本論文試圖利用量子力學之特性，推導出糾纏態之塌縮光子與糾纏態之間的Entanglement Correlation 之數學通式，並提出隨機導向之半量子金鑰分配協定，藉由近乎不誠實且具有完整量子能力的第三方協助使得兩個只具備部分量子能力之「古典」參與者安全地分享金鑰。本研究發現上述性質亦可延伸至隨機導向之半量子私密比較，藉由近乎不誠實且具有完整量子能力的第三方協助使得兩位僅擁有部分量子能力之古典參與者能安全地互相比較私密訊息。該近乎不誠實之第三方除了不能與參與者合作之外，會利用各種攻擊手段來取得參與者之秘密訊息。
此外，為了使得協定更為實際，本論文亦提出「嚴謹」之半量子金鑰分配，藉由近乎不誠實且具有完整量子能力的第三方使得兩位擁有更少的量子設備之參與者們仍能安全地分配金鑰。

This thesis derives the entanglement correlations of collapsed Bell state qubits and original Bell states (ECCB) by utilizing the law of fundamental quantum mechanics, and proposes a mediated randomization-based semi-quantum key distribution protocol for sharing a session key with two classical users who have limited quantum capabilities, by the assistance of the almost dishonest quantum third-party (QTP), who has powerful quantum capabilities. Besides, we also find that the ECCB can be applied to semi-quantum private comparison in randomization-based environment with an almost-dishonest QTP, where the classical users can compare their secret information securely by the assistance of the almost-dishonest QTP. The almost-dishonest QTP can perform any kind of attack, other than conspiring with any classical user.
Further, for achieving practical application, this thesis also proposes a mediated strict semi-quantum key distribution protocol, where classical users can only have fewer quantum capabilities than the original semi-quantum environment, and the trustworthiness of the QTP is also almost-dishonest.

[1] P. W. Shor, "Algorithms for quantum computation: discrete logarithms and factoring," in Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on, 1994, pp. 124-134.
[2] Artur K. Ekert, "Quantum cryptography based on Bell’s theorem," Physical Review Letters, vol. 67, pp. 661-663, 08/05/ 1991.
[3] C. H. Bennett and G. Brassard, "Quantum Cryptography: Public Key Distribution and Coin Tossing," presented at the Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, 1984.
[4] Charles H. Bennett, Gilles Brassard, and N. David Mermin, "Quantum cryptography without Bell’s theorem," Physical Review Letters, vol. 68, pp. 557-559, 02/03/ 1992.
[5] A. Cabello, "Quantum key distribution without alternative measurements," Physical Review A, vol. 61, pp. art. no.-052312, May 2000.
[6] G. L. Long and X. S. Liu, "Theoretically efficient high-capacity quantum-key-distribution scheme," Physical Review A, vol. 65, p. 032302, 02/01/ 2002.
[7] 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.
[8] D. G. Song, "Secure key distribution by swapping quantum entanglement," Physical Review A, vol. 69, Mar 2004.
[9] X. H. Li, F. G. Deng, and H. Y. Zhou, "Efficient quantum key distribution over a collective noise channel," Physical Review A, vol. 78, Aug 2008.
[10] Chang Ho Hong, Jin O. Heo, Gyong Luck Khym, Jongin Lim, Suc-Kyung Hong, and Hyung Jin Yang, "quantum channels are sufficient for Multi-user Quantum Key Distribution protocol between users," Optics Communications, vol. 283, pp. 2644-2646, 6/15/ 2010.
[11] Nan-Run Zhou, Li-Jun Wang, Jie Ding, and Li-Hua Gong, "Quantum deterministic key distribution protocols based on the authenticated entanglement channel," Physica Scripta, vol. 81, p. 045009, 2010.
[12] Nan-Run Zhou, Li-Jun Wang, Jie Ding, Li-Hua Gong, and Xiang-Wu Zuo, "Novel Quantum Deterministic Key Distribution Protocols with Entangled States," International Journal of Theoretical Physics, vol. 49, pp. 2035-2044, 2010/09/01 2010.
[13] N. Zhou, G. Zeng, and J. Xiong, "Quantum key agreement protocol," Electronics Letters, vol. 40, p. 1149, 2004.
