由於無線通訊與行動裝置設備的普及化，讓使用者能透過無線網路，無所不在的漫遊來存取相關服務，因此，安全之漫遊通訊的身份辨識機制，更顯得重要。然而，近年有些學者提出具有使用者匿名性之行動漫遊服務的安全身份辨識機制，而這些機制都需要高運算量之公開金鑰密碼系統，並不適合用於資源受限的無線通訊環境。因此，基於混鈍對應(chaotic maps) 理論，Guo 等人在2013 年提出一個有效率且無須運用高次模指數計算及橢圓曲線之乘法計算(scalar multiplication on elliptic curves) 並具備使用者匿名性的相互認證機制應用於行動通訊服務。但是，在我們的研究下發現Guo 等人所提出之方法仍然需要高次模指數運算，且使用者無法任意更改智慧卡之密碼。因此，基於模平方根運算(modular square root，簡稱MSR)，我們提出一個快速且更適用於行動漫遊服務的安全身份辨識機制。此外，即使儲存於智慧卡的內容資訊遭竄改或竊取時，我們的機制仍然能確保使用者與遠端系統的安全。與其他相關研究的方法相比較，我們所提出的機制更適合應用於行動裝置與智慧卡之使用。

Nowadays, people have numerous opportunities to obtain services or resources from application servers by using their mobile devices through the wireless network.Hence, many mutual authentication protocols with user anonymity are proposed
in literature for providing secure roaming service through wireless communications.However, most of them are based on smart cards and have to establish public key cryptosystems in advance. To solve this problem, in 2013, based on chaotic maps,Guo et al. firstly proposed an efficient mutual authentication protocol with user anonymity using smart card for wireless communications. They claimed their protocol could refrain from consuming modular exponential operation and scalar multiplication on elliptic curves. Unfortunately, this study will demonstrate that their scheme still requires high modular exponential operations for the security issues, and does not allow changing password freely for the mobile user. Based on the modular square root, we propose an efficient remote user authentication protocol with smart cards for wireless communications. The proposed scheme is secure even if the secret
information stored in the smart card is compromised. Compared with other related protocols, our protocol is more suitable for mobile devices and smart-card users.

Chapter 1 Introduction.............................................................................. 1
Chapter 2 Background ............................................................................... 4
2.1 Security requirements for wireless communications ............................ 4
Chapter 3 Related Works.......................................................................... 7
3.1 Preliminary ......................................................................................... 7
3.2 Review of Guo et al.’s protocol ........................................................... 8
3.3 Weakness of Guo et al.’s protocol........................................................ 12
3.4 The improved scheme and security analysis........................................ 13
3.4.1 The improved scheme........................................................... 14
3.4.2 Security analysis................................................................... 17
3.5 Summaries........................................................................................... 19
Chapter 4 The proposed scheme.............................................................. 21
Chapter 5 Discussion.................................................................................. 26
5.1 Security analysis ................................................................................. 26
5.2 Performance comparison ..................................................................... 32
Chapter 6 Conclusions ............................................................................... 35

