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arXiv:1009.4796 · Physical Review A

Experimentally feasible security check for n -qubit quantum secret sharing

Stefan Schauer, Marcus Huber, Beatrix C. Hiesmayr

Published Dec 13, 2010• 51 citations
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Abstract

In this article we present a general security strategy for quantum secret sharing (QSS) protocols based on the scheme presented by Hillery, Bu\ifmmode \check{z}\else \v{z}\fi{}ek, and Berthiaume (HBB) [Phys. Rev. A 59, 1829 (1999)]. We focus on a generalization of the HBB protocol to $n$ communication parties thus including $n$-partite Greenberger-Horne-Zeilinger states. We show that the multipartite version of the HBB scheme is insecure in certain settings and impractical when going to large $n$. To provide security for such QSS schemes in general we use the framework presented by some of the authors [M. Huber, F. Mintert, A. Gabriel, B. C. Hiesmayr, Phys. Rev. Lett. 104, 210501 (2010)] to detect certain genuine $n$-partite entanglement between the communication parties. In particular, we present a simple inequality which tests the security.

DOI
10.1103/physreva.82.062311

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In this article we present a general security strategy for quantum secret sharing (QSS) protocols based on the scheme presented by Hillery, Bu\ifmmode \check{z}\else \v{z}\fi{}ek, and Berthiaume (HBB) [Phys. Rev. A 59, 1829 (1999)]. We focus on a generalization of the HBB protocol to nnn communication parties thus including nnn-partite Greenberger-Horne-Zeilinger states. We show that the multipartite version of the HBB scheme is insecure in certain settings and impractical when going to large nnn. To provide security for such QSS schemes in general we use the framework presented by some of the authors [M. Huber, F. Mintert, A. Gabriel, B. C. Hiesmayr, Phys. Rev. Lett. 104, 210501 (2010)] to detect certain genuine nnn-partite entanglement between the communication parties. In particular, we present a simple inequality which tests the security.
The participant attack is the most serious threat for quantum secret-sharing protocols. We present a method to analyze the security of quantum secret-sharing protocols against this kind of attack taking the scheme of Hillery, Bu\ifmmode \check{z}\else \v{z}\fi{}ek, and Berthiaume (HBB) [Phys. Rev. A 59, 1829 (1999)] as an example. By distinguishing between two mixed states, we derive the necessary and sufficient conditions under which a dishonest participant can attain all the information without introducing any error, which shows that the HBB protocol is insecure against dishonest participants. It is easy to verify that the attack scheme of Karlsson, Koashi, and Imoto [Phys. Rev. A 59, 162 (1999)] is a special example of our results. To demonstrate our results further, we construct an explicit attack scheme according to the necessary and sufficient conditions. Our work completes the security analysis of the HBB protocol, and the method presented may be useful for the analysis of other similar protocols.
In this work, we generalize the quantum-secret-sharing scheme of Hillery, Bu\ifmmode \check{z}\else \v{z}\fi{}ek, and Berthiaume [Phys. Rev. A 59, 1829 (1999)] into arbitrary multiparties. Explicit expressions for the shared secret bit is given. It is shown that in the Hillery-Bu\ifmmode \check{z}\else \v{z}\fi{}ek-Berthiaume quantum-secret-sharing scheme the secret information is shared in the parity of binary strings formed by the measured outcomes of the participants. In addition, we have increased the efficiency of the quantum-secret-sharing scheme by generalizing two techniques from quantum key distribution. The favored-measuring-basis quantum-secret-sharing scheme is developed from the Lo-Chau-Ardehali technique [H. K. Lo, H. F. Chau, and M. Ardehali, e-print quant-ph∕0011056] where all the participants choose their measuring-basis asymmetrically, and the measuring-basis-encrypted quantum-secret-sharing scheme is developed from the Hwang-Koh-Han technique [W. Y. Hwang, I. G. Koh, and Y. D. Han, Phys. Lett. A 244, 489 (1998)] where all participants choose their measuring basis according to a control key. Both schemes are asymptotically 100100%100 in efficiency, hence nearly all the Greenberger-Horne-Zeilinger states in a quantum-secret-sharing process are used to generate shared secret information.
In this paper, we present an (n, n) threshold quantum secret sharing scheme of secure direct communication using Greenberger-Horne-Zeilinger state. The present scheme is efficient in that all the Greenberger-Horne-Zeilinger states used in the quantum secret sharing scheme are used to generate shared secret messages except those chosen for checking eavesdropper. In our scheme, the measuring basis of communication parties is invariable and the classical information used to check eavesdropping needs only the results of measurements of the communication parties. Another nice feature of our scheme is that the sender transmit her secret messages to the receivers directly and the receivers recover the sender's secret by combining their results, different from the QSS scheme whose object is essentially to allow a sender to establish a shared key with the receivers. This feature of our scheme is similar to that of quantum secret direct communication.
Analyzing carefully an experimentally feasible non-entangled single qubit quantum secret sharing protocol and its modified version [Phys. Rev. Lett. 95, 230505 (2005); \textit{ibid}. 98, 028902 (2007)], it is found that both versions are insecure against coherent attacks though the original idea is so remarkable. To overcome this fatal flaw, here we propose a protocol with a distinct security checking strategy, which still involves single qubit operations only, making it possible to achieve better security of quantum secret sharing with current technology.
With the rapid progress of quantum cryptography, secret sharing has been developed in the quantum setting for achieving a high level of security, which is known as quantum secret sharing (QSS). The first QSS scheme was proposed by Hillery et al. in 1999 [Phys. Rev. A, Vol. 59, p.1829 (1999)] based on entangled Greenberger-Horne-Zeilinger (GHZ) states. However, only 50% of the entangled quantum states are effective for eavesdropping detection and secret splitting in the original scheme. In this paper, we introduce a possible method, called measurement-delay strategy, to improve the qubit efficiency of the GHZ-based QSS scheme. By using this method, the qubit efficiency of the improved QSS scheme can reach 100% for both security detection and secret distribution. The improved QSS scheme can be implemented experimentally based on current technologies.