An Introduction to Mathematical Cryptography 1st edition by Jeffrey Hoffstein, Jill Pipher, Joseph Silverman ISBN 1441926747 978-1441926746

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ISBN 10: 1441926747
ISBN 13: 978-1441926746
Author: Jeffrey Hoffstein, Jill Pipher, Joseph Silverman

ThecreationofpublickeycryptographybyDi?eandHellmanin1976andthe subsequent invention of the RSA public key cryptosystem by Rivest, Shamir, and Adleman in 1978 are watershed events in the long history of secret c- munications. It is hard to overestimate the importance of public key cr- tosystems and their associated digital signature schemes in the modern world of computers and the Internet. This book provides an introduction to the theory of public key cryptography and to the mathematical ideas underlying that theory. Public key cryptography draws on many areas of mathematics, including number theory, abstract algebra, probability, and information theory. Each of these topics is introduced and developed in su?cient detail so that this book provides a self-contained course for the beginning student. The only prerequisite is a ?rst course in linear algebra. On the other hand, students with stronger mathematical backgrounds can move directly to cryptographic applications and still have time for advanced topics such as elliptic curve pairings and lattice-reduction algorithms. Amongthemanyfacetsofmoderncryptography,thisbookchoosestoc- centrate primarily on public key cryptosystems and digital signature schemes. This allows for an in-depth development of the necessary mathematics – quired for both the construction of these schemes and an analysis of their security. The reader who masters the material in this book will not only be well prepared for further study in cryptography, but will have acquired a real understanding of the underlying mathematical principles on which modern cryptography is based.

An Introduction to Mathematical Cryptography 1st Table of contents:

Chapter 1: Basic Number Theory

1.1. Divisibility and Primes
1.2. The Division Algorithm and the Euclidean Algorithm
1.3. Modular Arithmetic
1.4. The Chinese Remainder Theorem
1.5. Prime Factorization
1.6. Greatest Common Divisors and the Extended Euclidean Algorithm
1.7. Applications of Modular Arithmetic in Cryptography

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Chapter 2: Classical Cryptosystems

2.1. Substitution Ciphers: Caesar Cipher
2.2. Vigenère Cipher
2.3. The Playfair Cipher
2.4. The Enigma Machine
2.5. Block Ciphers and Stream Ciphers
2.6. Cryptanalysis of Classical Cryptosystems

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Chapter 3: Public-Key Cryptography

3.1. The Basics of Public-Key Cryptography
3.2. The RSA Algorithm
3.3. The Security of RSA
3.4. Diffie-Hellman Key Exchange
3.5. ElGamal Cryptosystem
3.6. Public-Key Cryptography and Modular Arithmetic

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Chapter 4: Elliptic Curve Cryptography

4.1. Introduction to Elliptic Curves
4.2. The Group Law on Elliptic Curves
4.3. Elliptic Curve Cryptography
4.4. The Elliptic Curve Discrete Logarithm Problem
4.5. Applications of Elliptic Curves in Cryptography
4.6. Cryptographic Protocols Based on Elliptic Curves

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Chapter 5: Number Theoretic Algorithms

5.1. Primality Testing
5.2. The Sieve of Eratosthenes
5.3. Miller-Rabin Primality Test
5.4. Integer Factorization Algorithms
5.5. Pollard’s Rho Algorithm
5.6. The General Number Field Sieve (GNFS)

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Chapter 6: Hash Functions and Digital Signatures

6.1. Introduction to Hash Functions
6.2. Properties of Cryptographic Hash Functions
6.3. Popular Hash Functions: MD5, SHA-1, and SHA-2
6.4. Digital Signatures and Their Applications
6.5. The RSA Signature Scheme
6.6. Digital Signature Algorithms (DSA)
6.7. Hash-Based Digital Signatures

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Chapter 7: Advanced Cryptographic Protocols

7.1. Zero-Knowledge Proofs
7.2. Secure Multi-Party Computation
7.3. Homomorphic Encryption
7.4. Digital Cash and Electronic Payments
7.5. Identity-Based Cryptography

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Chapter 8: Cryptanalysis

8.1. Introduction to Cryptanalysis
8.2. Brute-Force Attacks
8.3. Attacks on Classical Ciphers: Frequency Analysis
8.4. Attacks on Public-Key Systems
8.5. Attacks on Hash Functions
8.6. Side-Channel Attacks
8.7. Cryptographic Vulnerabilities in Real-World Systems

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Chapter 9: Modern Cryptographic Systems

9.1. Block Ciphers: AES and the Advanced Encryption Standard
9.2. Stream Ciphers and RC4
9.3. The Design and Analysis of Secure Cryptographic Systems
9.4. Modern Digital Signature Schemes
9.5. Quantum Cryptography: Challenges and Opportunities

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Chapter 10: Applications of Cryptography

10.1. Cryptography in Network Security
10.2. Public-Key Infrastructure (PKI)
10.3. Secure Sockets Layer (SSL) and Transport Layer Security (TLS)
10.4. Cryptography in Email and File Encryption
10.5. Cryptography in Blockchain and Cryptocurrencies
10.6. Privacy and Security in Digital Communication

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Chapter 11: Future Directions in Cryptography

11.1. Post-Quantum Cryptography
11.2. Lattice-Based Cryptography
11.3. Homomorphic Encryption and Privacy-Preserving Computation
11.4. Advanced Cryptographic Protocols and Their Implications

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