指數運算在有限乘法群中, 或者在橢圓曲線中稱為純量乘法,
在很多密碼系統中是最耗費時間的運算, 如在 RSA 或DES.
在本篇論文我們一開始先介紹一些過去被研究過的指數運算技巧,
然後我們提出一個想法可以加快兩個指數相成的計算, 如 $c=a^xb^y$ (或者是在橢圓曲線上是 $C=xA+yB$).
此想法藉著調整 Shamir method 中非零位元的計算次序,
使得非零位元能夠被最大限度的對齊以降低其計算量.

The computation of modular exponentiation in a finite multiplication group,
or scalar multiplication in elliptic curves,
is the most time-consuming operations for many cryptosystems, such as RSA or DSA.
In this thesis we first introduce some researched techniques for the exponentiation, then
we propose an idea to speed up the computation for pairs of integers, e.g. $c=a^xb^y$, or $C=xA+yB$ in elliptic curves, by adjusting the computing sequence of
the Shamir method and shifting the two integer's nonzero bits. So that the number of matched
nonzero bits is maximized to reduce the computing cost.

Abstract (in Chinese) .......................................... I
Abstract (in English) .......................................... II
Contents ....................................................... III
List of Tables ................................................. V
Chapter 1 Introduction ........................................ 1
Chapter 2 Binary and Window Methods ........................... 4
2.1 Binary method ........................................... 4
2.2 The Shamir Method ....................................... 5
2.3 -ary Method ........................................... 6
2.4 Sliding Window -ary Method ............................ 6
2.5 Efficient Squaring of Large Integers .................... 7
Chapter 3 Signed-Digit Expansions ............................ 9
3.1 Morain and Olivos' Addition-Subtraction Chains .......... 9
3.2 Signed Binary Expansions ................................ 10
3.3 The Nonadjacent Form (NAF) .............................. 11
3.4 The Montgomery Inverse .................................. 12
3.5 Joint Sparse Form ....................................... 13
3.6 Simple Joint Sparse Form and It's Higher Dimensions ..... 15
3.6.1 Simple Joint Sparse Form ............................ 15
3.6.2 Higher Dimensions ................................... 17
3.7 Optimal Left-to-Right Binary Signed-Digit Recoding ...... 18
Chapter 4 Block-Shift Computing ............................... 21
4.1 The Block-Shift Computing (BSC) Algorithm ............... 21
4.2 Simulation and Analysis ................................. 27

Chapter 5 Conclusion and Future Work .......................... 31
Reference ...................................................... 32

[1] D.E. Knuth. The art of computing programming. Vol.II, Addison-Wesley, 1969.

[2] T. ElGamal. A public-key cryptosystem and a signature scheme based on discrete logarithms. IEEE Transactions on Information Theory IT-31 (1985), pp. 469-472.

[3] D.M. Gordon. A Survey of Fast Exponentiation Methods.
Journal of Algorithms, Vol.27, pp.129-146, 1998.

[4] S.M. Yen and C.S. Laih. The fast cascade exponentiation algorithm and its application on cryptography. Advances in Cryptology-Auscypt'92, New York, Springer-Verlag, pp.447-456, 1993.

[5] Dan Zuras. On squaring and multiplying large integers.
pp 260-271, IEEE Conference, 1993.

[6] Dan Zuras. More on squaring and multiplying large integers. Vul. 48, no. 3, pp. 899-908, IEEE Trans. Comp., 1994.

[7] D. E. Knuth. The art of computer programming/seminumerical algorithms. Volume 2. Addison-Wesley, Second edition, 1973.

[8] J. Olivos. On vectorical addition chains. Journal of Algorithms, 2:13-21, 1981.

[9] A. Schonhage. A lower bound for the length of addition chains. In Theoretical Computer Science, voulme 1, pages 1-12. 1975.

[10] F. Morain and J. Olivos. Speeding up the computations on an elliptic curve using addition-subtraction chains.
Informatique theorique et Applications/Theoritical Informatics and Applications,
24(6):531-544, 1990.

[11] D.,. Gordon. A survey of fast exponentiation methods.
J. Algorithms, vol. 27, pp. 129-146, 1998.

[12] B.S. Kaliski. The Montgomery inverse and its applications. IEEE Trans. on Computers, Vol.44, No.8, pp.1064-1065, Aug. 1995.

[13] J. A. Solinas. Low-weight binary representations for pairs of integers. Tech. Report CORR 2001-41, University of Waterloo, 2001, manuscript.

[14] Peter J. Grabner, Clemens Heuberger, and Helmut Prodinger. Distribution results for low-weight binary representations for pairs of integers. Theoretical Computer Science, to appear.

[15] W. C. Yang, P. Y. Hsieh, and C. S. Laih. Efficient Squaring of Large Integers. The Institute of Electronics Information and Communication Engineers (IEICE)
Transactions on Fundamentals, vol. E87-A, no. 5, May 2004. (EI、SCI).

[16] M. Joye, and S. M. Yen. Optimal left-to-right binary signed-digit recoding. IEEE Transactions on Computers, (7):740-748, 2000.