Emerging Security Algorithms and Techniques

Khaleel Ahmad, M. N. Doja, Nur Izura Udzir, Manu Pratap Singh, Khaleel Ahmad, M. N. Doja, Nur Izura Udzir, Manu Pratap Singh

• Publisher: Chapman and Hall/CRC
• Year: 2019
• Print ISBN: 9780815361459
• eBook ISBN: 9781351021685
• Topic: Computer Science
• Subtopic: Computer Science General
• Index: Computer Science

Cyber security is the protection of information systems, hardware, software, and information as well from theft, damages, interruption or misdirection to any of these resources. In other words, cyber security focuses on protecting computers, networks, programs and data (in use, in rest, in motion) from unauthorized or unintended access, change or destruction. Therefore, strengthening the security and resilience of cyberspace has become a vital homeland security mission.

Cyber security attacks are growing exponentially. Security specialists must occupy in the lab, concocting new schemes to preserve the resources and to control any new attacks. Therefore, there are various emerging algorithms and techniques viz. DES, AES, IDEA, WAKE, CAST5, Serpent Algorithm, Chaos-Based Cryptography McEliece, Niederreiter, NTRU, Goldreich–Goldwasser–Halevi, Identity Based Encryption, and Attribute Based Encryption.

There are numerous applications of security algorithms like cyber security, web security, e-commerce, database security, smart card technology, mobile security, cloud security, digital signature, etc.

The book offers comprehensive coverage of the most essential topics, including:

• Modular Arithmetic, Finite Fields
• Prime Number, DLP, Integer Factorization Problem
• Symmetric Cryptography
• Asymmetric Cryptography
• Post-Quantum Cryptography
• Identity Based Encryption
• Attribute Based Encryption
• Key Management
• Entity Authentication, Message Authentication
• Digital Signatures
• Hands-On "SageMath"

This book serves as a textbook/reference book for UG, PG, PhD students, Teachers, Researchers and Engineers in the disciplines of Information Technology, Computer Science and Engineering, and Electronics and Communication Engineering.

1

Modular Arithmetic

Afroz

Maulana Azad National Urdu University

CONTENTS

• 1.1 Introduction to Number Theory
  • 1.1.1 Integers
  • 1.1.2 Integer Arithmetic
  • 1.1.3 Arithmetic Operations
• 1.2 Modular Arithmetic
  • 1.2.1 Examples
• 1.3 Additive Inverse
  • 1.3.1 Examples
• 1.4 Multiplicative Inverse
  • 1.4.1 Examples
• 1.5 Matrix
  • 1.5.1 Examples
• 1.6 Linear Congruence
  • 1.6.1 Solution of Linear Congruence
• 1.7 Prime and Relative Prime Numbers
  • 1.7.1 Prime Numbers
  • 1.7.2 Relatively Prime Numbers
  • 1.7.3 Examples
• 1.8 Greatest Common Divisor (Euclid’s Algorithm, Bézout’s Algorithm, and Extended Euclid’s Algorithm)
  • 1.8.1 Greatest Common Divisor
    • 1.8.1.1 Examples
  • 1.8.2 Euclid’s Algorithm
    • 1.8.2.1 Example
  • 1.8.3 Bézout’s Theorem
  • 1.8.4 Extended Euclid’s Algorithm
    • 1.8.4.1 Examples
• 1.9 Conclusion
• References

1.1 Introduction to Number Theory

Number theory is a branch of pure mathematics which helps to analyze and study the integers. It is partly experimental and partly theoretical. The experimental part leads to questions and suggests the ways to answer the questions. The theoretical part tries to device an argument that gives a conclusive answer to the questions. It examines and accumulates numerical data, and then finds the relationships between different types of numbers (Anglin, 1995). Number theory is also called “The Queen of Mathematics.”

1.1.1 Integers

The positive, negative, and whole numbers with zero together are known as an integer, i.e., … −3, −2, −1, 0, 1, 2, 3 …, etc. An integer can be represented along a number line by an arrow (Gay, 2007). A positive number is represented by an arrow pointing to the right. A negative number is represented by an arrow pointing to the left. The total value of these numbers is represented by the length of the arrow.

1.1.2 Integer Arithmetic

The integer arithmetic is indispensable at most every area such as cryptology, computer graphics, mathematics, and several other engineering areas.

1.1.3 Arithmetic Operations

The most basic integer operations are as follows:

1. Addition
2. Subtraction
3. Multiplication
4. Division.

Thus, the mathematical operations of the above integers can be performed as follows.

1. Addition: It states that the addition of two integers is always an integer.
   1. 2 + 5 = 7

      

   2. −7 + 4 = −3

      

2. Subtraction: It states that the subtraction of two integers is again an integer.
   1. 6 − 4 = 2

      

3. Multiplication: It states that multiplying two integers always results in an integer.

   

4. Division...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Perface
• Editors
• Contributors
• 1. Modular Arithmetic
• 2. Finite Fields
• 3. Prime Number
• 4. Discrete Logarithm Problem
• 5. Integer Factorization Problem
• 6. Symmetric Algorithms I
• 7. Symmetric Algorithms II
• 8. Asymmetric Cryptography
• 9. Post-Quantum Cryptography
• 10. Identity-Based Encryption
• 11. Attribute-Based Encryption
• 12. Key Management
• 13. Entity Authentication
• 14. Message Authentication
• 15. Digital Signatures
• 16. Applications
• 17. Hands-On “SageMath”
• Index