[14] Song-Kong Chong and Tzonelih Hwang, "Quantum key agreement protocol based on BB84," Optics Communications, vol. 283, pp. 1192-1195, 2010.
[15] Song-Kong Chong, Chia-Wei Tsai, and Tzonelih Hwang, "Improvement on “Quantum Key Agreement Protocol with Maximally Entangled States”," International Journal of Theoretical Physics, vol. 50, pp. 1793-1802, 2011.
[16] Run-Hua Shi and Hong Zhong, "Multi-party quantum key agreement with bell states and bell measurements," Quantum Information Processing, vol. 12, pp. 921-932, 2013/02/01 2013.
[17] Bin Liu, Fei Gao, Wei Huang, and Qiao-yan Wen, "Multiparty quantum key agreement with single particles," Quantum Information Processing, vol. 12, pp. 1797-1805, 2013/04/01 2013.
[18] Zhiwei Sun, Cai Zhang, Banghai Wang, Qin Li, and Dongyang Long, "Improvements on “multiparty quantum key agreement with single particles”," Quantum Information Processing, vol. 12, pp. 3411-3420, 2013.
[19] Xun-Ru Yin, Wen-Ping Ma, and Wei-Yan Liu, "Three-Party Quantum Key Agreement with Two-Photon Entanglement," International Journal of Theoretical Physics, vol. 52, pp. 3915-3921, 2013.
[20] Wei Huang, Qiao-Yan Wen, Bin Liu, Fei Gao, and Ying Sun, "Quantum key agreement with EPR pairs and single-particle measurements," Quantum Information Processing, vol. 13, pp. 649-663, 2014/03/01 2014.
[21] Mark Hillery, Vladimír Bužek, and André Berthiaume, "Quantum secret sharing," Physical Review A, vol. 59, p. 1829, 1999.
[22] Yongmin Li, Kuanshou Zhang, and Kunchi Peng, "Multiparty secret sharing of quantum information based on entanglement swapping," Physics Letters A, vol. 324, pp. 420-424, 4/26/ 2004.
[23] Zhan-jun Zhang, Yong Li, and Zhong-xiao Man, "Multiparty quantum secret sharing," Physical Review A, vol. 71, p. 044301, 04/11/ 2005.
[24] Run-hua Shi and Hong Zhong, "Multiparty quantum secret sharing with the pure entangled two-photon states," Quantum Information Processing, vol. 11, pp. 161-169, 2012/02/01 2012.
[25] Shih-Min Hung, Sheng-Liang Hwang, Tzonelih Hwang, and Shih-Hung Kao, "Multiparty quantum private comparison with almost dishonest third parties for strangers," Quantum Information Processing, vol. 16, p. 36, 2017.
[26] Bin Liu, Di Xiao, Wei Huang, Heng-Yue Jia, and Ting-Ting Song, "Quantum private comparison employing single-photon interference," Quantum Information Processing, vol. 16, p. 180, 2017.
[27] Ji Zhao-Xu and Ye Tian-Yu, "Multi-party quantum private comparison based on the entanglement swapping of d-level cat states and d-level Bell states," Quantum Information Processing, vol. 16, p. 177, 2017.
[28] B. S. Shi, J. Li, J. M. Liu, X. F. Fan, and G. C. Guo, "Quantum key distribution and quantum authentication based on entangled state," Physics Letters A, vol. 281, pp. 83-87, Mar 19 2001.
[29] H. C. Shih, K. C. Lee, and T. Hwang, "New Efficient Three-Party Quantum Key Distribution Protocols," Ieee Journal of Selected Topics in Quantum Electronics, vol. 15, pp. 1602-1606, Nov-Dec 2009.
[30] Brian Julsgaard, Jacob Sherson, J Ignacio Cirac, Jaromír Fiurášek, and Eugene S Polzik, "Experimental demonstration of quantum memory for light," Nature, vol. 432, pp. 482-486, 2004.
[31] Xing-Can Yao, Tian-Xiong Wang, Ping Xu, He Lu, Ge-Sheng Pan, Xiao-Hui Bao, et al., "Observation of eight-photon entanglement," Nature Photonics, vol. 6, pp. 225-228, 2012.
