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.
Table of Contents:
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
About the Author :
Alfred Basta, PhD, CMMC CCA, Six-Sigma Black Belt, PMP, CISM, CHPSE, CRISC, CPENT, OSCP is a professor of Mathematics, Cybersecurity, and Computer Science. He is a curriculum advisor for multiple colleges and universities. He has authored a variety of papers and publications in advanced Mathematics, Cybersecurity and Computer Science.
Stephan DeLong, PhD, is an Associate Professor of Mathematics at Tidewater Community College, where he has taught since 1993. He earned a Master of Science degree in Mathematics at Lehigh University and is the author of several textbooks in mathematics.
Stavros Basta, CIPP/US, CISM, CPENT, LPTM, CEH, SecurityX, PMP is a distinguished cybersecurity consultant and accomplished grant writer whose expertise bridges the theoretical foundations of mathematics with practical cybersecurity applications. With extensive experience in the field, Stavros has established himself as a thought leader in the integration of mathematical principles into cybersecurity frameworks. As a trusted advisor, he serves on numerous global advisory boards dedicated to advancing cybersecurity curricula and shaping privacy legislation across international jurisdictions. His multidisciplinary approach to cybersecurity, informed by rigorous mathematical analysis, offers readers a unique perspective on the quantitative underpinnings of modern digital security protocols.
