出版日期:2016-4
ISBN:9787030474309
作者:Wang Xiaoyun,Xu Guangwu,Wang Mingqiang,Meng Xiaomeng
页数:240页
书籍目录
Preface to Mathematics Monograph Series
Foreword
Preface
Acknowledgments
CHAPTER 1 Divisibility of Integers
1.1 THE CONCEPT OF DIVISIBILITY
1.2 THE GREATEST COMMON DIVISOR AND THE LEAST COMMON MULTIPLE
1.3 THE EUCLIDEAN ALGORITHM
1.4 SOLVING LINEAR DIOPHANTINE EQUATIONS
1.5 PRIME FACTORIZATION OF INTEGERS
CHAPTER 2 Congruences
2.1 CONGRUENCES
2.2 RESIDUE CLASSES AND SYSTEMS OF RESIDUES
2.3 EULER'S THEOREM
2.4 WILSON'S THEOREM
CHAPTER 3 Congruence Equations
3.1 BASIC CONCEPTS OF CONGRUENCES OF HIGH DEGREES
3.2 LINEAR CONGRUENCES
3.3 SYSTEMS OF LINEAR CONGRUENCE EQUATIONS AND THE CHINESE REMAINDER THEOREM
3.4 GENERAL CONGRUENCE EQUATIONS
3.5 QUADRATICRESIDUES
3.6 THE LEGENDRE SYMBOL AND THE JACOBI SYMBOL
CHAPTER 4 Exponents and Primitive Roots
4.1 EXPONENTS AND THEIR PROPERTIES
4.2 PRIMITIVE ROOTS AND THEIR PROPERTIES
4.3 INDICES，CONSTRUCTION OF REDUCED SYSTEM OF RESIDUES
4.4 NTH POWER RESIDUES
CHAPTER 5 Some Elementary Results for Prime Distribution
5.1 INTRODUCTION TO THE BASIC PROPERTIES OF PRIMES AND THE MAIN RESULTS OF PRIME NUMBER DISTRIBUTION
5.2 PROOF OF THE EULER PRODUCT FORMULA
5.3 PROOF OF AWEAKER VERSION OF THE PRIME NUMBER THEOREM
5.4 EQUIVALENT STATEMENTS OF THE PRIME NUMBER THEOREM
CHAPTER 6 Simple Continued Fractions
6.1 SIMPLE CONTINUED FRACTIONS AND THEIR BASIC PROPERTIES
6.2 SIMPLE CONTINUED FRACTION REPRESENTATIONS OF REAL NUMBERS
6.3 APPLICATION OF CONTINUED FRACTION IN CRYPTOGRAPHY—ATTACK TO RSA WITH SMALL DECRYPTION EXPONENTS
CHAPTER 7 Basic Concepts
7.1 MAPS
7.2 ALGEBRAIC OPERATIONS
7.3 HOMOMORPHISMS AND ISOMORPHISMS BETWEEN SETS WITH OPERATIONS
7.4 EQUIVALENCE RELATIONS AND PARTITIONS
CHAPTER 8 GroupTheory
8.1 DEFINITIONS
8.2 CYCLIC GROUPS
8.3 SUBGROUPS AND COSETS
8.4 FUNDAMENTAL HOMOMORPHISM THEOREM
8.5 CONCRETE EXAMPLES OF FINITE GROUPS
CHAPTER 9 Rings and Fields
9.1 DEFINITION OF A RING
9.2 INTEGRAL DOMAINS， FIELDS， AND DIVISION RINGS
9.3 SUBRINGS，IDEALS， AND RING HOMOMORPHISMS
9.4 CHINESE REMAINDER THEOREM
9.5 EUCLIDEAN RINGS
9.6 FINITE FIELDS
9.7 FIELD OF FRACTIONS
CHAPTER 10 Some Mathematical Problems in Public Key Cryptography
10.1 TIME ESTIMATION AND COMPLEXITY OF ALGORITHMS
10.2 INTEGER FACTORIZATION PROBLEM
10.3 PRIMALITY TESTS
10.4 THE RSA PROBLEM AND THE STRONG RSA PROBLEM
10.5 QUADRATIC RESIDUES
10.6 THE DISCRETE LOGARITHM PROBLEM
CHAPTER 11 Basics of Lattices
11.1 BASIC CONCEPTS
11.2 SHORTEST VECTOR PROBLEM
11.3 LATTICE BASIS REDUCTION ALGORITHM
11.4 APPLICATIONS OF LLL ALGORITHM
References
Further Reading
Index
作者简介
《公钥密码学的数学基础(英文版)》内容新颖，结构合理，注重严格的理论分析与启发性的数值实验相结合。适合从事数学、物理、天文和工程技术的科研工作者和研究生参考。
数论与代数结构这门课是数学学院信息安全专业的一门专业基础课。通过该课程的学习，让学生掌握密码学所需要的重要的数学基础理论，熟悉密码体制中常用的数学基本算法及其复杂性理论。
具体分为下面几个方面的内容：1.整除：整除的基本理论，辗转相除法；2.同余：同余、剩余类的基本理论，同余方程，Euler定理；3.原根：指标的基本理论，原根基本定理；4.群、环、域基本理论；5.群、环、域进一步的理论，扩域、有限域的理论；6.基本算法、及其复杂性理论；7.格理论。