臺灣博碩士論文加值系統

由於量子電腦和量子演算法的迅速發展，使得許多安全性基於數學難題的傳統密碼協定被證明可在多項式時間內破解。有鑑於此，量子密碼學利用量子物理的特性設計通訊協定，且可達到無條件安全。量子通訊協定具有兩項優點：(1) 協定之安全性係植基於量子物理的特性，故不受量子電腦的威脅；(2) 可偵測通訊過程中是否存在竊聽者。然而，現存的量子通訊協定必須假設量子傳輸過程中無雜訊干擾的影響，即假設量子通道為理想通道。若無此假設，於偵測竊聽者的過程中發現錯誤時，將無法分辨此錯誤是竊聽或雜訊干擾所造成。因此，攻擊者可藉此發動攻擊，利用雜訊干擾所造成的錯誤隱藏其攻擊所產生的錯誤。讓通訊雙方在檢查竊聽者存在的討論中，誤以為錯誤的量測結果是通道雜訊所造成，而非竊聽者干擾造成。因此，如何在雜訊干擾的環境中建立安全的通訊協定將是未來量子密碼學研究的重點議題。
本論文利用量子物理的特性與可抗雜訊干擾的量子態設計可抗集合雜訊之量子安全通訊協定。本論文專注於研究可抗雜訊干擾的量子糾結態與其編碼方法。首先，針對二種量子糾結態，包括：GHZ糾結態（GHZ state）與GHZ-like糾結態 (GHZ-like state)，提出編碼方法，且利用提出的編碼方法建置量子金鑰分配協定、量子直接通訊協定與確定式量子通訊協定。最後，本論文利用貝爾糾結態（Bell state）提出一量子對話通訊協定。

Recently, due to the rapid development of quantum computers and quantum computings, it is believed that many difficult mathematical problems can easily be resolved in near future with the presence of quantum computers. Accordingly, the various well known cryptosystems whose security is based on these mathematical problems can easily be broken and then they become insecure. Eventually, that makes the researches of the quantum computation as well as quantum cryptography busy in order to resolve the aforesaid issues. By using the quantum phenomena, e.g., the no-cloning theorem and the measurement uncertainty principle, many quantum security protocols have been proposed. Compared to the classical communication protocols, the quantum security protocols have the following advantages: (1) Its security is based on quantum phenomena. Therefore it is secure against quantum computers; and (2) The existence of eavesdroppers can also be detected. However, the existing quantum security protocols assumed that the quantum channels are ideal. That is, the quantum channel is assumed to be free from any kind of noise. In reality, however, with the existence of noises, an eavesdropper can disguise his/her attack in noises in order to avoid being detected in eavesdropping check process. Hence, in real applications, quantum security protocols have to be designed to combat the noises, which is an important research direction.
In view of this fact, this thesis attempts to employ the properties of the quantum mechanics as well as decoherece-free states to develop fault tolerant quantum secure communication protocols in quantum environments. This thesis investigates the properties of decoherece-free entangled states and its coding functions. Firstly, two entangled states, GHZ states and GHZ-like states, are considered. Three coding functions are developed using these entangled states, and these are used to establish the fault tolerant quantum key distribution protocol, the fault tolerant quantum direct communication protocol, and the fault tolerant deterministic quantum communication protocol, respectively. Finally, this thesis proposes a fault tolerant quantum dialogue protocol based on Bell states.