[32] Xi-Lin Wang, Luo-Kan Chen, Wei Li, H-L Huang, Chang Liu, Chao Chen, et al., "Experimental ten-photon entanglement," Physical Review Letters, vol. 117, p. 210502, 2016.
[33] Michel Boyer, Dan Kenigsberg, and Tal Mor, "Quantum key distribution with classical Bob," in Quantum, Nano, and Micro Technologies, 2007. ICQNM'07. First International Conference on, 2007, pp. 10-10.
[34] Michel Boyer, Ran Gelles, Dan Kenigsberg, and Tal Mor, "Semiquantum key distribution," Physical Review A, vol. 79, p. 032341, 2009.
[35] Xiangfu Zou, Daowen Qiu, Lvzhou Li, Lihua Wu, and Lvjun Li, "Semiquantum-key distribution using less than four quantum states," Physical Review A, vol. 79, p. 052312, 2009.
[36] Wang Jian, Zhang Sheng, Zhang Quan, and Tang Chao-Jing, "Semiquantum key distribution using entangled states," Chinese Physics Letters, vol. 28, p. 100301, 2011.
[37] K. F. Yu, C. W. Yang, C. H. Liao, and T. Hwang, "Authenticated semi-quantum key distribution protocol using Bell states," Quantum Information Processing, vol. 13, pp. 1457-1465, Jun 2014.
[38] Michel Boyer, Dan Kenigsberg, and Tal Mor, "Quantum key distribution with classical Bob," in Quantum, Nano, and Micro Technologies, 2007. ICQNM'07. First International Conference on, 2007, pp. 10-10.
[39] Chuan-Ming Li, Kun-Fei Yu, Shih-Hung Kao, and Tzonelih Hwang, "Authenticated semi-quantum key distributions without classical channel," Quantum Information Processing, vol. 15, pp. 2881-2893, 2016.
[40] Chen Xie, Lvzhou Li, and Daowen Qiu, "A novel semi-quantum secret sharing scheme of specific bits," International Journal of Theoretical Physics, vol. 54, pp. 3819-3824, 2015.
[41] Michel Boyer and Tal Mor, "Comment on “Semiquantum-key distribution using less than four quantum states”," Physical Review A, vol. 83, p. 046301, 2011.
[42] Xiangfu Zou and Daowen Qiu, "Reply to “Comment on ‘Semiquantum-key distribution using less than four quantum states’”," Physical Review A, vol. 83, p. 046302, 2011.
[43] Lvzhou Li, Daowen Qiu, and Paulo Mateus, "Quantum secret sharing with classical Bobs," Journal of Physics A: Mathematical and Theoretical, vol. 46, p. 045304, 2013.
[44] Xiangfu Zou, Daowen Qiu, Shengyu Zhang, and Paulo Mateus, "Semiquantum key distribution without invoking the classical party’s measurement capability," Quantum Information Processing, vol. 14, pp. 2981-2996, 2015.
[45] Yi-You Nie, Yuan Hua Li, Zi-Sheng Wang, et al., "Semi-quantum information splitting using GHZ-type states," Quantum Information Processing, vol. 12, pp.437-448, 2013.
[46] Zhi-Wei Sun, Rui-Gang Du, Dong-Yang Long, et al., "Semi-quantum key distribution with limited classical Bob," International Journal of Quantum Information, vol. 11, No. 11, 2013.
[47] B Clifford Neuman and Theodore Ts'o, "Kerberos: An authentication service for computer networks," IEEE Communications magazine, vol. 32, pp. 33-38, 1994.
[48] Nigel Jefferies, Chris Mitchell, and Michael Walker, "A proposed architecture for trusted third party services," in Cryptography: Policy and Algorithms, 1996, pp. 98-104.
[49] Xiu-Bo Chen, Gang Xu, Xin-Xin Niu, Qiao-Yan Wen, and Yi-Xian Yang, "An efficient protocol for the private comparison of equal information based on the triplet entangled state and single-particle measurement," Optics communications, vol. 283, pp. 1561-1565, 2010.
[50] Sheng-Liang Huang, Tzonelih Hwang, and Prosanta Gope, "Multi-party quantum private comparison protocol with an almost-dishonest third party using ghz states," International Journal of Theoretical Physics, vol. 55, pp. 2969-2976, 2016.
