在密碼學, a^xb^y mod n 是最重要也是最耗時的計算。該問題可以用 Classical binary method 來計算之, 爾後的研究便是建立在此基礎上加速運算效率。其中, Binary signed-digit representation recoding algorithm、稀鬆格式 (Sparse Form)、 DJM 查表式編碼、 聯合稀鬆格式 (Joint Sparse Form)等等有效降低非零位元的數量。
而另一種便是預先計算建表法 (Pre-computing algorithm) 先將部份的結果預算起來並儲存, 藉由移位和處理部份碼一起計算來減少計算的次數。 所以增加運算速度是建立在減少計算數量。聯合稀鬆格式編碼方法並非是查表式編碼演算法來將原始碼轉變為聯合稀鬆格式。
在本論文中, 我們先提供一個對於聯合稀鬆格式的查表式編碼演算法來簡化整個編碼概念, 並使用該演算法建構一個有限狀態機來表示編碼的過程。根據有限狀態機, 我們證明聯合稀鬆格式的平均聯合漢明權位是 0.5n, 其中 n 逼近無限大。最後, 我們證明聯合稀鬆格式在 SS_1 方法和 DS_1 方法的平均聯合漢明權位分別是是 0.469n 和0.438n。

In Cryptography, computing a^xb^y mod n is the most important and the most time-consuming calculation The problem can be solved by classical binary method. Later research is based on this basis to increase computational efficiency. Furthermore, Binary signed-digit representation recoding algorithm, the Sparse Form, the DJM recoding method, and the Joint Sparse Form can be used to decrease the number of multiplication by aligning more non-zero bits. Another method is to pre-compute and store the part of the results to decrease the number of computations by shifting bits. Joint Sparse Form recording method is not a table driven algorithm in converting source codes into joint sparse form. In this paper, we first proposed a table driven algorithm for joint sparse form to simply recording concept. This algorithm can be constructed a finite state machine to denote the recording procedure. According to this finite state machine, we show that the average joint Hamming weight among joint sparse form is 0.5n when n approaches infinity. Finally, we show that the average joint Hamming weights of SS1 method and DS1 method among joint sparse form are 0.469n and 0.438n by using a similar method, respectively.

1 緒論 .................................................................................. 5
1.1 前言 ................................................................................. 5
1.2 論文架構 ......................................................................... 6
2 傳統快速計算演算法 ........................................................ 7
2.1 二進位方法(Binary Method) ........................................ 7
2.2 x^A + y^B 計算方法 ....................................................... 9
3 編碼演算法 .......................................................................11
3.1 Binary Signed-digit Code ..........................................11
3.2 Sparse Form(SF) .......................................................12
3.3 DJM 編碼 ......................................................................14
3.4 Joint Sparse Form(JSF) ............................................16
4 非同步快速演算法 ..........................................................18
4.1 SS1 Method .................................................................19
4.2 DS1 Method .................................................................21
4.3 效率分析 .......................................................................23
5 聯合稀鬆格式查表演算法 ..............................................24
5.1 想法起源 .......................................................................25
5.2 研究分析 .......................................................................27
5.3 推論與證明 ...................................................................35
6 效率分析 ..........................................................................35
6.1 SS1 Method for Joint Sparse Form ........................35
6.2 DS1 Method for Joint Sparse Form ........................39
6.3 SS1 Method for DJM ..................................................43
7 結論 ..................................................................................46

[1] D. E. Knuth The art of computing programming. Vol.II, Addison-
Wesley, 1969.
[2] J. A. Solinas. Low-weight binary representations for pairs of integers.
University of Waterloo, Tech. Rep. CORR 2001-41, 2001.
[3] Steven Arno and Ferrell S. Wheeler. Signed Digit Representations of
Minimal Hamming Weight. IEEE Transactions on Computers, vol 42,
no. 8, 1993.
[4] X. Ruan and R. S. Katti. Left-to-right optimal signed-binary representation
of a pair of integers. IEEE Transactions on Computers, vol. 54,
no. 2, pp. 124-131, 2005.
[5] M. Joye and S. M. Yen. Optimal left-to-right binary signed-digit recoding.
IEEE Transactions on Computers, vol. 49, no. 7, pp. 740-748,
2003.
[6] V.S. Dimitrov, G.A. Jullien, and W.C. Miller. Complexity and fast
algorithms for multi-exponentiation. IEEE Transactions on Computers,
vol.49. 141-147, Feb.2000.
[7] Wu-Chuan Yang, D.J. Guan, Chi-Sung Laih. Fast Multi-Computation
with Asynchronous Strategy. PKC 2005, LNCS 3386, pp.138-153,2005.
[8] Wu-Chuan Yang. 官於多指數運算負數快速演算法之深入探討. 2007 Information
Security Conference.