中文摘要 I
Abstract II
誌謝 IV
Contents V
List of Tables IX
List of Figures X
Chapter 1 Introduction 1
1.1 Quantum Key Distribution Protocols 1
1.2 Quantum Message Communication Protocols 2
1.3 Motivation and Contributions 6
1.4 Thesis Structure 10
Chapter 2 Reviews of Quantum Cryptography 11
2.1 Quantum Properties 11
2.1.1 The Qubit and Its Properties 11
2.1.2 Unitary Operations 13
2.1.3 Entangled States and Their Properties 15
2.1.4 Efficiency Evaluation Function 18
2.2 Quantum Properties immune to Collective Noises 19
2.2.1 The Properties of Collective Noises 19
2.2.2 Quantum States immune to Collective Noises 20
2.2.2.1 The Logical Qubit immune to Collective-dephasing Noise 20
2.2.2.2 The Logical Qubit immune to Collective-rotation Noise 23
Chapter 3 New Techniques of Entangled States immune to Collective Noises 26
3.1 Coding Functions for 4-particle Entangled States 26
3.1.1 Coding Function for a 4-particle GHZ State 26
3.1.2 Coding Function for a 4-particle GHZ-like State 28
3.2 Entanglement Swapping via 3-particle Entangled States 31
3.3 Coding Functions for 3-particle Entangled States 33
3.3.1 Coding Function for a GHZ State 33
3.3.2 Coding Function for a GHZ-like State 35
Chapter 4 QKD Protocols over Collective Noise Channels 38
4.1 The Proposed QKD Protocol via 4-particle Entangled States 38
4.1.1 QKD over a Collective-dephasing Noise Channel 38
4.1.2 QKD over a Collective-rotation Noise Channel 41
4.1.3 Security Analysis 41
4.1.3.1 Security against Ordinary Eavesdropping 42
4.1.3.2 Security against Trojan Horse Attacks 42
4.2 The Proposed QKD Protocol via 3-particle Entangled States 43
4.2.1 QKD over a Collective-dephasing Noise Channel 43
4.2.2 QKD over a Collective-rotation Noise Channel 45
4.2.3 Security Analysis 45
4.2.3.1 Security against Intercept-and-resend Attack 45
4.2.3.2 Security against Entangle-and-measure Attack 46
4.2.3.3 Security under a Lossy Quantum Channel 47
4.3 Comparison 48
Chapter 5 QMC Protocols over Collective Noise Channels 51
5.1 Quantum Direct Communication (QDC) Protocols 51
5.1.1 The Proposed Two-step QDC Protocols 51
5.1.1.1 Fault Tolerant QDC immune to Collective-dephasing Noise 51
5.1.1.2 Fault Tolerant QDC immune to Collective-rotation Noise 53
5.1.2 Security Analysis 53
5.1.3 Comparison 55
5.2 Deterministic Quantum Communication (DQC) Protocols 56
5.2.1 The Proposed DQC Protocol via 3-particle Entangled States 56
5.2.1.1 DQC Protocol over a Collective-dephasing Noise Channel 56
5.2.1.2 DQC Protocol over a Collective-rotation Noise Channel 58
5.2.2 Security Analysis 58
5.2.2.1 Security Analysis of Information Leakage 59
5.2.2.2 Security against Ordinary Eavesdropping 59
5.2.3 Comparison 60
5.3 Quantum Dialogue (QD) Protocols 62
5.3.1 The Proposed QD Protocol via Bell States 62
5.3.1.1 QD immune to Collective-dephasing Noise 62
5.3.1.2 QD immune to Collective-rotation Noise 65
5.3.2 Security Analysis 66
5.3.2.1 Security Analysis of Information Leakage 66
5.3.2.2 Security against Intercept-and-resend Attack 67
5.3.2.3 Security against Entangle-and-measure Attack 68
5.3.2.4 Security against Correlation-elicitation (CE) Attack 69
5.3.2.5 Security under a Lossy Quantum Channel 71
5.3.2 Comparison 72
Chapter 6 Conclusions 73
Bibliography 76

[1]C. H. Bennett and G. Brassard, Quantum Cryptography: Public Key Distribution and Coin Tossing, presented at the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, 1984.
[2]C. H. Bennett, Quantum cryptography using any two nonorthogonal states, Physical Review Letters, vol. 68, pp. 3121-3124, 1992.
[3]C. H. Bennett, G. Brassard, and N. D. Mermin, Quantum cryptography without Bell’s theorem, Physical Review Letters, vol. 68, pp. 557-559, 1992.
[4]F. G. Deng, G. L. Long, Y. Wang, and L. Xiao, Increasing the efficiencies of random-choice-based quantum communication protocols with delayed measurement, Chinese Physics Letters, vol. 21, pp. 2097-2100, 2004.
