Cybersecurity and Applied Mathematics

Leigh Metcalf, William Casey

• Publisher: Syngress
• Year: 2016
• Print ISBN: 9780128044520
• eBook ISBN: 9780128044995
• Topic: Computer Science
• Subtopic: Cyber Security
• Index: Computer Science

Cybersecurity and Applied Mathematics explores the mathematical concepts necessary for effective cybersecurity research and practice, taking an applied approach for practitioners and students entering the field. This book covers methods of statistical exploratory data analysis and visualization as a type of model for driving decisions, also discussing key topics, such as graph theory, topological complexes, and persistent homology.Defending the Internet is a complex effort, but applying the right techniques from mathematics can make this task more manageable. This book is essential reading for creating useful and replicable methods for analyzing data.- Describes mathematical tools for solving cybersecurity problems, enabling analysts to pick the most optimal tool for the task at hand- Contains numerous cybersecurity examples and exercises using real world data- Written by mathematicians and statisticians with hands-on practitioner experience

Chapter 1

Introduction

Abstract

In this chapter we discuss the purpose of the book and the mathematical underpinnings of it.

Keywords

Introduction; Models; Mathematical techniques

The practice of cybersecurity involves diverse data sets, including DNS, malware samples, routing data, network traffic, user interaction, and more. There is no “one size fits all” analysis scheme for this data, a new method must be created for each data set. The best methods have a mathematical basis to them.

A mathematical model of a system is an abstract description of that system that uses mathematical concepts. We want to take the systems in cybersecurity and create mathematical models of them in order to analyze the systems, make predictions of the system, derive information about the system, or other goals, depending on the system. This book is designed to give you the basic understanding of the mathematical concepts that are most relevant to designing models for cybersecurity models.

Cybersecurity is often about finding the needle in the needlestack. Finding that one bit that looks almost, but not quite like, everything else. In a network that can generate gigabytes of traffic a day, discovering that small amount of anomalous traffic that is associated with malware is a difficult proposition. Similarly, finding the one set of maliciously registered domains in the hundreds of million of domain names is not an easy process.

There are a wide variety of mathematical techniques that can be used to create methods to analyze cybersecurity data. These techniques are the underpinnings that essentially “make it work.” Statistics cares about the origin of the data, how it was collected, and what assumptions you can make about the data. Mathematical techniques, such as graph theory, are developed on the structure known as a “graph,” and work no matter what they are used to model. That is the beauty of math.

The point of this book is not to spend time going through proofs as to why the various mathematical techniques work, but rather to give an introduction into the areas themselves. Careful consideration was taken in the chapters to include the description of “what” things are and “how” they work, but to not overwhelm the reader with the “why.” The “why” is not always relevant to understanding the “what” or “how.” This book is designed for the cybersecurity analyst who wishes to create new techniques that have a secure foundation in math.

The content is designed to cover various areas that are used in cybersecurity today, to give the reader a firm basis in understanding how they can be applied in creating new analysis methods as well as to enable the reader to achieve greater understanding of current methods. The reader is expected to have studied calculus in order to understand the concepts in the book.

Chapter 2

Metrics, similarity, and sets

Abstract

In this chapter we cover an introduction to set theory, with common operations such as subset, intersection, union, set difference, complement and symmetric difference with examples from cybersecurity data. Set functions are also discussed, which leads us directly to the definition of a metric. We cover the variations of metric, including pseudometric, quasimetric, and semimetric. Similarities are also discussed. We then illustrate the metric on various sets, including strings, sets, Internet and cybersecurity specific metrics.

Keywords

Sets; Set operations; Functions; Metric; Similarity

The human eye can discern differences between two objects, but cannot necessarily quantify that difference. For example, a red apple and a green apple are obviously different, but still similar in that they are both apples. If we consider a red apple and a computer, they are obviously completely different. We can only say “similar but different” or “obviously different.”

We need to quantify this difference in a reasonable way. To this end, we create a framework that standardizes the properties that a distance should satisfy. Before we cover the distance, we begin with the basics of set theory. Distances are inherently defined on a set, so the knowledge of some basics of set theory is useful. The chapter concludes with relevant examples of distances.

2.1 Introduction to Set Theory

We have lots of words for collections of things, a deck of cards, a flock of birds, a pod of whales, a murder of ravens, or a fleet of automobiles. The underlying theme of these words is collection of things that have a property in common; or as we say it in math, a set of things such as a set of birds or a set of domains. A set of domains would not contain a URL, becaus...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Biography
• Chapter 1: Introduction
• Chapter 2: Metrics, similarity, and sets
• Chapter 3: Probability models
• Chapter 4: Introduction to data analysis
• Chapter 5: Graph theory
• Chapter 6: Game theory
• Chapter 7: Visualizing cybersecurity data
• Chapter 8: String analysis for cyber strings
• Chapter 9: Persistent homology
• Appendix: Introduction to linear algebra
• Bibliography
• Index