Cover -- HalfTitle Page -- Title Page -- Copyright -- Contents -- 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 -- 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 -- 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 -- 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 -- 4.5.1 Solving Simultaneous Congruences -- 4.5.2 The Euler Totient Function -- 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 -- 4.10.1 Choosing the Primes p and q

4.10.2 Finding the Pair (d, e) -- 4.10.3 Computing me mod N and cd mod N -- 4.11 RSA and the Future -- 4.12 Other Applications of Modular Arithmetic -- 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 -- 5.7.1 Groups of Symmetries -- 5.7.2 Permutation 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 -- 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 -- 6.6 Unique Factorisation Domains-Key Properties -- 6.7 When Are Two Rings the Same Ring? -- 6.8 Fields -- 6.9 Coding Theory -- 6.9.1 Coding and Block Codes -- 6.9.2 Hamming Codes via a Party Trick -- 6.9.3 Codes via Rings