[5]G. Long and X. Liu, Theoretically efficient high-capacity quantum-key-distribution scheme, Physical Review A, vol. 65, p. 032302, 2002.
[6]C. Li, H. S. Song, L. Zhou, and C. F. Wu, A random quantum key distribution achieved by using Bell states, Journal of Optics B-Quantum and Semiclassical Optics, vol. 5, pp. 155-157, Apr 2003.
[7]D. G. Song, Secure key distribution by swapping quantum entanglement, Physical Review A, vol. 69, Mar 2004.
[8]H. Xin, L. Shu-Min, and W. An-Min, A modified protocol of quantum key distribution based on entanglement swapping, Chinese Physics Letters, vol. 24, pp. 2479-2481, Sep 2007.
[9]G. Gao, Quantum key distribution by comparing Bell states, Optics Communications, vol. 281, pp. 876-879, Feb 2008.
[10]S. Schauer and M. Suda, A novel attack strategy on entanglement swapping QKD protocols, International Journal of Quantum Information, vol. 6, pp. 841-858, Aug 2008.
[11]H. Yuan, J. Song, L. F. Han, K. Hou, and S. H. Shi, Improving the total efficiency of quantum key distribution by comparing Bell states, Optics Communications, vol. 281, pp. 4803-4806, Sep 2008.
[12]G. Gao, Quantum Key Distribution Scheme with High Efficiency, Communications in Theoretical Physics, vol. 51, pp. 820-822, May 2009.
[13]N. Zhou, L. Wang, L. Gong, X. Zuo, and Y. Liu, Quantum deterministic key distribution protocols based on teleportation and entanglement swapping, Optics Communications, vol. 284, pp. 4836-4842, Sep 2011.
[14]H. K. Lo and H. F. Chau, Unconditional security of quantum key distribution over arbitrarily long distances, Science, vol. 283, pp. 2050-2056, Mar 26 1999.
[15]P. W. Shor and J. Preskill, Simple Proof of Security of the BB84 Quantum Key Distribution Protocol, Physical Review Letters, vol. 85, pp. 441-444, 2000.
[16]H. K. Lo, A simple proof of the unconditional security of quantum key distribution, Journal of Physics A-Mathematical and General, vol. 34, pp. 6957-6967, Sep 7 2001.
[17]D. Mayers, Unconditional security in quantum cryptography, Journal of the Acm, vol. 48, pp. 351-406, May 2001.
[18]E. Biham, M. Boyer, P. O. Boykin, T. Mor, and V. Roychowdhury, A proof of the security of quantum key distribution, Journal of Cryptology, vol. 19, pp. 381-439, Oct 2006.
[19]H. Inamori, N. Lutkenhaus, and D. Mayers, Unconditional security of practical quantum key distribution, European Physical Journal D, vol. 41, pp. 599-627, Mar 2007.
[20]Y. F. Chung, Z. Y. Wu, and T. S. Chen, Unconditionally secure cryptosystems based on quantum cryptography, Information Sciences, vol. 178, pp. 2044-2058, Apr 15 2008.
[21]K. Tamaki and T. Tsurumaru, Security Proof of Quantum Key Distribution, IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, vol. E93A, pp. 880-888, May 2010.
[22]H. Lu, C.-H. F. Fung, X. Ma, and Q.-Y. Cai, Unconditional security proof of a deterministic quantum key distribution with a two-way quantum channel, Physical Review A, vol. 84, Oct 31 2011.
[23]C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, Experimental quantum cryptography, Journal of Cryptology, vol. 5, pp. 3-28, 1992.
[24]R. Hughes, G. Luther, G. Morgan, C. Peterson, and C. Simmons, Quantum Cryptography over Underground Optical Fibers, presented at the Advances in Cryptology - CRYPTO '96, 16th Annual International Cryptology Conference, Santa Barbara, California, USA, August 18-22, 1996, Proceedings, Santa Barbara, California, USA, 1996.
[25]W. K. Wootters and W. H. Zurek, A Single Quantum Cannot Be Cloned, Nature, vol. 299, pp. 802-803, 1982.
