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Group-Based Cryptography

Author: Alexei Myasnikov; Alexander Ushakov; Manuel Castellet
Publisher: Basel : Birkhäuser Basel, 2008.
Series: Advanced Courses in Mathematics - CRM Barcelona, Centre de Recerca Matemàtica.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Database:WorldCat
Summary:
This book is about relations between three different areas of mathematics and theoretical computer science: combinatorial group theory, cryptography, and complexity theory. It is explored how non-commutative (infinite) groups, which are typically studied in combinatorial group theory, can be used in public key cryptography. It is also shown that there is a remarkable feedback from cryptography to combinatorial group  Read more...
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Genre/Form: Electronic books
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Alexei Myasnikov; Alexander Ushakov; Manuel Castellet
ISBN: 3764388269 9783764388263
OCLC Number: 990521328
Language Note: English.
Notes: Description based upon print version of record.
Description: 1 online resource (191 p.)
Contents: ""Contents""; ""Preface""; ""Introduction""; ""Part I Background on Groups, Complexity, and Cryptography""; ""Chapter 1 Background on Public Key Cryptography""; ""1.1 From key establishment to encryption""; ""1.2 The Diffie-Hellman key establishment""; ""1.3 The ElGamal cryptosystem""; ""1.4 Authentication""; ""Chapter 2 Background on Combinatorial Group Theory""; ""2.1 Basic definitions and notation""; ""2.2 Presentations of groups by generators and relators""; ""2.3 Algorithmic problems of group theory""; ""2.4 Nielsen�s and Schreier�s methods""; ""2.5 Tietze�s method"" ""2.6 Normal forms""""Chapter 3 Background on Computational Complexity""; ""3.1 Algorithms""; ""3.2 Computational problems""; ""3.3 The worst case complexity""; ""Part II Non-commutative Cryptography""; ""Chapter 4 Canonical Non-commutative Cryptography""; ""4.1 Protocols based on the conjugacy search problem""; ""4.2 Protocols based on the decomposition problem""; ""4.3 A protocol based on the factorization search problem""; ""4.4 Stickel�s key exchange protocol""; ""4.5 The Anshel-Anshel-Goldfeld protocol""; ""4.6 Authentication protocols based on the conjugacy problem"" ""4.7 Relations between different problems""""Chapter 5 Platform Groups""; ""5.1 Braid groups""; ""5.2 Thompson�s group""; ""5.3 Groups of matrices""; ""5.4 Small cancellation groups""; ""5.5 Solvable groups""; ""5.6 Artin groups""; ""5.7 Grigorchuk�s group""; ""Chapter 6 Using Decision Problems in Public Key Cryptography""; ""6.1 The Shpilrain-Zapata scheme""; ""6.2 Public key encryption and encryption emulation attacks""; ""Part III Generic Complexity and Cryptanalysis""; ""Chapter 7 Distributional Problems and the Average-Case Complexity""; ""7.1 Distributional computational problems"" ""7.2 Average case complexity""""Chapter 8 Generic Case Complexity""; ""8.1 Generic Complexity""; ""8.2 Generic versus average case complexity""; ""Chapter 9 Generic Complexity of NP-complete Problems""; ""9.1 The linear generic time complexity of subset sum problem""; ""9.2 A practical algorithm for subset sum problem""; ""9.3 3-Satisfiability""; ""Part IV Asymptotically Dominant Properties and Cryptanalysis""; ""Chapter 10 Asymptotically Dominant Properties""; ""10.1 A brief description""; ""10.2 Random subgroups and generating tuples""; ""10.3 Asymptotic properties of subgroups"" ""10.4 Groups with generic free basis property""""10.5 Quasi-isometrically embedded subgroups""; ""Chapter 11 Length-Based and Quotient Attacks""; ""11.1 Anshel-Anshel-Goldfeld scheme""; ""11.2 Length-based attacks""; ""11.3 Computing the geodesic length in a subgroup""; ""11.4 Quotient attacks""; ""Bibliography""; ""Abbreviations and Notation""; ""Index""
Series Title: Advanced Courses in Mathematics - CRM Barcelona, Centre de Recerca Matemàtica.
Other Titles: Advanced Courses in Mathematics - CRM Barcelona

Abstract:

This book is about relations between three different areas of mathematics and theoretical computer science: combinatorial group theory, cryptography, and complexity theory. It is explored how non-commutative (infinite) groups, which are typically studied in combinatorial group theory, can be used in public key cryptography. It is also shown that there is a remarkable feedback from cryptography to combinatorial group theory because some of the problems motivated by cryptography appear to be new to group theory, and they open many interesting research avenues within group theory. Then, complexity theory, notably generic-case complexity of algorithms, is employed for cryptanalysis of various cryptographic protocols based on infinite groups, and the ideas and machinery from the theory of generic-case complexity are used to study asymptotically dominant properties of some infinite groups that have been applied in public key cryptography so far. Its elementary exposition makes the book accessible to graduate as well as undergraduate students in mathematics or computer science.

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