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 基於離散對數問題的數位簽章，在驗證處理時需要使用到雙指數的運算。關於指數的快速運算，在質數有限域上，由於反元素運算成本過高，不能使用有號位數表示式，因此無法降低漢明密度。利用二元有限域上的最佳正規基底進行運算，能有效降低平方與反元素的計算成本，所以可以將指數編碼成有號位數表示式，進而降低漢明密度。本論文提出雙指數的由左至右重編碼法以進行雙指數運算，增進簽章驗證處理的效能。
 The verification process of the digital signature based on the discrete logarithm problems requires operations of exponent pairs. About the fast exponentiation operations over the prime finite field, because the cost of the inversion operation is high, the signed digit representation cannot be used, and then it is unable to reduce the Hamming density. The optimal normal basis over the binary finite field can reduce the cost of the square and the inversion operations efficiently, so we can recode the exponent into the signed digit representation to reduce the Hamming density. In this thesis, we propose a left-to-right exponent pair recoding method that can be used for operations of exponent pairs to increase the performance of the signature verification process.
 致謝 i摘要 iiAbstract iii目錄 iv表目錄 v圖目錄 vi1 緒論 11.1 研究背景與動機 11.2 研究目的 21.3 論文架構 22 相關研究 42.1 有號位數表示式 42.1.1 非毗鄰型式 42.1.2 相互交替型式 52.2 同步指數運算 72.3 正規基底 93 研究方法 134 效率分析 185 結論與未來的研究方向 23參考文獻 24
 [1]Bailey, D.V. and Paar, C., “Efficient arithmetic in finite field extensions with application in elliptic curve cryptography,” Journal of Cryptology, vol. 14, no. 3, pp. 153-176, 2001.[2]Bailey, D.V. and Paar, C., “Optimal extension fields for fast arithmetic in public-key algorithms,” Advances in Cryptology – CRYPTO’98, LNCS vol. 1462, pp. 472-485, 1998.[3]Diffie, W. and Hellman, M., “New directions in cryptography,” IEEE Transactions on Information Theory, vol. 22, no. 6, pp. 644-654, 1976.[4]ElGamal, T., “A public key cryptosystem and a signature scheme based on discrete logarithms,” IEEE Transactions on Information Theory, vol. 31, no. 4, pp. 469-472, 1985.[5]Gao, S., von zur Gathen, J. and Panario, D., “Gauss periods and fast exponentiation in finite fields,” Proceedings of the Second Latin American Symposium on Theoretical Informatics, LNCS vol. 911, pp. 311-322, 1995.[6]Gao, S., von zur Gathen, J. and Panario, D., “Gauss periods: orders and cryptographical applications,” Mathematics of Computation, vol. 67, no. 221, pp. 343-352, 1998.[7]Gao, S. and Vanstone, S., “On orders of optimal normal basis generators,” Mathematics of Computation, vol. 64, no. 221, pp. 1227-1233, 1995.[8]Gollmann, D., Han, Y. and Mitchell, C.J., “Redundant integer representations and fast exponentiation,” Designs, Codes and Cryptography, vol. 7, no. 1-2, pp. 135-151, 1996.[9]Gordon, D.M., “A survey of fast exponentiation methods,” Journal of Algorithms, vol.27, no. 1, pp. 129-146, 1998.[10]IEEE P1363, “Standard specifications for public key cryptography”, available at http://grouper.ieee.org/groups/1363/, 2000.[11]Joye, M., and Yen, S.M., “Optimal left-to-right binary signed digit recoding,” IEEE Transactions on Computers, vol. 49, no. 7, pp. 740-748, 2000.[12]Kog, C.K., “Analysis of sliding window techniques for exponentiation,” Computers and Mathematics with Applications, vol. 30, no. 10, pp. 17-24, 1995.[13]Kwon, S., “Signed digit representation with NAF and balanced ternary form and efficient exponentiation in GF(qn) using a Gaussian normal basis of type II,” Workshop on Information Security Applications 2004, LNCS vol. 3325, pp. 332-344, 2004.[14]Moller, B., “Algorithms for multi-exponentiation,” Selected Areas in Cryptography – SAC 2001, LNCS vol. 2259, pp. 165-180, 2001.[15]Nealon, G.J., “ElGamal-type signature schemes in modular arithmetic and Galois fields,” Rochester Institute of Technology Department of Computer Science, available at http://www.cs.rit.edu/~gjn3855/msprj/final_report.pdf, 2005.[16]Okeya, K., Schmidt-Samoa, K., Spahn, C. and Takagi, T., “Signed binary representations revisited,” Advances in Cryptology – CRYPTO’04, LNCS vol. 3152, pp. 123-139, 2004.[17]Proos, J., “Joint sparse forms and generating zero columns when combing,” Technical Report CORR 2003-23, Center for Applied Cryptographic Research, University of Waterloo, 2003.[18]Reitwiesner, G.W., “Binary arithmetic,” Advances in Computers, vol. 1, pp. 231-308, 1960.[19]Rivest, R.L., Shamir, A. and Adleman, L., “A method for obtaining digital signatures and public key cryptosystems,” Communications of the ACM, vol. 21, no. 2, pp. 120-126, 1978.[20]Solinas, J.A., “Low-weight binary representations for pairs of integers,” Technical Report CORR 2001-41, University of Waterloo, manuscript, available at http://www.cacr.math.uwaterloo.ca/techreports/2001/corr2001-41.ps, 2001.[21]王志文，「橢圓曲線多點運算之快速演算法」，東吳大學資訊科學研究所，碩士論文，2003。[22]黃昱軫，「同步橢圓曲線點乘法之純量表示式」，東吳大學資訊科學研究所，碩士論文，2006。
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 1 橢圓曲線多點運算之快速演算法 2 同步橢圓曲線點乘法之純量表示式

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