# Non-commutative cryptography and complexity of group-theoretic problems

- Responsibility
- Alexei Myasnikov, Vladimir Shpilrain, Alexander Ushakov ; with an appendix by Natalia Mosina.
- Language
- English.
- Imprint
- Providence, R.I. : American Mathematical Society, c2011.
- Physical description
- xiv, 385 p. : ill. ; 26 cm.
- Series
- Mathematical surveys and monographs ; no. 177.

## Access

## Creators/Contributors

- Author/Creator
- Myasnikov, Alexei G., 1955-
- Contributor
- Shpilrain, Vladimir, 1960-
- Ushakov, Alexander.
- Mosina, Natalia.

## Contents/Summary

- Bibliography
- Includes bibliographical references and index.
- Publisher's Summary
- 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.

(source: Nielsen Book Data)

## Subjects

- Subject
- Combinatorial group theory.
- Cryptography.
- Computer algorithms.
- Number theory.
- Information and communication, circuits -- Communication, information -- Cryptography.
- Group theory and generalizations -- Special aspects of infinite or finite groups -- Word problems, other decision problems, connections with logic and automata.
- Computer science -- Theory of computing -- Analysis of algorithms and problem complexity.
- Information and communication, circuits -- Communication, information -- Authentication and secret sharing.
- Number theory -- Finite fields and commutative rings (number-theoretic aspects) -- Algebraic coding theory; cryptography.

## Bibliographic information

- Publication date
- 2011
- Series
- Mathematical surveys and monographs ; v. 177
- ISBN
- 0821853600 (alk. paper)
- 9780821853603 (alk. paper)