## Book description

Cryptography has become essential as bank transactions, credit card infor-mation, contracts, and sensitive medical information are sent through inse-cure channels. This book is concerned with the mathematical, especially algebraic, aspects of cryptography. It grew out of many courses presented by the authors over the past twenty years at various universities and covers a wide range of topics in mathematical cryptography. It is primarily geared towards graduate students and advanced undergraduates in mathematics and computer science, but may also be of interest to researchers in the area.

Besides the classical methods of symmetric and private key encryption, the book treats the mathematics of cryptographic protocols and several unique topics such as

• Group-Based Cryptography
• Gröbner Basis Methods in Cryptography
• Lattice-Based Cryptography

1. Cover
2. Title
4. Preface
5. Table of content
6. 1&#8195;&#8195;Basic Ideas of Cryptography
7. 2&#8195;&#8195;Symmetric Key Cryptosystems
8. 3&#8195;&#8195;Cryptanalysis and Complexity
1. 3.1   Cryptanalysis and Cryptanalytic Attacks
2. 3.2   Statistical Methods
3. 3.3   Cryptographic Security
4. 3.3.1     Security Proofs
5. 3.4   Perfect Security and the One-Time Pad
6. 3.5   Complexity of Algorithms
7. 3.6   Exercises
9. 4&#8195;&#8195;Cryptographic Protocols
1. 4.1   Cryptographic Protocols
2. 4.2   Cryptographic Hash Functions
3. 4.3   Authentication Protocols
4. 4.4   Digital Signatures
5. 4.5   Secret Sharing Schemes
6. 4.6   Zero-Knowledge Proofs
7. 4.7   Exercises
10. 5&#8195;&#8195;Elementary Number Theoretic Techniques
11. 6&#8195;&#8195;Some Number Theoretic Algorithms
1. 6.1   Algorithms for Public Key Cryptography
2. 6.2   Quadratic Residues and Square Roots
3. 6.3   Modular Square Roots
4. 6.4   Products of Two Primes
5. 6.5   The Discrete Log Problem
6. 6.6   Primality Testing
7. 6.7   Exercises
12. 7&#8195;&#8195;Public Key Cryptography
1. 7.1   Public Key Cryptography
2. 7.2   Standard Model for Public Key Encryption
3. 7.3   The Diffie-Hellman Key Exchange and Protocol
4. 7.4   ElGamal Encryption
5. 7.5   The RSA Algorithm and Protocol
6. 7.6   Rabin Encryption
7. 7.7   Session Keys and Mixed Encryption
8. 7.8   The RSA Signature Method
9. 7.9   Exercises
13. 8&#8195;&#8195;Elliptic Curve Cryptography
1. 8.1   The ElGamal and Elliptic Curve Encryption System
2. 8.2   Elliptic Curves
3. 8.3   Elliptic Curve Cryptography
4. 8.4   Cryptoanalysis of Elliptic Curve Cryptosystems
5. 8.5   The MOV-Algorithm
6. 8.6   The Elliptic Curve Digital Signature
7. 8.7   Exercises
14. 9&#8195;&#8195;Basic Concepts from Group Theory
15. 10&#8195;&#160;Group Based Cryptography
1. 10.1      Group Based Methods
2. 10.2      The Magnus Method
3. 10.3      Free Group Cryptosystems
4. 10.4    Cryptographic Protocols Using Groups
5. 10.5    Non-Abelian Digital Signatures
7. 10.7    A Secret Sharing Scheme
8. 10.8    Exercises
16. 11&#8195;&#160;Braid Group Cryptography
1. 11.1    Cryptographic Platforms and Platform Groups
2. 11.2    The Ko-Lee and AAG Protocols
3. 11.3    Some Other Group Based Cryptosystems
4. 11.4    The Shamir Three-Pass
5. 11.5    Hard Group Theoretic Properties
6. 11.6    Braid Group Cryptography
7. 11.7    The Braid Groups
8. 11.8    Cryptanalysis of Braid Group Cryptosystems
9. 11.9    Some Other Braid Group Based Protocols
10. 11.10    Exercises
17. 12&#8195;&#160;Further Applications
18. 13&#8195;&#160;Commutative Grobner Basis Methods
1. 13.1    Commutative Grobner Bases
2. 13.2    Commutative Grobner Basis Cryptosystems
3. 13.3    Algebraic Attacks Using Grobner Bases
4. 13.4    Exercises
19. 14&#8195;&#160;Non-Commutative Grobner Basis Methods
20. 15&#8195;&#160;Lattice-Based Cryptography
1. 15.1    Lattice-Based Cryptography
2. 15.2    General Cryptoprimitives
3. 15.3    Lattices and Their Properties
4. 15.4    Hard Lattice Problems
5. 15.5    Lattice Reduction and Babai’s Algorithm
6. 15.6    Main Lattice Based Cryptosystems
7. 15.7    Security Proofs
8. 15.8    Exercises
21. Bibliography
22. Index

## Product information

• Title: A Course in Mathematical Cryptography
• Author(s): Gilbert Baumslag, Benjamin Fine, Martin Kreuzer, Gerhard Rosenberger
• Release date: June 2015
• Publisher(s): De Gruyter
• ISBN: 9783110386165