The Mathematics of Encryption : An Elementary Introduction

Includes index

Includes bibliography (325-327) and references

1.1.Ancient Times 1.2.Cryptography During the Two World Wars 1.3.Postwar Cryptography, Computers, and Security 1.4.Summary 1.5.Problems 2.1.Ancient Cryptography 2.2.Substitution Alphabet Ciphers 2.3.The Caesar Cipher 2.4.Modular Arithmetic 2.5.Number Theory Notation 2.6.The Affine Cipher 2.7.The Vigenère Cipher 2.8.The Permutation Cipher 2.9.The Hill Cipher 2.10.Summary 2.11.Problems 3.1.Setting the Stage 3.2.Some Counting 3.3.Enigma's Security 3.4.Cracking the Enigma 3.5.Codes in World War II 3.6.Summary 3.7.Appendix: Proofs by Induction 3.8.Problems 4.1.Breaking the Caesar Cipher 4.2.Function Preliminaries 4.3.Modular Arithmetic and the Affine Cipher 4.4.Breaking the Affine Cipher 4.5.The Substitution Alphabet Cipher 4.6.Frequency Analysis and the Vigenère Cipher 4.7.The Kasiski Test 4.8.Summary 4.9.Problems 5.1.Breaking the Permutation Cipher Contents note continued: 5.2.Breaking the Hill Cipher 5.3.Running Key Ciphers 5.4.One-Time Pads 5.5.Summary 5.6.Problems 6.1.Binary Numbers and Message Streams 6.2.Linear Feedback Shift Registers 6.3.Known-Plaintext Attack on LFSR Stream Ciphers 6.4.LFSRsum 6.5.BabyCSS 6.6.Breaking BabyCSS 6.7.BabyBlock 6.8.Security of BabyBlock 6.9.Meet-in-the-Middle Attacks 6.10.Summary 6.11.Problems 7.1.The Perfect Code Cryptography System 7.2.KidRSA 7.3.The Euclidean Algorithm 7.4.Binary Expansion and Fast Modular Exponentiation 7.5.Prime Numbers 7.6.Fermat's little Theorem 7.7.Summary 7.8.Problems 8.1.RSA 8.2.RSA and Symmetric Encryption 8.3.Digital Signatures 8.4.Hash Functions 8.5.Diffie-Hellman Key Exchange 8.6.Why RSA Works 8.7.Summary 8.8.Problems 9.1.Introduction 9.2.Error Detection and Correction Riddles 9.3.Definitions and Setup 9.4.Examples of Error Detecting Codes Contents note continued: 9.5.Error Correcting Codes 9.6.More on the Hamming (7,4) Code 9.7.From Parity to UPC Symbols 9.8.Summary and Further Topics 9.9.Problems 10.1.Steganography Messages You Don't Know Exist 10.2.Steganography in the Computer Age 10.3.Quantum Cryptography 10.4.Cryptography and Terrorists at Horne and Abroad 10.5.Summary 10.6.Problems 11.1.Introduction 11.2.Brute Force Factoring 11.3.Fermat's Factoring Method 11.4.Monte Carlo Algorithms and FlT Primality Test 11.5.Miller-Rabin Test 11.6.Agrawal-Kayal-Saxena Primality Test 11.7.Problems 12.1.Chapter 1: Historical Introduction 12.2.Chapter 2: Classical Cryptography: Methods 12.3.Chapter 3: Enigma and Ultra 12.4.Chapter 4: Classical Cryptography: Attacks I 12.5.Chapter 5: Classical Cryptography: Attacks II 12.6.Chapter 6: Modern Symmetric Encryption 12.7.Chapter 7: Introduction to Public-Channel Cryptography Contents note continued: 12.8.Chapter 8: Public-Channel Cryptography 12.9.Chapter 9: Error Detecting and Correcting Codes 12.10.Chapter 10: Modern Cryptography 12.11.Chapter 11: Primality Testing and Factorization.

How quickly can you compute the remainder when dividing 109837⁹⁷ 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.