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論文名稱 Title |
另一種左移最大公因數演算法的分析 Analysis of Another Left Shift Binary GCD Algorithm |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
40 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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口試日期 Date of Exam |
2009-06-17 |
繳交日期 Date of Submission |
2009-07-14 |
關鍵字 Keywords |
自我測試、反元素、最大公因數 Self-test, Modular inverse, GCD |
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統計 Statistics |
本論文已被瀏覽 5682 次,被下載 1156 次 The thesis/dissertation has been browsed 5682 times, has been downloaded 1156 times. |
中文摘要 |
一般來說, 計算反元素在資訊安全領域中是非常重要的, 許多加解密及簽章演算法都會利用到反元素的計算. 在 2007年, Liu, Horng, and Liu 提出一種可以計算反元素像計算最大公因數一樣簡單的演算法. 這篇論文分析了另外一種也可以做到計算反元素像計算最大公因數一樣簡單的演算法, 而且會比1996年Shallit and Sorenson 分析過的LSBGCD 需要更少的bit 運算量. 最後, 這篇論文證明了最大公因數的演算法也有自我測試/修復(self-testing/correcting)的性質. |
Abstract |
In general, to compute the modular inverse is very important in information security, many encrypt/decrypt and signature algorithms always need to use it. In 2007, Liu, Horng, and Liu proposed a variation on Euclidean algorithm, which can calculate the modular inverses as simple as calculate GCDs. This paper analyzes another type of left-shift binary GCD algorithm, which is suitable for the variation and that needs the fewer bit-operations than LSBGCD, which is analyzed by Shallit, and Sorenson. |
目次 Table of Contents |
Chapter 1 Introduction 1 Chapter 2 Review of Euclidean algorithm and its variants 3 Chapter 3 Another LSBGCD algorithm and its variation 7 3.1 Another LSBGCD algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Correctness proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Variation of another LSBGCD algorithm . . . . . . . . . . . . . . . . . . . . 9 3.4 Correctness of variation of left-shift like GCD algorithm . . . . . . . . . . . 10 3.4.1 Proof of the variation of the standard left shift GCD algorithm . . . . 10 3.4.2 Proof of the variation of the LSBGCD algorithm . . . . . . . . . . . 11 3.4.3 Proof of the variation of our algorithm . . . . . . . . . . . . . . . . . 14 Chapter 4 Analysis of Our Algorithm 16 4.1 The first way to estimation the probability of the output of our algorithm is least remainder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 The second way to estimation the probability of the output of our algorithm is least remainder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2.1 The probability of our algorithm after one iteration has the least remainder in each of four cases . . . . . . . . . . . . . . . . . . . . . . 18 4.2.2 Probability of (A;B) in the one of four cases . . . . . . . . . . . . . 21 4.3 Experiment result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Chapter 5 Self-Testing/Correcting of GCD Algorithm 26 Chapter 6 Conclusion 28 |
參考文獻 References |
[1] D. E. Knuth, The art of computer programming, vol. 2, 3rd ed. 1997. [2] J. Sorenson, “Two fast gcd algorithms,” Journal of algorithms, vol. 16, 1994. [3] J. Shallit and J. Sorenson, “Analysis of a left-shift binary gcd algorithm,” Algorithmic Number Theory, vol. 877, 1994. [4] H.-Y. L. Chao-Liang Liu, Gwoboa Horng, “Computing the modular inverses is as simple as computing the gcds,” Finite Fields and Their Applications, vol. 14. [5] R. J. Lipton, “New directions in testing,” DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 2, 1991. [6] M. Blum, M. Luby, and R. Rubinfeld, “Self-testing/correcting with applications to numerical problems,” Proceedings of the twenty-second annual ACM symposium on Theory of computing, 1990. [7] P. Gemmell, R. Lipton, R. Rubinfeld, M. Sudan, and A. Wigderson, “Selftesting/ correcting for polynomials and for approximate functions,” 23th ACM STOC Conference Proceedings, 1991. [8] A. Goupil and J. Palicot, “Variation on variation on euclid’s algorithm,” IEEE Signal Processing Letters, vol. 11. [9] L. Calvez, S. Azou, and P. Vilb′e, “Variation on euclid’s algorithm for polynomials,” Electronics Letters 22nd, vol. 33, no. 11. [10] W. C. Yang, D. Guan, and C. S. Laih, “Fast multicomputation with asychronous strategy,” IEEE Transactions on Computers, vol. 56, no. 2, 2007. [11] J. Sorenson, “An analysis of lehmer’s euclidean gcd algorithm,” Proceedings of the 1995 international symposium on Symbolic and algebraic computation, 1995. [12] T. Jebelean, “A generalization of the binary gcd algorithm,” Proceedings of the 1993 international symposium on Symbolic and algebraic computation, 1993. [13] J. P. Soreson, “An analysis of the generalized binary gcd algorithm,” Fields Insitute Communications, vol. 41, 2004. [14] T. Jebelean, “Comparing several gcd algorithms,” 11th IEEE Symposium on Computer Arithmetic, 1993. [15] H. Brunner, A. Curiger, and M. Hofstetter, “On computing multiplicative inverses in gf(2m),” IEEE Transactions on Compurters, vol. 42, no. 8, 1993. [16] R. L′orencz, “New algorithm for classical modular inverse,” Cryptographic Hardware and Embedded Systems - CHES 2002, vol. 2523, 2003. [17] M. Joye and P. Paillier, “Gcd-free algorithms for computing modular inverses,” Cryptographic Hardware and Embedded Systems - CHES 2003, vol. 2779, 2003. |
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