[26]C. E. Shannon, Communication theory of secrecy systems, Bell System Technical Journal, vol. 28, pp. 656-715, 1949.
[27]P. W. Shor, Algorithms for quantum computation: discrete logarithms and factoring, in 35th Annual Symposium on the Foundations of Computer Science, Los Alamitos, Calif., 1994, pp. 124-134.
[28]P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Journal on Computing, vol. 26, pp. 1484-1509, Oct 1997.
[29]P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Review, vol. 41, pp. 303-332, Jun 1999.
[30]R. Jozsa, Quantum factoring, discrete logarithms, and the hidden subgroup problem, Computing in Science ＆ Engineering, vol. 3, pp. 34-43, Mar-Apr 2001.
[31]J. Proos and C. Zalka, Shor's discrete logarithm quantum algorithm for elliptic curves, Quantum Information ＆ Computation, vol. 3, pp. 317-344, Jul 2003.
[32]K. Boström and T. Felbinger, Deterministic Secure Direct Communication Using Entanglement, Physical Review Letters, vol. 89, p. 187902, 2002.
[33]F.-G. Deng, G. Long, and X.-S. Liu, Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block, Physical Review A, vol. 68, p. 042317, 2003.
[34]Q.-Y. Cai, The “Ping-Pong Protocol Can Be Attacked without Eavesdropping, Physical Review Letters, vol. 91, p. 109801(1), 2003.
[35]F.-G. Deng, X.-H. Li, C.-Y. Li, P. Zhou, and H.-Y. Zhou, Eavesdropping on the `ping-pong' quantum communication protocol freely in a noise channel, Chinese Physics, vol. 16, p. 277, 2007.
[36]F.-G. Deng and G. Long, Secure direct communication with a quantum one-time pad, Physical Review A, vol. 69, p. 052319, 2004.
[37]G.-L. Long, F.-G. Deng, C. Wang, X.-H. Li, K. Wen, and W.-Y. Wang, Quantum secure direct communication and deterministic secure quantum communication, Frontiers of Physics in China, vol. 2, pp. 251-272, 2007.
[38]A. D. Zhu, Y. Xia, Q. B. Fan, and S. Zhang, Secure direct communication based on secret transmitting order of particles, Physical Review A, vol. 73, Feb 2006.
[39]C. Wang, F. G. Deng, and G. L. Long, Multi-step quantum secure direct communication using multi-particle Green-Horne-Zeilinger state (vol 253, pg 15, 2005), Optics Communications, vol. 262, pp. 134-134, Jun 1 2006.
[40]X.-H. Li, C.-Y. Li, F.-G. Deng, P. Zhou, Y.-J. Liang, and H.-Y. Zhou, Quantum secure direct communication with quantum encryption based on pure entangled states, Chinese Physics, vol. 16, p. 2149, 2007.
[41]C. Wang, F.-G. Deng, Y.-S. Li, X.-S. Liu, and G. L. Long, Quantum secure direct communication with high-dimension quantum superdense coding, Physical Review A, vol. 71, p. 044305, 2005.
[42]Y. Yang and Q. Wen, Threshold quantum secure direct communication without entanglement, Science in China Series G-Physics Mechanics ＆ Astronomy, vol. 51, pp. 176-183, Feb 2008.
[43]G. D. Kang and M. F. Fang, A Scheme to Share Information via Employing Discrete Algorithm to Quantum States, Communications in Theoretical Physics, vol. 55, pp. 239-243, Feb 2011.
[44]K. Shimizu and N. Imoto, Communication channels secured from eavesdropping via transmission of photonic Bell states, Physical Review A, vol. 60, pp. 157-166, Jul 1999.
[45]A. Beige, B. G. Englert, C. Kurtsiefer, and H. Weinfurter, Secure communication with a publicly known key, Acta Physica Polonica A, vol. 101, pp. 357-368, Mar 2002.
[46]A. Beige, B. G. Englert, C. Kurtsiefer, and H. Weinfurter, Secure communication with a publicly known key (vol 101, pg 357, 2002), Acta Physica Polonica A, vol. 101, pp. 901-901, Jun 2002.
