Cover: A Course in Mathematical Cryptography - De Gruyter

A Course in Mathematical Cryptography

An Introduction

Gilbert Baumslag Benjamin Fine Martin Kreuzer Gerhard Rosenberger(Author)

De Gruyter (Publisher)

1st Edition

Published on 16. June 2015

XIII, 376 pages

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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

Reviews / Votes

"The book under review concerns mathematical cryptography: it presents old and new concepts and methods, which clearly show how mathematics is crucial for modern purposes. [...] Surely this book will provide an important service for the community: it will be useful for graduate students who want to be introduced to the subject, for postgraduate students who want to be acquainted with the different aspects of cryptography, and for researchers who want to know recent results in the area."
Patrizia Longobardi in: Mathematical Reviews Clippings MR3558843 94A60 October 2017

"Es werden alle relevanten Themen für eine Lehrveranstaltung besprochen."
Prof. Dr. Jörg Hettel, Markus Höfler, Hochschule Kaiserslautern

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Content

Intro
Contents
Preface
1 Basic Ideas of Cryptography
1.1 Mathematical Cryptography
1.2 Cryptography, Cryptanalysis and Cryptosystems
1.3 A Very Brief History of Cryptography
1.4 Encryption and Number Theory
1.5 Public Key Cryptography
1.6 Cryptosystems and the Key Space
1.7 Cryptographic Protocols
1.8 Exercises
2 Symmetric Key Cryptosystems
2.1 Mixed Encryption
2.2 Block Ciphers
2.3 Stream Ciphers
2.4 Feistel Networks, DES and AES
2.5 One-Way Functions and Trapdoors
2.6 Exercises
3 Cryptanalysis and Complexity
3.1 Cryptanalysis and Cryptanalytic Attacks
3.2 Statistical Methods
3.3 Cryptographic Security
3.3.1 Security Proofs
3.4 Perfect Security and the One-Time Pad
3.4.1 Vigènere Encryption and Polyalphabetic Ciphers
3.4.2 Breaking a Protocol
3.5 Complexity of Algorithms
3.6 Exercises
4 Cryptographic Protocols
4.1 Cryptographic Protocols
4.2 Cryptographic Hash Functions
4.3 Authentication Protocols
4.4 Digital Signatures
4.5 Secret Sharing Schemes
4.5.1 The Shamir Secret Sharing Scheme
4.5.2 Alternatives for Secret Sharing Protocols
4.5.3 Verifying Secret Sharing Protocols (VSS)
4.6 Zero-Knowledge Proofs
4.7 Exercises
5 Elementary Number Theoretic Techniques
5.1 Cryptography and Number Theory
5.2 Modular Arithmetic
5.3 Units and the Multiplicative Group Z*n
5.4 The Field Zp and Finite Fields
5.5 Finite Abelian Groups
5.6 Cyclic Groups and Primitive Elements
5.7 The Chinese Remainder Theorem
5.8 Exercises
6 Some Number Theoretic Algorithms
6.1 Algorithms for Public Key Cryptography
6.2 Quadratic Residues and Square Roots
6.3 Modular Square Roots
6.4 Products of Two Primes
6.5 The Discrete Log Problem
6.5.1 Shank's Baby Step Giant Step Algorithm (BSGS)
6.5.2 Pollard's ?-Algorithm
6.5.3 The Index Calculus Method
6.6 Primality Testing
6.6.1 Sieving Methods
6.6.2 Fermat's Primality Testing
6.6.3 Pseudoprimes and Probabilistic Primality Testing
6.6.4 Miller-Rabin Primality Testing
6.6.5 Mersenne Primes and the Lucas-Lehmer Test
6.7 Exercises
7 Public Key Cryptography
7.1 Public Key Cryptography
7.2 Standard Model for Public Key Encryption
7.3 The Diffie-Hellman Key Exchange and Protocol
7.4 ElGamal Encryption
7.4.1 Generalizations of ElGamal
7.5 The RSA Algorithm and Protocol
7.5.1 The RSA Cryptosystem
7.5.2 RSA as a Block Cipher
7.5.3 Practical Implementation of RSA
7.5.4 Feasibility of the RSA Algorithm
7.5.5 Security of RSA
7.5.6 Cryptanalysis of RSA
