臺灣博碩士論文加值系統

在簽章方面，使用數位簽署的方式是將來的趨勢。又使用橢圓曲線密碼系統做加解密有位元數少的優點。因此，許多學者有鑑於此，紛紛的利用這些優點，提出他們的橢圓曲線密碼系統數位簽署方案。其中，Chung等人提出橢圓曲線密碼系統數位簽署方案，他們的論文確實有降低RSA數位簽署系統複雜度的優點。Chung等人數位簽署方案的安全性是建立在橢圓曲線離散對數問題上。後來，Yang等人提出一篇論文，文中指出Chung等人數位簽署方案的安全性有問題。他們使用一個Diophantine方程式，用來迴避橢圓曲線離散對數問題，因而能破解簽署者的密碼，致使Chung等人數位簽署案系統變得的不安全。因此我們提出一個新的植基於橢圓曲線密碼系統數位簽署方案。
本橢圓曲線數位簽署方案乃植基於橢圓曲線離散對數問題的防護功能而設計。我們闡述了我們數位簽署案的簽署方法與步驟。接著，我們也說明了驗證者的驗證方法與步驟，並且做了安全性的分析，證明我們提出數位簽署方案是安全的。在複雜度效能方面，我們比較了Nyang等人的數位簽署方案與Chung等人簽署案系統兩組方案的複雜度，證明我們提出數位簽署方案，相較於兩組為低。因此，我們的橢圓曲線密碼系統數位簽署方案是一個具有安全性高且複雜度低，實用可行的數位簽署方案。

It is a trend with digital signature form in the future and the elliptic curve cryptosystem have the advantages to encipher and decipher with fewer bits. Hence, many scholars proposed their digital signature scheme by using the advantages. In the schemes, Chung et al. proposed an ID-based digital signature scheme on elliptic curve cryptosystem. Their scheme really reduces the complexity of RSA digital signature. After, Yang et al. proposed an article of weakness of Chung et al.’s scheme. Yang et al. use the linear Diophantine equation successfully to avoid touching the elliptic curve discrete logarithm problem (ECDLP) in Chung et al.’s scheme. The signer’s secret code is obtained by an attacker. As the result, the Chung’s scheme becomes insecure.
Our digital signature scheme is protected for designing based on the elliptic curve discrete logarithm problem (ECDLP). We described the method and process of signature and verification in our scheme. Also, we make the analysis of the security and complexity in our scheme. We prove our scheme that is safe and the performance complexity is lower to compare Nyang et al.’s scheme and Chung et al.’s scheme. So, we propose a practical feasible elliptic curve digital signature system in condition of high security and low complexity.

中文摘要 i
英文摘要 ii
誌 謝 iii
目 錄 iv
圖 目 錄 vi
表 目 錄 vii
第一章　　緒論 1
1.1 研究動機 1
1.2 研究背景 4
1.3 研究目的 4
1.4 論文架構 5
第二章　　文獻探討 6
2.1 數位簽署 6
2.2 傳統的數位簽署 9
2.2.1 RSA數位簽署 9
2.2.2 ElGamal 數位簽署 11
2.2.3 總結 13
2.3 橢圓曲線與赫序函數的研究 14
2.3.1橢圓曲線之簡介 15
2.3.1.1橢圓曲線之定義 15
2.3.1.2橢圓曲線之圖形 15
2.3.1.3橢圓曲線之加法公式 17
2.3.1.4橢圓曲線之加法實例 17
2.3.2橢圓曲線離散對數問題 17
2.3.3 赫序函數 18
2.4 Chung等人數位簽署案 19
2.4.1 簽署者的簽署過程 19
2.4.2 驗證者的驗證過程 20
2.4.3 安全性分析 21
2.4.4 複雜度分析 22
2.5 Yang等人提出Chung等人簽署案的弱點論文 24
2.5.1 Diophantine方程式的介紹 24
2.5.2破解Chung等人數位簽署案的實例 24
2.5.3結論 26
第三章　　我們的橢圓曲線密碼系統數位簽署方案 27
3.1 簽署者α簽署過程 27
3.2 驗證者β驗證過程 29
3.3實例說明 30
3.3.1 簽署者α的簽署過程 30
3.3.2 驗證者β的驗證過程 32
第四章 我們的數位簽署複雜度與安全性分析 34
4.1 複雜度分析 34
4.2 安全性分析 35
第五章 結 論 43
參考文獻 45

