**Mathematical Surveys and Monographs**

Volume: 177;
2011;
385 pp;
Hardcover

MSC: Primary 94; 20; 68; 11;

Print ISBN: 978-0-8218-5360-3

Product Code: SURV/177

List Price: $110.00

Individual Member Price: $88.00

**Electronic ISBN: 978-1-4704-1404-7
Product Code: SURV/177.E**

List Price: $110.00

Individual Member Price: $88.00

#### Supplemental Materials

# Non-commutative Cryptography and Complexity of Group-theoretic Problems

Share this page
*Alexei Myasnikov; Vladimir Shpilrain; Alexander Ushakov*

with an Appendix by Natalia Mosina

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.

#### Table of Contents

# Table of Contents

## Non-commutative Cryptography and Complexity of Group-theoretic Problems

- Cover Cover11 free
- Title page i2 free
- Contents v6 free
- Preface xiii14 free
- Introduction 116 free
- Part I. Background on groups, complexity, and cryptography 520 free
- Background on public key cryptography 722
- Background on combinatorial group theory 1328
- Background on computational complexity 2540
- Part II. Non-commutative cryptography 3954
- Canonical non-commutative cryptography 4156
- Platform groups 5570
- More protocols 7388
- Using decision problems in public key cryptography 7994
- Authentication 91106
- Part III. Generic complexity and cryptanalysis 113128
- Distributional problems and the average case complexity 117132
- Generic case complexity 131146
- Generic complexity of NP-complete problems 141156
- Generic complexity of undecidable problems 147162
- Strongly, super, and absolutely undecidable problems 155170
- Part IV. Asymptotically dominant properties and cryptanalysis 167182
- Asymptotically dominant properties 171186
- Length based and quotient attacks 179194
- Part V. Word and conjugacy search problems in groups 197212
- Word search problem 199214
- Conjugacy search problem 247262
- Part VI. Word problem in some special classes of groups 283298
- Free solvable groups 285300
- Compressed words 309324
- Probabilistic group-based cryptanalysis 325340
- Bibliography 369384
- Abbreviations and notation 381396
- Index 383398 free
- Back Cover Back Cover1402

#### Readership

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

#### 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