir’s SS into the area of quantum. In QSS, the secret is split, distributed and reconstructed by quantum operations. QSS provides more perfect security based on the quantum theory such as uncertainty principle and no-cloning theorem. Similar to SS schemes, the existing QSS schemes8,9,10,11,12,13,14,15,16,17,18,19,20,21 do not consider the rational behavior of participants. However, it is natural for the last participant, if he is rational, to generate the secret and quit with it alone. Thus, rational participants in QSS would always prefer not to provide their shares, making the conventional QSS schemes fail. It should be emphasized those approaches suggested in RSS, such as signed share or verifiable SS, are based on unproven assumptions such as the intractability of integer factorization. In the quantum domain, participants and adversaries are always assumed to have unbounded computational power. As a result, these methods are inadequate for the design of rational quantum secret sharing (RQSS). In addition, there are other technical hurdles to be overcome for the design of RQSS, for example, the existing quantum signature schemes22,23 fail to deal with the entanglement among distributed shares, and a participant cannot generate copies of his share due to the no-cloning theorem. Designing a workable RQSS is challenging but valuable, and it is also the main objective of this paper. In our proposal, the shared secret is assumed to be a d-dimensional quantum state. Some basic quantum operations, such as the quantum Fourier transform and quantum-controlled-not, are employed. Unlike our previous work24 and other QSS schemes, here the issue of “rationality” is focused. Game theory is introduced to analyze the rational behavior of participants, based on the concepts of rationality, fairness and Nash equilibrium. As with most of the QSS schemes7,8,9,10,11,12, the threshold structure of our scheme is (n, n) structure, meaning that all the n participants compose the only qualified set and any subset with fewer than n participants is a forbidden set. Our design can avoid rational participants to deviate from the protocol since an unfaithful act will not gain higher utility than a faithful one. As a result, the achieved Nash equilibrium corresponds to the case when all the rational participants perform the protocol faithfully, and eventually, the shared qua (...truncated)


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Huawang Qin, Wallace K. S. Tang, Raylin Tso. Rational quantum secret sharing, Scientific Reports, 2018, Issue: 8, DOI: 10.1038/s41598-018-29051-z