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