[1] Levente Buttyan, Constant Gbaguidi, Sebastian Staamann, and Uwe Wilhelm. Extensions to an authentication technique proposed for the global mobility network. Communications, IEEE Transactions on, 48(3):373–376, 2000.
[2] Cheng Guo, Chin Chen Chang, and Chin Yu Sun. Chaotic maps-based mutual authentication and key agreement using smart cards for wireless communications. Journal of information Hiding and Multimedia Signal Processing, 4(2): 99–109, 2013.
[3] Carmit Hazay and Yehuda Lindell. Constructions of truly practical secure protocols using standardsmartcards. In Proceedings of the 15th ACM conference on Computer and communications security, pages 491–500. ACM, 2008.
[4] Daojing He, Maode Ma, Yan Zhang, Chun Chen, and Jiajun Bu. A strong user authentication scheme with smart cards for wireless communications. Computer Communications, 34(3):367–374, 2011.
[5] L Kocarev, J Makraduli, and P Amato. Public-key encryption based on chebyshev polynomials. Circuits, Systems and Signal Processing, 24(5):497–517, 2005.
[6] Ramanujachary Kumanduri, Cristina Romero, Cristina Romero, and R Kumanduri. Number theory with computer applications. Prentice Hall New Jersey, 1998.
[7] Cheng Chi Lee, Min Shiang Hwang, and I En Liao. Security enhancement on a new authentication scheme with anonymity for wireless environments. Industrial Electronics, IEEE Transactions on, 53(5):1683–1687, 2006.
[8] Ji Seon Lee, Jik Hyun Chang, and Dong Hoon Lee. Security flaw of authentication scheme with anonymity for wireless communications. Communications Letters, IEEE, 13(5):292–293, May 2009.
[9] Chun Ta Li and Cheng Chi Lee. A novel user authentication and privacy preserving scheme with smart cards for wireless communications. Mathematical and Computer Modelling, 55(1):35–44, 2012.
[10] Chu Hsing Lin and Chia Yin Lee. Cryptanalysis of a new authentication scheme with anonymity for wireless environments. In Proceedings of the Second International Conference on Advances in Mobile Multimedia, Bali, Indonesia, pages 399–402, 2004.
[11] Hyeran Mun, Kyusuk Han, Yan Sun Lee, Chan Yeob Yeun, and Hyo Hyun Choi. Enhanced secure anonymous authentication scheme for roaming service in global mobility networks. Mathematical and Computer Modelling, 55(1):214–222, 2012.
[12] M. O. Rabin. Digitalized signatures and public-key functions as intractable as factorization. Technical report, Cambridge, MA, USA, 1979.
[13] Shigefusa Suzuki and Kazuhiko Nakada. An authentication technique based on distributed security management for the global mobility network. Selected Areas in Communications, IEEE Journal on, 15(8):1608–1617, 1997.
[14] Zhi-Jia Tzeng and Wen Guey Tzeng. Authentication of mobile users in third generation mobile systems. Wireless Personal Communications, 16(1):35–50, 2001.
[15] Xingyuan Wang and Jianfeng Zhao. An improved key agreement protocol based on chaos. Communications in Nonlinear Science and Numerical Simulation, 15(12):4052–4057, 2010.
[16] HC Williams. A modification of the rsa public-key encryption procedure (corresp.). Information Theory, IEEE Transactions on, 26(6):726–729, 1980.
[17] Chia Chun Wu, Wei Bin Lee, and Woei Jiunn Tsaur. A secure authentication scheme with anonymity for wireless communications. Communications Letters, IEEE, 12(10):722–723, 2008.
[18] Jing Xu, Wen Tao Zhu, and Deng Guo Feng. An efficient mutual authentication and key agreement protocol preserving user anonymity in mobile networks. Computer Communications, 34(3):319–325, 2011.
[19] Guomin Yang, Duncan S Wong, and Xiaotie Deng. Anonymous and authenticated key exchange for roaming networks. Wireless Communications, IEEE Transactions on, 6(9):3461–3472, 2007.
[20] Eun Jun Yoon and Il Soo Jeon. An efficient and secure diffie–hellman key agreement protocol based on chebyshev chaotic map. Communications in Nonlinear Science and Numerical Simulation, 16(6):2383–2389, 2011.
[21] Eun Jun Yoon, Kee Young Yoo, and Keum Sook Ha. A user friendly authentication scheme with anonymity for wireless communications. Computers & Electrical Engineering, 37(3):356–364, 2011.
[22] Linhua Zhang. Cryptanalysis of the public key encryption based on multiple chaotic systems. Chaos, Solitons & Fractals, 37(3):669–674, 2008.
[23] Tao Zhou and Jing Xu. Provable secure authentication protocol with anonymity for roaming service in global mobility networks. Computer Networks, 55(1):205–213, 2011.
[24] Jianming Zhu and Jianfeng Ma. A new authentication scheme with anonymity for wireless environments. Consumer Electronics, IEEE Transactions on, 50(1):231–235, 2004.