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

Author: Alexei G. Myasnikov

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 402

ISBN-13: 0821853600

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Examines the relationship between three different areas of mathematics and theoretical computer science: combinatorial group theory, cryptography, and complexity theory. It explores how non-commutative (infinite) groups 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.

Foundations of Free Noncommutative Function Theory
Foundations of Free Noncommutative Function Theory

Author: Dmitry S. Kaliuzhnyi-Verbovetskyi

Publisher: American Mathematical Soc.

Published: 2014-11-19

Total Pages: 194

ISBN-13: 1470416972

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In this book the authors develop a theory of free noncommutative functions, in both algebraic and analytic settings. Such functions are defined as mappings from square matrices of all sizes over a module (in particular, a vector space) to square matrices over another module, which respect the size, direct sums, and similarities of matrices. Examples include, but are not limited to, noncommutative polynomials, power series, and rational expressions. Motivation and inspiration for using the theory of free noncommutative functions often comes from free probability. An important application area is "dimensionless" matrix inequalities; these arise, e.g., in various optimization problems of system engineering. Among other related areas are those of polynomial identities in rings, formal languages and finite automata, quasideterminants, noncommutative symmetric functions, operator spaces and operator algebras, and quantum control.

Complexity and Randomness in Group Theory
Complexity and Randomness in Group Theory

Author: Frédérique Bassino

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2020-06-08

Total Pages: 386

ISBN-13: 3110667029

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This book shows new directions in group theory motivated by computer science. It reflects the transition from geometric group theory to group theory of the 21st century that has strong connections to computer science. Now that geometric group theory is drifting further and further away from group theory to geometry, it is natural to look for new tools and new directions in group theory which are present.

Computational and Combinatorial Group Theory and Cryptography
Computational and Combinatorial Group Theory and Cryptography

Author: Benjamin Fine

Publisher: American Mathematical Soc.

Published: 2012

Total Pages: 210

ISBN-13: 0821875639

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This volume contains the proceedings of the AMS Special Session on Computational Algebra, Groups, and Applications, held April 30-May 1, 2011, at the University of Nevada, Las Vegas, Nevada, and the AMS Special Session on the Mathematical Aspects of Cryptography and Cyber Security, held September 10-11, 2011, at Cornell University, Ithaca, New York. Over the past twenty years combinatorial and infinite group theory has been energized by three developments: the emergence of geometric and asymptotic group theory, the development of algebraic geometry over groups leading to the solution of the Tarski problems, and the development of group-based cryptography. These three areas in turn have had an impact on computational algebra and complexity theory. The papers in this volume, both survey and research, exhibit the tremendous vitality that is at the heart of group theory in the beginning of the twenty-first century as well as the diversity of interests in the field.

The Compressed Word Problem for Groups
The Compressed Word Problem for Groups

Author: Markus Lohrey

Publisher: Springer Science & Business Media

Published: 2014-04-04

Total Pages: 193

ISBN-13: 1493907484

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The Compressed Word Problem for Groups provides a detailed exposition of known results on the compressed word problem, emphasizing efficient algorithms for the compressed word problem in various groups. The author presents the necessary background along with the most recent results on the compressed word problem to create a cohesive self-contained book accessible to computer scientists as well as mathematicians. Readers will quickly reach the frontier of current research which makes the book especially appealing for students looking for a currently active research topic at the intersection of group theory and computer science. The word problem introduced in 1910 by Max Dehn is one of the most important decision problems in group theory. For many groups, highly efficient algorithms for the word problem exist. In recent years, a new technique based on data compression for providing more efficient algorithms for word problems, has been developed, by representing long words over group generators in a compressed form using a straight-line program. Algorithmic techniques used for manipulating compressed words has shown that the compressed word problem can be solved in polynomial time for a large class of groups such as free groups, graph groups and nilpotent groups. These results have important implications for algorithmic questions related to automorphism groups.

Finitely Presented Groups
Finitely Presented Groups

Author: Volker Diekert

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2024-10-07

Total Pages: 322

ISBN-13: 3111474275

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This book contains surveys and research articles on the state-of-the-art in finitely presented groups for researchers and graduate students. Overviews of current trends in exponential groups and of the classification of finite triangle groups and finite generalized tetrahedron groups are complemented by new results on a conjecture of Rosenberger and an approximation theorem. A special emphasis is on algorithmic techniques and their complexity, both for finitely generated groups and for finite Z-algebras, including explicit computer calculations highlighting important classical methods. A further chapter surveys connections to mathematical logic, in particular to universal theories of various classes of groups, and contains new results on countable elementary free groups. Applications to cryptography include overviews of techniques based on representations of p-groups and of non-commutative group actions. Further applications of finitely generated groups to topology and artificial intelligence complete the volume. All in all, leading experts provide up-to-date overviews and current trends in combinatorial group theory and its connections to cryptography and other areas.

Computational Methods and Mathematical Modeling in Cyberphysics and Engineering Applications 1
Computational Methods and Mathematical Modeling in Cyberphysics and Engineering Applications 1

Author: Dmitri Koroliouk

Publisher: John Wiley & Sons

Published: 2024-04-16

Total Pages: 452

ISBN-13: 1394284330

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Mathematical methods in engineering are characterized by a wide range of techniques for approaching various problems. Moreover, completely different analysis techniques can be applied to the same problem, which is justified by the difference in specific applications. Therefore, the study of the analyses and solutions of specific problems leads the researcher to generate their own techniques for the analysis of similar problems continuously arising in the process of technical development. Computational Methods and Mathematical Modeling in Cyberphysics and Engineering Applications contains solutions to specific problems in current areas of computational engineering and cyberphysics.

Trusted Computing and Information Security
Trusted Computing and Information Security

Author: Huanguo Zhang

Publisher: Springer

Published: 2019-01-08

Total Pages: 420

ISBN-13: 981135913X

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This book constitutes the refereed proceedings of the Chinese Conference on Trusted Computing and Information Security, CTCIS 2018, held in Wuhan, China, in October 2018. The 24 revised full papers presented were carefully reviewed and selected from 73 submissions. The papers are centered around cryptography, systems security, trusted computing, information security, and network security.