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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, NY, and Alexander Ushakov, Stevens Institute of Technology, Hoboken, NJ
with an Appendix by Natalia Mosina

Mathematical Surveys and Monographs
2011; 385 pp; hardcover
Volume: 177
ISBN-10: 0-8218-5360-0
ISBN-13: 978-0-8218-5360-3
List Price: US$110
Member Price: US$88
Order Code: SURV/177
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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.


"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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