7.6 Rabin Encryption
7.6.1 Quadratic residues and Rabin Encryption
7.6.2 The Rabin Cryptosystem
7.6.3 Security Equivalence of the Rabin Cryptosystem
7.7 Session Keys and Mixed Encryption
7.8 The RSA Signature Method
7.9 Exercises
8 Elliptic Curve Cryptography
8.1 The ElGamal and Elliptic Curve Encryption System
8.2 Elliptic Curves
8.2.1 Fields and Field Extensions
8.2.2 Elliptic Curves
8.2.3 Elliptic Curve Groups
8.2.4 The Order of an Elliptic Curve Group
8.2.5 Calculating Points in Elliptic Curve Groups
8.3 Elliptic Curve Cryptography
8.4 Cryptoanalysis of Elliptic Curve Cryptosystems
8.5 The MOV-Algorithm
8.6 The Elliptic Curve Digital Signature
8.7 Exercises
9 Basic Concepts from Group Theory
9.1 Groups and Group Theory
9.2 Cosets and Normal Subgroups
9.3 Examples of Groups
9.4 Generators and Group Presentations
9.5 Free Groups and Group Presentations
9.6 Group Presentations
9.6.1 The Modular Group
9.7 Presentations of Subgroups
9.8 Group Decision Problems
9.9 Group Amalgams
9.10 Exercises
10 Group Based Cryptography
10.1 Group Based Methods
10.2 The Magnus Method
10.2.1 The Wagner-Magyarik Method
10.3 Free Group Cryptosystems
10.3.1 An Implementation Within the Classical Modular Group
10.3.2 A Variation Using the Magnus Representation
10.4 Cryptographic Protocols Using Groups
10.5 Non-Abelian Digital Signatures
10.6 Password Security
10.6.1 The Strong Generic Free Group Property
10.6.2 Security Analysis of the Group Randomizer Protocols
10.6.3 Actual Implementation of a Group Randomizer System Protocol
10.7 A Secret Sharing Scheme
10.8 Exercises
11 Braid Group Cryptography
11.1 Cryptographic Platforms and Platform Groups
11.2 The Ko-Lee and AAG Protocols
11.2.1 The Ko-Lee Protocol
11.2.2 The Anshel-Anshel-Goldfeld Protocol
11.3 Some Other Group Based Cryptosystems
11.4 The Shamir Three-Pass
11.5 Hard Group Theoretic Properties
11.6 Braid Group Cryptography
11.7 The Braid Groups
11.7.1 The Artin Presentation
11.7.2 Normal Forms Within Bn
11.7.3 The Pure Braid Group for Bn
11.7.4 Linear Representations of Bn
11.8 Cryptanalysis of Braid Group Cryptosystems
11.8.1 Attacks on the Conjugacy Search Problem
11.8.2 Length Based Attacks
11.8.3 Representation Theoretic Attacks
11.8.4 Braid Group Security Summary
11.9 Some Other Braid Group Based Protocols
11.10 Exercises
12 Further Applications
12.1 Finitely Presented Groups and Cryptography
12.2 Group Theory for Access Control
12.3 Public Key Control Groups
12.4 Diophantine Control Security groups
12.5 The Social Security Control Groups
12.6 Further Extensions of Diffie-Hellman and RSA
12.7 Exercises
13 Commutative Gröbner Basis Methods
13.1 Commutative Gröbner Bases
13.2 Commutative Gröbner Basis Cryptosystems
13.3 Algebraic Attacks Using Gröbner Bases
13.3.1 The Gröbner Basis Attack
13.3.2 The Integer Programming Attack
13.3.3 The SAT Attack
13.4 Exercises
14 Non-Commutative Gröbner Basis Methods
14.1 Non-Commutative Gröbner Bases
14.2 Elimination and its Applications
14.3 Gröbner Bases of Modules
14.4 Non-Commutative GB Cryptosystems
14.5 Exercises
15 Lattice-Based Cryptography
15.1 Lattice-Based Cryptography
15.2 General Cryptoprimitives
15.3 Lattices and Their Properties
15.3.1 The Geometry of Numbers
15.4 Hard Lattice Problems
15.5 Lattice Reduction and Babai's Algorithm
15.6 Main Lattice Based Cryptosystems
15.6.1 Ajtai's Hash Function and Cryptosystem
15.6.2 The Ajtai-Dwork Cryptosystem
15.6.3 The GGH Cryptosystem
15.6.4 NTRU Cryptosystem
15.7 Security Proofs
15.8 Exercises
Bibliography
Index

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