[1] 韓亮、賴溪松、張真誠，近代密碼學及其應用，松崗電腦圖書，1995 年。
[2] 粘添壽、吳順裕，資訊與網路安全技術，旗標圖書，2006 年。
[3] 華羅庚，數論導引，凡異出版社，五版，1993年。
[4] 張煥國等，橢圓曲線密碼學導論，電子工業出版社，2005 年 8 月。
[5] 胥鳴偉、印林生，數論，高等教育出版社，2009 年 6 月。
[6] 吳鋌、董軍武、王明強，橢圓曲線及其在密碼學中的應用--導引，科學出版社，2007 年 12 月。
[7] 顏松遠，橢圓曲線，大連理工大學出版社，2011 年 5 月。
[8] 楊松諺、上官飛鳳， “Java Security全方位解決方案 ”，碁峰資訊圖書，2004 年 4 月。
[9] L. E. Dickson, History of the Theory of Numbers Volume II: Diophantine Analysis, Bronx, Chelsea Publishing, New York, vol. 2, 1992.
[10]J. Gathen and J. Gerhard, Modern Computer Algebra, 2nd ed., Cambridge University Press, 2003.
[11]A. W. Knapp, Elliptic Curve, Princeton University Prece, 1992.
[12]H. Liang, Hanging on the Small Stream Loose, Modern Cryptography and Application, Loose Hillock Computer Books, 1995.
[13]A. J. Menezes, Elliptic Curve Public Key Cryptosystems, Kluwer Academic Publishers, Boston, 1997.
[14] L. J. Mordell, Diophantine Equations, Academic Press, New York, 1969.
[15]K. H. Rosen, Elementary Number Theory and Its Applications, 2nd edition, Addison Wesley, 1988.
[16]J. H. Silverman, the Arithmetic of Elliptic Curves, New York, Springer, 1986.
[17]M. R. Schroeder, Number Theory in Science and Communication, 2nd edition, Spring-verlag. Berlin, 1986.
[18]S. Y. Wu, Information and Network Safe Practice, Flag Marks Books, 2006.

[19]Y. F. Chung, K. H. Huang, F. Lai, and T. S. Chen, “ID-based Digital Signature Scheme on Elliptic Curve Cryptosystem”, Computer Standards and Interfaces, vol. 29, pp. 601-604, 2007.
[20]W. Diffie and M. E. Hellman, “New Direction in Cryptography”, IEEE Transactions on information Theory, vol. IT-22, no. 6, pp. 644-654, Nov. 1976.
[21]T. ElGamal, “A Public Key Cryptosystem and a Signature Scheme based on Discrete Logarithms”, IEEE Transactions on Information Theory, vol. IT-31, no. 4. July 1985.
[22]N. Koblitz, “Elliptic Curve Cryptosystems”, Mathematics of Computation, vol. 48, pp. 203-209, 1987.
[23]N. Koblitz, A. J. Menezes, and S. A. Vanstone, “The State of Elliptic Curve Cryptography”, Designs, Codes and Cryptography, vol. 19, pp. 173-193, 2000.
[24] A. J. Menezes, T. Okamoto, and S. A.Vanstone, “Reducing Elliptic Curve Logarithms to Logarithms in a Finite Field”, IEEE Transactions on Information Theory, vol. 39, 5, pp. 1639-1646, 1993.
[25] “Proposed Federal Information Processing Standard for Digital Signature Standard (DSS)”, Federal Register, vol. 56, 169, pp. 42980-42982, Aug. 30, 1991.
[26] R. L. Rivest, A. Shamir, and L. Adleman “A Method for Obtaining Digital Signatures and Public Key Cryptosystems”, Communications of ACM, vol. 21, 2, pp. 120-126, 1978.
[27]C. E. Shannon, “Communication Theory of Secret Systems”, Bell System Technical Journal, vol. 28, no.4, pp. 656-715, 1949.
[28] “The Digital Signatures Standard Proposed by NIST”, Communication of ACM, vol. 35, 7, pp. 36-40, July 1992.
[29] S. A. Vanstone, “Elliptic Curve Cryptosystem - the Answer to Strong, Fast Public-key Cryptography for Securing Constrained Environments”, Information Security Tech. Rep., vol. 12, 2, pp. 78-87, 1997.
[30] J. H. Wen, M. C. Wu, and T. S. Chen, “A Novel Elliptic Curve Dynamic Access Control System”, IEICE Transactions on Communications, vol. E90-B, 8, pp. 1979-1987, 2007.
[31]S. M. Yen, “Improved Digital Signatures Algorithm”, IEEE Transactions on Computers, vol. 44, 5, pp. 729-730, May 1995.

[32]張惟淙、楊中皇，“結合 Java Card 的 ECDSA 數位簽章軟體設計與實現”，第十一屆資訊管理暨實務研討會，實踐大學資訊管理學系，2005年12月10日，頁2839-2849。
[33]D. H. Nyang and J. S. Song, “Knowledge-proof based Versatile Smart Card Verification Protocol”, ACM SIGCOMM Computer Communication Review, July 2000, vol. 30, no, 3, pp.39-44.
[34]C. P. Schnorr, “Efficient Identification and Signatures for Smart Cards”, Lecture Notes in Computer Science 435, advances in cryptology: crypto’89, Springer Verlag, Berlin, 1990, pp.339-351.
[35]J. H. Yang and C. C. Chang, “Cryptanalysis of ID-based Digital Signature Scheme on Elliptic Curve Cryptosystem”, ISDA, 2008 Eighth International Conference on Intelligent Systems Design and Applications, Nov 2008, vol. 3, pp.3-5.