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Non-commutative Cryptography and Complexity of Group-theoretic Problems

Alexei Myasnikov Stevens Institute of Technology, Hoboken, NJ
Vladimir Shpilrain City College of New York, New York, NY
Alexander Ushakov Stevens Institute of Technology, Hoboken, NJ
with an Appendix by Natalia Mosina
Available Formats:
Hardcover ISBN: 978-0-8218-5360-3
Product Code: SURV/177
List Price: $117.00 MAA Member Price:$105.30
AMS Member Price: $93.60 Electronic ISBN: 978-1-4704-1404-7 Product Code: SURV/177.E List Price:$110.00
MAA Member Price: $99.00 AMS Member Price:$88.00
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AMS Member Price: $140.40 Click above image for expanded view Non-commutative Cryptography and Complexity of Group-theoretic Problems Alexei Myasnikov Stevens Institute of Technology, Hoboken, NJ Vladimir Shpilrain City College of New York, New York, NY Alexander Ushakov Stevens Institute of Technology, Hoboken, NJ with an Appendix by Natalia Mosina Available Formats:  Hardcover ISBN: 978-0-8218-5360-3 Product Code: SURV/177  List Price:$117.00 MAA Member Price: $105.30 AMS Member Price:$93.60
 Electronic ISBN: 978-1-4704-1404-7 Product Code: SURV/177.E
 List Price: $110.00 MAA Member Price:$99.00 AMS Member Price: $88.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$175.50 MAA Member Price: $157.95 AMS Member Price:$140.40
• Book Details

Mathematical Surveys and Monographs
Volume: 1772011; 385 pp
MSC: Primary 94; 20; 68; 11;

This book is about relations between three different areas of mathematics and theoretical computer science: combinatorial group theory, cryptography, and complexity theory. It explores how non-commutative (infinite) groups, which are typically studied in combinatorial group theory, can be used in public-key cryptography. It also shows that there is 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.

In particular, a lot of emphasis in the book is put on studying search problems, as compared to decision problems traditionally studied in combinatorial 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.

This book also describes new interesting developments in the algorithmic theory of solvable groups and another spectacular new development related to complexity of group-theoretic problems, which is based on the ideas of compressed words and straight-line programs coming from computer science.

Graduate students and research mathematicians interested in the relations between group theory, cryptography, and complexity theory.

• Chapters
• Introduction
• Part 1. Background on groups, complexity, and cryptography
• 1. Background on public key cryptography
• 2. Background on combinatorial group theory
• 3. Background on computational complexity
• Part 2. Non-commutative cryptography
• 4. Canonical non-commutative cryptography
• 5. Platform groups
• 6. More protocols
• 7. Using decision problems in public key cryptography
• 8. Authentication
• Part 3. Generic complexity and cryptanalysis
• 9. Distributional problems and the average case complexity
• 10. Generic case complexity
• 11. Generic complexity of NP-complete problems
• 12. Generic complexity of undecidable problems
• 13. Strongly, super, and absolutely undecidable problems
• Part 4. Asymptotically dominant properties and cryptanalysis
• 14. Asymptotically dominant properties
• 15. Length based and quotient attacks
• Part 5. Word and conjugacy search problems in groups
• 16. Word search problem
• 17. Conjugacy search problem
• Part 6. Word problem in some special classes of groups
• 18. Free solvable groups
• 19. Compressed words
• Appendix A. Probabilistic group-based cryptanalysis

• Reviews

• The world of cryptography is evolving; new improvements constantly open new opportunities in public-key cryptography. Cryptography inspires new group-theoretic problems and leads to important new ideas. The book includes exciting new improvements in the algorithmic theory of solvable groups. Another exceptional new development is the authors' analysis of the complexity of group-theoretic problems.

MAA Reviews
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Volume: 1772011; 385 pp
MSC: Primary 94; 20; 68; 11;

This book is about relations between three different areas of mathematics and theoretical computer science: combinatorial group theory, cryptography, and complexity theory. It explores how non-commutative (infinite) groups, which are typically studied in combinatorial group theory, can be used in public-key cryptography. It also shows that there is 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.

In particular, a lot of emphasis in the book is put on studying search problems, as compared to decision problems traditionally studied in combinatorial 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.

This book also describes new interesting developments in the algorithmic theory of solvable groups and another spectacular new development related to complexity of group-theoretic problems, which is based on the ideas of compressed words and straight-line programs coming from computer science.

Graduate students and research mathematicians interested in the relations between group theory, cryptography, and complexity theory.

• Chapters
• Introduction
• Part 1. Background on groups, complexity, and cryptography
• 1. Background on public key cryptography
• 2. Background on combinatorial group theory
• 3. Background on computational complexity
• Part 2. Non-commutative cryptography
• 4. Canonical non-commutative cryptography
• 5. Platform groups
• 6. More protocols
• 7. Using decision problems in public key cryptography
• 8. Authentication
• Part 3. Generic complexity and cryptanalysis
• 9. Distributional problems and the average case complexity
• 10. Generic case complexity
• 11. Generic complexity of NP-complete problems
• 12. Generic complexity of undecidable problems
• 13. Strongly, super, and absolutely undecidable problems
• Part 4. Asymptotically dominant properties and cryptanalysis
• 14. Asymptotically dominant properties
• 15. Length based and quotient attacks
• Part 5. Word and conjugacy search problems in groups
• 16. Word search problem
• 17. Conjugacy search problem
• Part 6. Word problem in some special classes of groups
• 18. Free solvable groups
• 19. Compressed words
• Appendix A. Probabilistic group-based cryptanalysis
• The world of cryptography is evolving; new improvements constantly open new opportunities in public-key cryptography. Cryptography inspires new group-theoretic problems and leads to important new ideas. The book includes exciting new improvements in the algorithmic theory of solvable groups. Another exceptional new development is the authors' analysis of the complexity of group-theoretic problems.

MAA Reviews
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