[47]F. L. Yan and X. Q. Zhang, A scheme for secure direct communication using EPR pairs and teleportation, The European Physical Journal B - Condensed Matter and Complex Systems, vol. 41, pp. 75-78, 2004.
[48]T. Gao, F. L. Yan, and Z. X. Wang, Quantum secure conditional direct communication via EPR pairs, International Journal of Modern Physics C, vol. 16, pp. 1293-1301, Aug 2005.
[49]Z. X. Man, Z. J. Zhang, and Y. Li, Deterministic secure direct communication by using swapping quantum entanglement and local unitary operations, Chinese Physics Letters, vol. 22, pp. 18-21, Jan 2005.
[50]X.-H. Li, F.-G. Deng, C.-Y. Li, Y.-J. Liang, P. Zhou, and H.-Y. Zhou, Deterministic secure quantum communication without maximally entangled states, Journal of the Korean Physical Society, vol. 49, pp. 1354-1359, Oct 2006.
[51]H.-J. Cao and H.-S. Song, Quantum Secure Direct Communication with W State, Chinese Physics Letters, vol. 23, p. 290, 2006.
[52]L. Dong, X.-M. Xiu, Y.-J. Gao, and F. Chi, Quantum Secure Communication Using a Class of Three-Particle W State, Communications in Theoretical Physics, vol. 50, p. 359, 2008.
[53]L. Dong, X.-M. Xiu, Y.-J. Gao, and F. Chi, Quantum Secure Direct Communication Using W State, Communications in Theoretical Physics, vol. 49, p. 1495, 2008.
[54]X.-M. Xiu, H.-K. Dong, L. Dong, Y.-J. Gao, and F. Chi, Deterministic secure quantum communication using four-particle genuine entangled state and entanglement swapping, Optics Communications, vol. 282, pp. 2457-2459, 6/15/ 2009.
[55]M. Lucamarini and S. Mancini, Secure Deterministic Communication without Entanglement, Physical Review Letters, vol. 94, p. 140501, 04/14/ 2005.
[56]J. Wang, Q. Zhang, and C.-J. Tang, Quantum secure direct communication based on order rearrangement of single photons, Physics Letters A, vol. 358, pp. 256-258, 10/23/ 2006.
[57]C.-Y. Li, X.-H. Li, F.-G. Deng, and H.-Y. Zhou, Efficient quantum secure communication with a publicly known key, Chinese Physics B, vol. 17, pp. 2352-2355, Jul 2008.
[58]B. Nguyen, Quantum dialogue, Physics Letters A, vol. 328, pp. 6-10, 2004.
[59]Z. X. Man, Z. J. Zhang, and Y. Li, Quantum dialogue revisited, Chinese Physics Letters, vol. 22, pp. 22-24, Jan 2005.
[60]Z. X. Man, Y. J. Xia, and Z. J. Zhang, Secure deterministic bidirectional communication without entanglement, International Journal of Quantum Information, vol. 4, pp. 739-746, Aug 2006.
[61]Y. Yang and Q. Wen, Quasi-secure quantum dialogue using single photons, Science in China Series G: Physics, Mechanics and Astronomy, vol. 50, pp. 558-562, 2007.
[62]F. Gao, F. Guo, Q. Wen, and F. Zhu, Revisiting the security of quantum dialogue and bidirectional quantum secure direct communication, Science in China Series G: Physics, Mechanics and Astronomy, vol. 51, pp. 559-566, 2008.
[63]G.-F. Shi, X.-Q. Xi, X.-L. Tian, and R.-H. Yue, Bidirectional quantum secure communication based on a shared private Bell state, Optics Communications, vol. 282, pp. 2460-2463, 2009.
[64]G. Gao, Two quantum dialogue protocols without information leakage, Optics Communications, vol. 283, pp. 2288-2293, 2010.
[65]G.-F. Shi, X.-Q. Xi, M.-L. Hu, and R.-H. Yue, Quantum secure dialogue by using single photons, Optics Communications, vol. 283, pp. 1984-1986, 2010.
[66]X. H. Li, F. G. Deng, and H. Y. Zhou, Efficient quantum key distribution over a collective noise channel, Physical Review A, vol. 78, p. 022321, Aug 2008.
