隨著網際網路的快速發展，人類的食衣住行已經脫離不了網路，而透過網路進行的交易行為也越來越頻繁，因此資訊安全的領域也開始受到重視。本論文提到的RSA加解密系統是一個發展很久的加解密系統，透過簡單的模數指數運算，用來確保資料在傳遞過程中的安全性。
但隨著科技不斷的發展，這套系統還是有可能透過暴力法解開，為了確保足夠的安全性，所以RSA加解密系統通常要求位元數大於1024或2048位元以上，使得以軟體執行這套系統將會非常浪費時間。為了改善這個問題，於是我們透過硬體來實現這個系統。RSA加解密系統主要是由模數指數運算所組成，而模數指數運算的運算基礎是模數乘法運算。為了降低模數乘法器設計的困難度，便發展出蒙哥馬利演算法，這套演算法可以利用加法和移位來實現模數乘法運算。
但是傳統的蒙哥馬利演算法會因為安全性的問題而採用大位元數的運算，造成部分訊號有大量扇出(fanout)的問題，因此我們透過多重字組式的架構去解決這個問題。且為了能更快的執行完運算，我們結合了高基數的做法，並提出高基數之字組式蒙哥馬利乘法器的通用設計方法與規則，此外本論文採用hybrid-radix編碼方式以減少需要壓縮的運算元數目，使得壓縮樹層數與壓縮器的數量可以大量減少，降低高基數蒙哥馬利乘法器的延遲時間與硬體面積，如此設計出來的硬體能用更少的執行時間完成運算。

With the rapid development of the Internet, human can not live without the Internet today.Moreover,the trading behavior through the Internet is more frequently. Therefore,much attention has been paid to the field of information security.RSA encryption and decryption system mentioned in this thesis is a well-known system developed for a long time. It uses a simple modulus to ensure that the information in the transmission process is safe.
But with the development of science and technology, this system is possible to be broken through the violent method.To ensure adequate security,RSA encryption system usually requires the key length more than 1024 or 2048. But executing this system by software is time-consuming. In order to improve this problem, we use hardware design to implement this system. RSA encryption system is mainly composed of the modular exponentiation , and the modular exponentiation is composed of modular multiplication . And in order to reduce the difficulties in the hardware design,the Montgomery algorithm has been proposed. The algorithm can use the addition and shift to implement modular multiplication.
However, the traditional Montgomery algorithm has security problems,we need to use a large size of modulus for long-term security. But it causes the potential problem of high fan-out signals, we use word-based Montgomery architecture to improve the problem. To execute Montgomery multiplication more quickly, we combines the high-radix technique. Moreover ,we propose High-Radix Word-based generalized design method and regulation . We use hybrid-radix technique to reduce the numbers of partial product and Montgomery multiplication hardware delay and area.So that the design can use fewer execution cycles to complete the operation.

目錄
論文審定書 i
論文提要 ii
誌謝 iii
摘要 iv
Abstract v
第一章 緒論 1
1.1 研究動機 1
1.2 論文大綱 3
第二章 研究背景 4
2.1 RSA公開金鑰密碼系統 4
2.2 蒙哥馬利演算法 6
2.3 進位節省蒙哥馬利演算法 8
2.3.1 5-to-2 CSA蒙哥馬利演算法及架構 8
2.3.2 4-to-2 CSA蒙哥馬利演算法及架構 10
2.4 字組式蒙哥馬利演算法及架構 12
2.4.1 字組式蒙哥馬利演算法 13
2.4.2 字組式蒙哥馬利乘法器架構 15
2.4.3 字組式蒙哥馬利演算法的資料相依性 15
2.5 基數4蒙哥馬利架構.....................................................................................18
2.5.1 基數4蒙哥馬利乘法器 19
第三章 高基數之字組式蒙哥馬利乘法器 22
3.1 高基數蒙哥馬利演算法及硬體架構 22
3.2 High radix Booth 編碼 ………..25
3.3 改良式Radix-4蒙哥馬利乘法器..….……..................................................27
3.4 低延遲技術....................................................................................................30
3.4.1 預先計算之低延遲技術.................................................................................30
3.4.2 重新排程之低延遲技術................................................................................32
3.4.3 Tsai採用的低延遲技術.................................................................................35
第四章 提出的演算法及硬體架構設計 38
4.1 提出的高基數之字組式蒙哥馬利乘法演算法 38
4.2 演算法之效能分析 42
4.3 提出的高基數之字組式蒙哥馬利乘法器架構 43
4.3.1 提出的字組式蒙哥馬利乘法器整體架構 43
4.3.2 各種處理單元硬體設計 45
第五章 實驗數據 51
5.1 實驗步驟與方法 51
5.2 實驗結果 53
第六章 結論與未來研究方向………………………………………………………...58
6.1 結論 58
6.2 未來研究方向 58
參考文獻 59

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