,Table of Contents
Cover
Table of Contents
Title Page
Copyright
Dedication
List of Figures
List of Tables
About the Author
Preface: Why This Book?
Acknowledgements
Table of Abbreviations
About the Companion Site
Chapter 1: A Gentle Introduction
1.1 What Is This Book About?
1.2 Mathematics as Mathematicians See It
1.3 Theorems and Proofs
1.4 Abstract Algebra
1.5 What Do You Need to Know to Make Sense of
This Book?
1.6 Case Studies of Applications
Notes
Chapter 2: Sets, Functions and Relations
2.1 Why Are Sets Important?
2.2 Sets
2.3 Cartesian Products of Sets
2.4 Relations
, 2.5 Equivalence Relations and Equivalence Classes
2.6 Relations: A Detailed Example
2.7 Functions
2.8 Operations
Notes
Chapter 3: Numbers as We Know and Love Them
3.1 Where Does Mathematics Start?
3.2 The Natural Numbers and the Integers
3.3 Writing Down Numbers
3.4 Ordering the Integers
3.5 Induction
3.6 The Division Theorem
3.7 Prime Numbers and Common Factors
3.8 Unique Factorisation
3.9 The Euclidean Algorithm
3.10 The Rationals
3.11 The Real and Complex Numbers
3.12 Applying Complex Numbers—An Everyday
Example
Notes
Chapter 4: Modular Arithmetic on the Integers
4.1 Working Relative to a Modulus
4.2 Congruences: Making It More Mathematical
4.3 Parity Checks: Using Modulo 2 Arithmetic
4.4 Check Digits: A More Complex Example
4.5 Elementary Properties of ℤn
4.6 The Extended Euclidean Algorithm
4.7 Cryptography Ancient and Modern
, 4.8 RSA: How Does It Work?
4.9 Using RSA
4.10 Implementing RSA
4.11 RSA and the Future
4.12 Other Applications of Modular Arithmetic
Notes
Chapter 5: Groups
5.1 What Is a Group?
5.2 A First Example: The Integers
5.3 A Second Example: Modular Addition
5.4 But What About Modular Multiplication?
5.5 Subgroups and Lagrange’s Theorem
5.6 Proving Euler’s Theorem
5.7 Examples of Non-abelian Groups
5.8 When Are Two Groups the Same Group?
5.9 Combining Groups
5.10 Discrete Logarithms
5.11 Diffie–Hellman Key Agreement
5.12 Other Applications of Discrete Logarithms
5.13 The Threat Posed by Quantum Computing
5.14 Other Applications of Groups
Notes
Chapter 6: Rings and Fields
6.1 Two Operations, Not Just One!
6.2 So What Is a Ring?
6.3 Types of Rings
6.4 Combining Rings
6.5 Integral Domains—Some Key Properties
Cover
Table of Contents
Title Page
Copyright
Dedication
List of Figures
List of Tables
About the Author
Preface: Why This Book?
Acknowledgements
Table of Abbreviations
About the Companion Site
Chapter 1: A Gentle Introduction
1.1 What Is This Book About?
1.2 Mathematics as Mathematicians See It
1.3 Theorems and Proofs
1.4 Abstract Algebra
1.5 What Do You Need to Know to Make Sense of
This Book?
1.6 Case Studies of Applications
Notes
Chapter 2: Sets, Functions and Relations
2.1 Why Are Sets Important?
2.2 Sets
2.3 Cartesian Products of Sets
2.4 Relations
, 2.5 Equivalence Relations and Equivalence Classes
2.6 Relations: A Detailed Example
2.7 Functions
2.8 Operations
Notes
Chapter 3: Numbers as We Know and Love Them
3.1 Where Does Mathematics Start?
3.2 The Natural Numbers and the Integers
3.3 Writing Down Numbers
3.4 Ordering the Integers
3.5 Induction
3.6 The Division Theorem
3.7 Prime Numbers and Common Factors
3.8 Unique Factorisation
3.9 The Euclidean Algorithm
3.10 The Rationals
3.11 The Real and Complex Numbers
3.12 Applying Complex Numbers—An Everyday
Example
Notes
Chapter 4: Modular Arithmetic on the Integers
4.1 Working Relative to a Modulus
4.2 Congruences: Making It More Mathematical
4.3 Parity Checks: Using Modulo 2 Arithmetic
4.4 Check Digits: A More Complex Example
4.5 Elementary Properties of ℤn
4.6 The Extended Euclidean Algorithm
4.7 Cryptography Ancient and Modern
, 4.8 RSA: How Does It Work?
4.9 Using RSA
4.10 Implementing RSA
4.11 RSA and the Future
4.12 Other Applications of Modular Arithmetic
Notes
Chapter 5: Groups
5.1 What Is a Group?
5.2 A First Example: The Integers
5.3 A Second Example: Modular Addition
5.4 But What About Modular Multiplication?
5.5 Subgroups and Lagrange’s Theorem
5.6 Proving Euler’s Theorem
5.7 Examples of Non-abelian Groups
5.8 When Are Two Groups the Same Group?
5.9 Combining Groups
5.10 Discrete Logarithms
5.11 Diffie–Hellman Key Agreement
5.12 Other Applications of Discrete Logarithms
5.13 The Threat Posed by Quantum Computing
5.14 Other Applications of Groups
Notes
Chapter 6: Rings and Fields
6.1 Two Operations, Not Just One!
6.2 So What Is a Ring?
6.3 Types of Rings
6.4 Combining Rings
6.5 Integral Domains—Some Key Properties