[67]P. Zanardi and M. Rasetti, Noiseless Quantum Codes, Physical Review Letters, vol. 79, p. 3306, 1997.
[68]E. Knill, R. Laflamme, and L. Viola, Theory of Quantum Error Correction for General Noise, Physical Review Letters, vol. 84, p. 2525, 2000.
[69]D. A. Lidar, D. Bacon, J. Kempe, and K. Birgitta Whaley, Protecting quantum information encoded in decoherence-free states against exchange errors, Physical Review A, vol. 61, p. 052307, 2000.
[70]J. Kempe, D. Bacon, D. Lidar, and K. Whaley, Theory of decoherence-free fault-tolerant universal quantum computation, Physical Review A, vol. 63, p. 042307, 2001.
[71]A. M. Basharov, V. N. Gorbachev, A. I. Trubilko, and E. S. Yakovleva, One-way gates based on EPR, GHZ and decoherence-free states of W class, Physics Letters A, vol. 373, pp. 3410-3412, 2009.
[72]P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. White, Experimental verification of decoherence-free subspaces, Science, vol. 290, pp. 498-501, Oct 2000.
[73]X. H. Li, B. K. Zhao, Y. B. Sheng, F. G. Deng, and H. Y. Zhou, Fault Tolerant Quantum Key Distribution Based on Quantum Dense Coding with Collective Noise, International Journal of Quantum Information, vol. 7, pp. 1479-1489, Dec 2009.
[74]X. M. Xiu, L. Dong, Y. J. Gao, and F. Chi, Quantum key distribution protocols with six-photon states against collective noise, Optics Communications, vol. 282, pp. 4171-4174, Oct 15 2009.
[75]C. Y. Li and Y. S. Li, Fault-tolerate quantum key distribution over a collective-noise channel, International Journal of Quantum Information, vol. 8, pp. 1101-1109, Oct 2010.
[76]Q. Y. Cai, Eavesdropping on the two-way quantum communication protocols with invisible photons, Physics Letters A, vol. 351, pp. 23-25, 2006.
[77]F. G. Deng, P. Zhou, X. H. Li, C. Y. Li, and H. Y. Zhou, Robustness of two-way quantum communication protocols against trojan horse attack, Quantum Physics, 2005.
[78]F. G. Deng, X. H. Li, H. Y. Zhou, and Z. J. Zhang, Improving the security of multiparty quantum secret sharing against Trojan horse attack, Physical Review A, vol. 72, p. 044302, 2005.
[79]F. G. Deng, X. H. Li, H. Y. Zhou, and Z. J. Zhang, Improving the security of multiparty quantum secret sharing against Trojan horse attack (vol 72, art no 044302, 2005), Physical Review A, vol. 73, p. 049901, 2006.
[80]X. H. Li, F. G. Deng, and H. Y. Zhou, Improving the security of secure direct communication based on the secret transmitting order of particles, Physical Review A, vol. 74, p. 054302, 2006.
[81]H. Ge and W. Y. Liu, A new quantum secure direct communication protocol using decoherence-free subspace, Chinese Physics Letters, vol. 24, pp. 2727-2729, Oct 2007.
[82]C.-W. Yang and T. Hwang, Improved QSDC Protocol over a Collective-Dephasing Noise Channel, International Journal of Theoretical Physics, vol. 51, pp. 3941-3950, Dec 2012.
[83]S. J. Qin, F. Gao, Q. Y. Wen, and F. C. Zhu, Robust Quantum Secure Direct Communication over Collective Rotating Channel, Communications in Theoretical Physics, vol. 53, pp. 645-647, Apr 2010.
[84]H. Yuan, J. Song, Q. He, L. F. Han, K. Hou, X. Y. Hu, et al., Robust quantum secure direct communication and deterministic secure quantum communication over collective dephasing noisy channel, Communications in Theoretical Physics, vol. 50, pp. 627-632, Sep 15 2008.
