Non-commutative Cryptography and Complexity of Group-theoretic Problems
About this Title
Alexei Myasnikov, Stevens Institute of Technology, Hoboken, NJ, Vladimir Shpilrain, City College of New York, New York, NY and Alexander Ushakov, Stevens Institute of Technology, Hoboken, NJ
Publication: Mathematical Surveys and Monographs
Publication Year 2011: Volume 177
ISBNs: 978-0-8218-5360-3 (print); 978-1-4704-1404-7 (online)
MathSciNet review: 2850384
MSC: Primary 68-02; Secondary 11T71, 20F10, 68P25, 68Q25, 94-02, 94A60, 94A62
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.
Table of Contents
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
Part 5. Word and conjugacy search problems in groups
Part 6. Word problem in some special classes of groups