The Mathematics of Encryption An Elementary Introduction 1st Edition by Margaret Cozzens, Steven Miller ISBN 0821883216 9780821883211

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ISBN 10: 0821883216
ISBN 13: 9780821883211
Author: Margaret Cozzens, Steven Miller
How quickly can you compute the remainder when dividing $109837^{97}$ by 120143? Why would you even want to compute this? And what does this have to do with cryptography? Modern cryptography lies at the intersection of mathematics and computer sciences, involving number theory, algebra, computational complexity, fast algorithms, and even quantum mechanics. Many people think of codes in terms of spies, but in the information age, highly mathematical codes are used every day by almost everyone, whether at the bank ATM, at the grocery checkout, or at the keyboard when you access your email or purchase products online.

This book provides a historical and mathematical tour of cryptography, from classical ciphers to quantum cryptography. The authors introduce just enough mathematics to explore modern encryption methods, with nothing more than basic algebra and some elementary number theory being necessary. Complete expositions are given of the classical ciphers and the attacks on them, along with a detailed description of the famous Enigma system. The public-key system RSA is described, including a complete mathematical proof that it works. Numerous related topics are covered, such as efficiencies of algorithms, detecting and correcting errors, primality testing and digital signatures. The topics and exposition are carefully chosen to highlight mathematical thinking and problem solving. Each chapter ends with a collection of problems, ranging from straightforward applications to more challenging problems that introduce advanced topics. Unlike many books in the field, this book is aimed at a general liberal arts student, but without losing mathematical completeness.

The Mathematics of Encryption An Elementary Introduction 1st Edition Table of contents:

Chapter 1: Introduction to Cryptography

• What is Cryptography?
• Historical Context and Early Cryptography
• Modern Cryptography and Its Applications
• The Role of Mathematics in Cryptography

Chapter 2: Basic Concepts of Number Theory

• Divisibility and Prime Numbers
• Greatest Common Divisor (GCD) and Euclidean Algorithm
• Modular Arithmetic
• The Fundamental Theorem of Arithmetic

Chapter 3: Cryptographic Systems and Operations

• Substitution and Transposition Ciphers
• Classical Ciphers: Caesar Cipher, Vigenère Cipher
• Cryptanalysis and Breaking Ciphers
• The Role of Keys in Cryptographic Systems

Chapter 4: Modular Arithmetic and Cryptography

• Modulo Operation and Congruences
• Modular Inverses
• Fermat’s Little Theorem
• Applications of Modular Arithmetic in Cryptography

Chapter 5: Symmetric Key Cryptography

• Symmetric Key Encryption Basics
• The DES and AES Algorithms
• Block Ciphers and Stream Ciphers
• Modes of Operation and their Security

Chapter 6: Public Key Cryptography

• Introduction to Public Key Cryptography
• RSA Algorithm: Key Generation and Encryption/Decryption
• Security of RSA
• Other Public Key Cryptosystems

Chapter 7: Digital Signatures and Authentication

• Digital Signatures: How They Work
• Signing Algorithms and Authentication Protocols
• RSA Digital Signatures
• Applications of Digital Signatures in Modern Systems

Chapter 8: Elliptic Curve Cryptography

• Introduction to Elliptic Curves
• Cryptographic Applications of Elliptic Curves
• Elliptic Curve Diffie-Hellman and ECDSA
• Advantages of Elliptic Curve Cryptography

Chapter 9: Cryptographic Protocols

• Key Exchange Protocols: Diffie-Hellman
• Secure Communication: SSL/TLS
• Public Key Infrastructure (PKI)
• Zero-Knowledge Proofs

Chapter 10: Attacks on Cryptosystems

• Types of Cryptographic Attacks
• Brute-Force and Dictionary Attacks
• Side-Channel Attacks
• Attacks on RSA and AES
• Countermeasures and Improving Security

Chapter 11: Quantum Computing and Cryptography

• Introduction to Quantum Computing
• Shor’s Algorithm and Its Impact on Cryptography
• Quantum-Resistant Cryptography
• The Future of Cryptography in the Quantum Era

Appendices

• A: Review of Basic Algebra and Number Theory
• B: Further Reading and Resources
• C: Glossary of Cryptographic Terms
• D: Recommended Software and Tools for Cryptography

Index

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