[85]L. Dong, X. M. Xiu, Y. J. Gao, and F. Chi, Deterministic secure quantum communication against collective-dephasing noise by using EPR pairs and auxiliary photons, Optics Communications, vol. 282, pp. 1688-1690, Apr 15 2009.
[86]B. Gu, S. X. Pei, B. Song, and K. Zhong, Deterministic secure quantum communication over a collective-noise channel, Science in China Series G-Physics Mechanics ＆ Astronomy, vol. 52, pp. 1913-1918, Dec 2009.
[87]A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete?, Physical Review, vol. 47, pp. 0777-0780, May 1935.
[88]D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, Bell's theorem without inequalities, American Journal of Physics, vol. 58, pp. 1131-1143, 1990.
[89]W. Dür, G. Vidal, and J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Physical Review A, vol. 62, p. 062314, 11/14/ 2000.
[90]T. Hwang, C.-C. Hwang, C.-W. Yang, and C.-M. Li, Revisiting Deng et al.’s Multiparty Quantum Secret Sharing Protocol, International Journal of Theoretical Physics, vol. 50, pp. 2790-2798, 2011.
[91]C.-W. Yang and T. Hwang, Quantum dialogue protocols immune to collective noise, Quantum Information Processing, vol. 12, pp. 2131-2142, 2012/12/01 2013.
[92]C.-W. Yang, T. Hwang, and Y.-P. Luo, Enhancement on Quantum Blind Signature Based on Two-State Vector Formalism, Quantum Information Processing, vol. 12, pp. 109-117, 2013.
[93]B. Gu, L. L. Mu, L. G. Ding, C. Y. Zhang, and C. Q. Li, Fault tolerant three-party quantum secret sharing against collective noise, Optics Communications, vol. 283, pp. 3099-3103, Aug 1 2010.
[94]C. H. Bennett, G. Brassard, C. Crepeau, and U. M. Maurer, Generalized privacy amplification, IEEE Transactions on Information Theory, vol. 41, pp. 1915-1923, Nov 1995.
[95]C. H. Bennett, G. Brassard, and J. M. Robert, Privacy Amplification by Public Discussion, SIAM Journal on Computing, vol. 17, pp. 210-229, Apr 1988.
[96]Y. Sun, Q. Y. Wen, and F. C. Zhu, Improving the multiparty quantum secret sharing over two collective-noise channels against insider attack, Optics Communications, vol. 283, pp. 181-183, Jan 1 2010.
[97]F. Gao, Q.-Y. Wen, and F.-C. Zhu, Comment on: Quantum exam [Phys. Lett. A 350 (2006) 174], Physics Letters A, vol. 360, pp. 748-750, 2007.
[98]F. Gao, F.-Z. Guo, Q.-Y. Wen, and F.-C. Zhu, Comment on “Quantum key distribution for d-level systems with generalized Bell states, Physical Review A, vol. 72, p. 066301, 2005.
[99]F. Gao, F.-Z. Guo, Q.-Y. Wen, and F.-C. Zhu, Comment on “Quantum secret sharing based on reusable Greenberger-Horne-Zeilinger states as secure carriers, Physical Review A, vol. 72, p. 036302, 2005.
[100]S.-J. Qin, F. Gao, F.-Z. Guo, and Q.-Y. Wen, Comment on Two-way protocols for quantum cryptography with a nonmaximally entangled qubit pair, Physical Review A, vol. 82, p. 036301, 2010.
[101]F. Gao, S. Lin, Q.-Y. Wen, and F.-C. Zhu, A Special Eavesdropping on One-Sender Versus N -Receiver QSDC Protocol, Chinese Physics Letters, vol. 25, p. 1561, 2008.
[102]F. Gao, S.-J. Qin, Q.-Y. Wen, and F.-C. Zhu, Cryptanalysis of multiparty controlled quantum secure direct communication using Greenberger-Horne-Zeilinger state, Optics Communications, vol. 283, pp. 192-195, 2010.
[103]S.-J. Qin, F. Gao, Q.-Y. Wen, L.-M. Meng, and F.-C. Zhu, Cryptanalysis and improvement of a secure quantum sealed-bid auction, Optics Communications, vol. 282, pp. 4014-4016, 2009.