大整數因數分解一直是困難的計算問題，也造就許多公開密碼系統的設計。官大智教授撰寫「GQS 大整數因數分解」為實作「多個多項式二次篩法 (The Multiple Polynomial Quadratic Sieve)」演算法(簡稱為 MPQS)，並且在 2004 年成功的分解 RSA-130 的整數。在 MPQS 演算法的篩選 (sieve) 過程中，保留因數分解後還剩一個或兩個大質數的平滑數 (smooth number)分解式，可縮短篩選的時間，但會使最後分解基底 (factor base) 擴增的很大。本論文利用所保留下來的大質數，能盡量與平滑數分解式配對，可以有效地減緩分解基底擴增的情況，進而增快分解大整數的速度。本篇論文中，我們在一台 AIX Server 上實作此想法，並藉由觀察參數調整結果，做為改進「MPQS」效能的建議。

Large integer factoring problem is a difficult computing problem. The security of many public-key cryptograohy system depend on the large interger factoring problem. Dr. Guan implement 「The Multiple Polynomail Quadratic Sieve Algorithm」 and name the program 「GQS」. The program successfully factor RSA-130 interger in 2004. It can reduce the time of sieving that the MPQS algorithm retain the smooth number with one or two prime. But finally the size of factor basis is large. We use some of the prime retained by the MPQS algorithm to match with the smooth number and reduce the size of factor basis. And then we can reduce the time of factoring. In this paper, we implement our idea in a AIX server and the result of this paper can be a suggestion of the improvement of MPQS.

1 緒論.....................................................................................................................................................................................................................................1
1.1 前言..................................................................................................................................................................................................................................1
1.2 論文架構..........................................................................................................................................................................................................................2
2 Quadratic Sieve 演算法及相關數論介紹...........................................................................................................................................................................4
2.1 基礎數論介紹..................................................................................................................................................................................................................4
2.1.1 二次剩餘 (Quadratic Residue).......................................................................................................................................................................................4
2.1.2 雷建德符號 (Legendre symbol)....................................................................................................................................................................................4
2.1.3 加寇比符號 (Jacobi symbol).........................................................................................................................................................................................5
2.1.4 平滑數 (smooth number)...............................................................................................................................................................................................5
2.2 Kraitchik's Scheme............................................................................................................................................................................................................5
2.3 二次篩法 (Quadratic Sieve)..............................................................................................................................................................................................6
2.4 多個多項式二次篩法 (The Multiple Polynomial Quadratic Sieve)..................................................................................................................................10
2.5 自初始型多個多項式二次篩法 (The Self Initializing Variant of the Multiple Polynomial Quadratic Sieve)...................................................................13
2.6 大質數和二個大質數型多個多項式二次篩法 (The Large Prime and Double Large Prime Variation of the Multiple Polynomial Quadratic Sieve).....15
3 程式介紹.............................................................................................................................................................................................................................17
3.1 程式構想..........................................................................................................................................................................................................................17
3.2 程式架構..........................................................................................................................................................................................................................18
3.3 演算法..............................................................................................................................................................................................................................19
3.3.1 gqsinit 子程式...............................................................................................................................................................................................................19
3.3.2 gqssieve 子程式............................................................................................................................................................................................................20
3.3.3 gqsmerge 子程式..........................................................................................................................................................................................................21
3.3.4 gqscombine 子程式......................................................................................................................................................................................................21
3.3.5 gqsieve 子程式.............................................................................................................................................................................................................22
3.3.6 gqscycle 子程式............................................................................................................................................................................................................23
3.3.7 gqsreduce 子程式.........................................................................................................................................................................................................23
3.3.8 gqsfactor 子程式...........................................................................................................................................................................................................24
4 實驗步驟與結果................................................................................................................................................................................................................25
4.1 實作環境.........................................................................................................................................................................................................................25
4.2 實驗步驟.........................................................................................................................................................................................................................25
4.3 實驗結果.........................................................................................................................................................................................................................28
4.4 結果分析與說明.............................................................................................................................................................................................................31
5 結論與未來展望................................................................................................................................................................................................................32
參考文獻...............................................................................................................................................................................................................................33

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