餘數系統可以應用在密碼學、影像處理、數位訊號處理、平行處理及雲端運算上。餘數系統的基本概念是將一個大的數值轉換成若干個小的數值，所以它具有平行運算的性質，且因為不需要考慮進位的問題，所以能夠對大數運算提供很好的效率。然而，餘數系統對於正負號偵測、溢位偵測、數值比較以及除法會非常困難。只要有奇偶驗證的方法，上述的運算就能變的很有效率。

本論文是在餘數系統下，架構於模數組{2p-1,2p+1,2p^2-1}，進行有效率的奇偶驗證，其中p為正整數。給定一數X，X的餘數系統表示形式為(x_1,x_2,x_3)，其中x_1= X mod 2p-1, x_2= X mod 2p+1, x_3= X mod 2p^2-1。X的奇偶值會與F(x_1 + x_2 + x_3 + G(d)) 的結果相同 (0 為偶，1為奇，F(x) = x mod 2)。 其中，d=2p(x_2-x_1)+(2x_3-x_1-x_2)，如果d < 2(2p^2-1) 或 d < 0，G(d)=1；反之，G(d)=0。

The residue number system (RNS) can be applied to cryptography, image processing, digital filtering, parallel computation, and cloud computing. It represents a large integer using a set of smaller integers, so it has property of carry-free and parallel, and high-speed in addition, subtraction, and multiplication. However, the other RNS operations, such as number comparison, division, overflow detection, and sign detection is very difficult and needs significant amounts of time. With the parity detection technique, we can improve these operations to be efficiently.

In this paper, we provide a parity detection method basic on residue number system using three-moduli set {2p-1,2p+1,2p^2-1}, where p is a positive integer. Given RNS representation of X = (x_1,x_2,x_3 ) based on the three-moduli set where x_1= X mod 2p-1, x_2= X mod 2p+1, x_3= X mod 2p^2-1. The parity of X is equals to the result of (x_1 + x_2 + x_3 + G(d)) mod 2 (0 means even, 1 means odd) where d=2p(x_2-x_1)+(2x_3-x_1-x_2), G(d)=1, if d < 2m_3 or d < 0; otherwise, G(d)=0.

論文審定書 i
Acknowledgments iv
摘要 v
Abstract vi
List of Tables ix
Chapter 1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Residue Number System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Motivation and Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Chapter 2 Literature Review 5
2.1 Existing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Method of Lu and Chiang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Method of Chen and Hsueh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Chapter 3 Proposed Method 9
3.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 Theorem and Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 Program Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Chapter 4 Conclusion and Future Works 21
Bibliography 22

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