數論在密碼學上的應用

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姓名

黃博峙(Bo-Chih Huang) 查詢紙本館藏

畢業系所

數學系

論文名稱

數論在密碼學上的應用
(Number Theory in Cryptography - a General Survey)

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摘要(英)

The purpose of this thesis is to make a general survey of the development of cryptography and to discuss the mathematical background of cryptography. We first introduce the RSA cryptosystem which is based on the difficulty of factoring a large integer, and other cryptosystems based on the problem in number theory so called discrete logarithm problem. Then we describe some methods which can determine whether or not a given integer is a prime number and methods of efficient factoring large composite integers, and some methods of solving discrete logarithm problem on finite fields. In the second half of this thesis, we present some basic properties of elliptic curves and introduce a factorization method which is based on properties of elliptic curves over finite fields. Some elliptic curve cryptosystems will be introduced. In the final section of this thesis, we discuss some methods of counting the number of rational points on the elliptic curve over finite fields.

關鍵字(中)

★ 橢圓曲線
★ 離散對數
★ 整數分解
★ 質數檢定
★ 密碼學

關鍵字(英)

★ Factoring Integer
★ Primality Test
★ Cryptography
★ Discrete Logarithm Problem
★ Elliptic Curve

論文目次

1 Introduction. ----------------------------------------------3
2 RSA. -------------------------------------------------------5
3 Discrete Logarithm. ----------------------------------------6
4 Methods of Primality Test. ---------------------------------8
5 Methods of Factorization. ---------------------------------13
6 Methods of Solving Discrete Logarithm Problem over
Finite Fields. --------------------------------------------26
7 Introduction to Elliptic Curves. --------------------------29
8 Counting Points on Elliptic Curves over Finite Fields. ----38
9 Appendix. -------------------------------------------------51

參考文獻

[1] Neal Kobliz, A Course in Number Theory and Cryptography,
Springer-Verlag, New York, 1987.
[2] Neal Kobliz, Algebraic Aspects of Cryptography, Springer-
Verlag, New York, 1991.
[3] Joseph H. Silverman, John Tate, Rational Points on Elliptic
Curves, Springer-Verlag, New York, 1992.
[4] Joseph H. Silverman, The Arithmetic of Elliptic Curves,
Springer-Verlag, 1986.
[5] Serge Lang, Elliptic Functions, Springer-Verlag, 1987.
[6] Serge Lang, Elliptic Curves Diophantine Analysis, Springer-
Verlag, 1978.
[7] Alfred Menezes, Elliptic curve public key cryptosystem,
Kluwer Academic Pub-lishers, Boston, 1993.
[8] Henri Cohen, A Course in Computational Algebraic Number
Theory, Springer-Verlag, 1993.
[9] René Schoof, Elliptic Curves Over Finite Fields and
the Computation of Square Roots mod p, Math. Comp. 44
(1985),483-494.
[10] René Schoof, Counting points on elliptic curves over
fnite fields, Journal de Théorie des Nombres de Bordeaux
7 (1995), 219-254.
[11] Stephen Wolfram, The MATHEMATICA Book, 3rd Ed. Cambridge
University.

指導教授

夏良忠(Liang-Chung Hsia)

審核日期

2000-7-18

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