Mathematics in Cybersecurity

Balanced, in-depth exploration of mathematics in cybersecurity, supported by case studies and other learning features With clear and concise explanations, practical examples, case studies, and real-world applications, Mathematics in Cybersecurity is an essential learning aid for readers seeking an understanding of the fundamental concepts in the field. This comprehensive book delves into a wide range of topics, including sets, logic, binary and other number systems, logic gates, combinatorial logic circuits, equations and graphs, linear equations and matrices, sequences and series. Other topics covered include right triangle geometry and trigonometry, exponential and logarithmic equations, probability, statistics, graph theory, and cryptography. The book is an ideal companion resource for courses involving mathematical concepts in cybersecurity. It allows tailoring of the content to meet the needs of career education and online schools, ensuring it is comprehensible to students who may be learning remotely and relying on self-study. Mathematics in Cybersecurity also includes information on: Discrete math, set theory fundamentals, propositional logic, and number systems and conversions Digital logic circuits, linear equations and matrices, sequences and recursion, and trigonometry and pre-calculus Complex numbers, exponential and logarithmic functions, and probability and counting techniques Graph theory applications and coding for cryptography By adopting a balanced approach that combines traditional and contemporary ideas, Mathematics in Cybersecurity provides a readable and rigorous treatment of the subject matter for aspiring cybersecurity professionals, equipping them with the necessary tools to tackle complex cybersecurity challenges confidently and excel in their field.

About the Authors xix Acknowledgments xxi About the Companion Website xxiii 1 Introduction to Sets 1 1.1 Introduction 1 1.2 Types of Sets 5 1.3 Operations on Sets 8 1.4 Set Relations 11 2 Logic 17 2.1 Introduction to Logic 17 2.2 Propositional Logic 19 2.3 Logical Equivalence 26 2.4 Logical Inference and Proofs 30 2.5 Logic Gates 33 2.6 Predicate Logic 40 2.7 Applications of Logic 43 2.8 Summary 49 3 Introduction to Number Systems 51 3.1 What Is a Number System? 51 3.2 Binary Number System 52 3.3 Octal (Base-8) Number System 64 3.4 Hexadecimal (Hex) Number System 75 3.5 Ternary Number System 81 3.6 Generalization: Base-n Number Systems 84 3.7 Applications of Number Systems 85 3.8 Arithmetic in Other Number Systems 89 3.9 Floating-Point Representation 96 4 Straight-Line Equations and Graphs 103 4.1 Introduction to Straight Lines 103 4.2 Definition of a Straight Line 103 4.3 The Intercepts of a Line 105 4.4 Slope of a Line 106 4.5 The Equation of a Line 109 4.6 Standard Form of a Straight-Line Equation 112 4.7 The Primary Uses of the Three Forms of the Equation of a Line 113 4.8 Graphing Straight Lines 113 4.9 Graphing Using the Standard Form 114 4.10 Parallel and Perpendicular Lines 118 4.11 Applications of Straight-Line Equations: Modeling Linear Relationships 119 5 Introduction to Systems of Linear Equations 123 5.1 Definition and Notation of Systems of Linear Equations 123 5.2 Graphical Representation of a System of Linear Equations 123 5.3 The Solution to a System of Two Linear Equations in Two Variables 126 5.4 Applications That Lead to Systems of Linear Equations 139 5.5 Introduction to Matrices 148 6 Sequences and Series 177 6.1 Sequences 177 6.2 Arithmetic Sequences 181 6.3 Geometric Sequences 189 6.4 Fibonacci Sequence 192 6.5 The Principles of Mathematical Induction 196 6.6 Factorial Notation and Binomial Coefficients (Combinations) 199 7 Measuring an Angle 205 7.1 Measuring an Angle 205 7.2 Trigonometric Functions 214 7.3 Introduction to Trigonometric Identities 226 7.4 Trigonometric Functions of the Sum of Two Angles 230 7.5 Trigonometric Functions of the Difference of Two Angles 232 7.6 Trigonometric Functions of Double-Angles 234 7.7 Trigonometric Functions of Half-Angles 236 8 Introduction to Complex Numbers 241 8.1 Introduction to Complex Numbers 241 8.2 Addition and Subtraction of Complex Numbers 243 8.3 Multiplication and Division of Complex Numbers 248 8.4 Graphical Representation of a Complex Number 251 8.5 Exponential Form 254 8.6 Applications of Complex Numbers in Cybersecurity 257 9 Exponentials and Algorithms 259 9.1 Exponential Functions 259 9.2 Logarithms 271 9.3 The Properties of Exponential and Logarithmic Function Graphs 281 9.4 Solving Exponential and Logarithmic Equations 284 9.5 Applications of Exponential and Logarithmic Functions in Cybersecurity 289 10 What Do We Mean by Statistics? 293 10.1 Sampling 293 10.2 Statistical Graphs 297 10.3 The Measures of Central Tendency 304 10.4 The Measures of Dispersion 307 10.5 The Normal Distribution 311 10.6 The Binomial Distribution 323 10.7 Linear Correlation and Regression 329 11 Basic Concepts of Probability 341 11.1 Experiments, Outcomes, Sample Spaces, and Events 341 11.2 Axioms of Probability 348 11.3 Basic Rules of Probability 349 11.4 Counting Techniques and Probability 352 11.5 Random Variables 355 11.6 Probability Mass Function 355 11.7 Distributions of Discrete Data 356 11.8 Continuous Probability Distributions 367 11.9 The Central Limit Theorem 374 12 Graph Theory 377 12.1 What Is Graph Theory? 377 12.2 Exploration of Basic Concepts 381 12.3 Types of Graphs 395 12.4 Adjacency Matrix 408 12.5 Graph Traversals 413 12.6 Connectivity 418 12.7 Special Graphs 423 12.8 Graph Algorithms 432 13 What Is Cryptography? 445 13.1 What Is Cryptography? 445 13.2 Basic Cryptographic Concepts 446 13.3 Classical Ciphers 448 13.4 Modern Cryptography 459 13.5 Applications of Cryptography 467 13.6 Cryptanalysis and Cryptographic Attacks 469 13.7 Conclusion and Future Directions 471 13.8 A Closing Remark on a Nearly Unbreakable Code, Resolved After 50 Years 472 Index 477