[51] Tzonelih Hwang, Tzu-Han Lin, and Shih-Hung Kao, "Quantum entanglement establishment between two strangers," Quantum Information Processing, vol. 15, pp. 385-403, 2016.
[52] Wen-Han Chou, Tzonelih Hwang, Jun Gu, “Semi-quantum private comparison protocol under an almost-dishonest third party,” arxiv:1607.07961
[53] Double-CNOT attack on "Semi-quantum key distribution with limited classical Bob".
[54] Walter O Krawec, "Mediated semiquantum key distribution," Physical Review A, vol. 91, p. 032323, 2015.
[55] Zhi-Rou Liu, and Tzonelih Hwang, “Mediated semi-quantum key distribution without invoking quantum measurement “, Annalen Der Physik, vol.530, issue 4, 2018.
[56] Jian-Wei Pan, Dik Bouwmeester, Harald Weinfurter, and Anton Zeilinger, "Experimental entanglement swapping: Entangling photons that never interacted," Physical Review Letters, vol. 80, p. 3891, 1998.
[57] M Zukowski, A Zeilinger, MA Horne, and AK Ekert, "" Event-ready-detectors" Bell experiment via entanglement swapping," Physical Review Letters, vol. 71, pp. 4287-4290, 1993.
[58] Warren P Grice, "Arbitrarily complete Bell-state measurement using only linear optical elements," Physical Review A, vol. 84, p. 042331, 2011.
[59] Charles H Bennett, Gilles Brassard, and Jean-Marc Robert, "Privacy amplification by public discussion," SIAM journal on Computing, vol. 17, pp. 210-229, 1988.
[60] Charles H Bennett, Gilles Brassard, Claude Crépeau, and Ueli M Maurer, "Generalized privacy amplification," IEEE Transactions on Information Theory, vol. 41, pp. 1915-1923, 1995.
[61] F. G. Deng, Zhou, P., Li, X. H., Li, C. Y., Zhou, H. Y., "Robustness of two-way quantum communication protocols against trojan horse attack," Quantum Physics, 2005.
[62] Q. Y. Cai, "Eavesdropping on the two-way quantum communication protocols with invisible photons," Physics Letters A, vol. 351, pp. 23-25, Feb 20 2006.
[63] 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, Jan 2013.
[64] Yong-gang Tan, Hua Lu, and Qing-yu Cai, "Comment on “Quantum key distribution with classical Bob”," Physical review letters, vol. 102, p. 098901, 2009.
[65] 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, Oct 2005.
[66] X. H. Li, F. G. Deng, and H. Y. Zhou, "Improving the security of secure direct vcommunication based on the secret transmitting order of particles," Physical Review A, vol. 74, Nov 2006.
[67] Michel Boyer, Dan Kenigsberg, and Tal Mor, "Boyer, kenigsberg, and mor reply," Physical Review Letters, vol. 102, p. 098902, 2009.
[68] Fu-Guo Deng, Gui Lu Long, and Xiao-Shu Liu, "Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block," Physical Review A, vol. 68, p. 042317, 2003.
[69] Fei Gao, FenZhuo Guo, QiaoYan Wen, and FuChen Zhu, "Comparing the efficiencies of different detect strategies in the ping-pong protocol," Science in China Series G: Physics Mechanics and Astronomy, vol. 51, pp. 1853-1860, 2008.
[70] Yi-Ping Luo, Ching-Ying Lin, and Tzonelih Hwang, "Efficient quantum dialogue using single photons," Quantum information processing, vol. 13, pp. 2451-2461, 2014.
[71] Jason Lin and Tzonelih Hwang, "An enhancement on Shi et al.'s multiparty quantum secret sharing protocol," Optics Communications, vol. 284, pp. 1468-1471, 2011.
[72] Jason Lin and Tzonelih Hwang, "New circular quantum secret sharing for remote agents," Quantum information processing, pp. 1-13, 2013.
[73] Fei-Gao, Su-Juan Qin, Fen-Zhuo Guo, Qiao-Yan Wen, “Dense-coding attack on three-party quantum key distribution protocols,” IEEE Journal of Quantum Electronics, vol. 47, issue: 5, 2011.