Author: Maria Isabel Gonzalez VascoPublish On: 2015-04-01

Group theoretic problems have propelled scientific achievements across a wide range of fields, including mathematics, physics, chemistry, and the life sciences.

Author: Maria Isabel Gonzalez Vasco

Publisher: CRC Press

ISBN: 9781584888376

Category: Computers

Page: 244

View: 470

Group theoretic problems have propelled scientific achievements across a wide range of fields, including mathematics, physics, chemistry, and the life sciences. Many cryptographic constructions exploit the computational hardness of group theoretical problems, and the area is viewed as a potential source of quantum-resilient cryptographic primitives

This book is about relations between three different areas of mathematics and theoretical computer science: combinatorial group theory, cryptography, and complexity theory.

Author: Alexei Myasnikov

Publisher: Springer Science & Business Media

ISBN: 9783764388263

Category: Mathematics

Page: 183

View: 664

This book is about relations between three different areas of mathematics and theoretical computer science: combinatorial group theory, cryptography, and complexity theory. It is explored how non-commutative (infinite) groups, which are typically studied in combinatorial group theory, can be used in public key cryptography. It is also shown that there is a 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. 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. Its elementary exposition makes the book accessible to graduate as well as undergraduate students in mathematics or computer science.

This book is about relations between three different areas of mathematics and theoretical computer science: combinatorial group theory, cryptography, and complexity theory.

Author: Alexei G. Myasnikov

Publisher: American Mathematical Soc.

ISBN: 9780821853603

Category: Mathematics

Page: 385

View: 157

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.

Springer 2004. MR2075209(2005f:94084) 3. B. Fine, M. Habeeb, D. Kahrobaei,
G. Rosenberger, Aspects of Non-Abelian Group Based cryptography: A Survey
and Open Problems, JP Journal of Algebra, Number Theory and Applications,
Vol.

Author: Benjamin Fine

Publisher: American Mathematical Soc.

ISBN: 9780821875636

Category: Mathematics

Page: 199

View: 936

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.

Author: María Isabel González VascoPublish On: 2020-04-22

The protocol is derived from a two-party solution by means of a protocol compiler
presented by Abdalla et al. at TCC 2007, evidencing the possibility of
meaningfully integrating cryptographic and group-theoretic tools in cryptographic
protocol ...

Author: María Isabel González Vasco

Publisher: MDPI

ISBN: 9783039288021

Category: Mathematics

Page: 164

View: 433

Cryptography lies at the heart of most technologies deployed today for secure communications. At the same time, mathematics lies at the heart of cryptography, as cryptographic constructions are based on algebraic scenarios ruled by group or number theoretical laws. Understanding the involved algebraic structures is, thus, essential to design robust cryptographic schemes. This Special Issue is concerned with the interplay between group theory, symmetry and cryptography. The book highlights four exciting areas of research in which these fields intertwine: post-quantum cryptography, coding theory, computational group theory and symmetric cryptography. The articles presented demonstrate the relevance of rigorously analyzing the computational hardness of the mathematical problems used as a base for cryptographic constructions. For instance, decoding problems related to algebraic codes and rewriting problems in non-abelian groups are explored with cryptographic applications in mind. New results on the algebraic properties or symmetric cryptographic tools are also presented, moving ahead in the understanding of their security properties. In addition, post-quantum constructions for digital signatures and key exchange are explored in this Special Issue, exemplifying how (and how not) group theory may be used for developing robust cryptographic tools to withstand quantum attacks.

We start with a cryptographic primitive – the discrete logarithm problem. The
standard reference for public-key cryptography is Hoffstein et. al. [5]. Definition
3.1 (The discrete logarithm problem). Let G = 〈g〉 be a finite cyclic group of
prime ...

Author: Delaram Kahrobaei

Publisher: American Mathematical Soc.

ISBN: 9780821898598

Category: Mathematics

Page: 123

View: 622

This volume contains the proceedings of the AMS Special Session on Algorithmic Problems of Group Theory and Their Complexity, held January 9-10, 2013 in San Diego, CA and the AMS Special Session on Algorithmic Problems of Group Theory and Applications to Information Security, held April 6-7, 2013 at Boston College, Chestnut Hill, MA. Over the past few years the field of group-based cryptography has attracted attention from both group theorists and cryptographers. The new techniques inspired by algorithmic problems in non-commutative group theory and their complexity have offered promising ideas for developing new cryptographic protocols. The papers in this volume cover algorithmic group theory and applications to cryptography.

This book is concerned with the mathematical, especially algebraic, aspects of cryptography.

Author: Gilbert Baumslag

Publisher: Walter de Gruyter GmbH & Co KG

ISBN: 9783110386165

Category: Computers

Page: 389

View: 225

The subject of this book is mathematical cryptography. By this we mean the mathematics involved in cryptographic protocols. As the field has expanded, using both commutative and noncommutative algebraic objects as cryptographic platforms, a book describing and explaining all these mathematical methods is of immeasurable value.

This unique text provides a geometric approach to group theory and linear algebra, bringing to light the interesting ways in which these subjects interact.

Author: James B. Carrell

Publisher: Springer

ISBN: 9780387794280

Category: Mathematics

Page: 410

View: 503

This unique text provides a geometric approach to group theory and linear algebra, bringing to light the interesting ways in which these subjects interact. Requiring few prerequisites beyond understanding the notion of a proof, the text aims to give students a strong foundation in both geometry and algebra. Starting with preliminaries (relations, elementary combinatorics, and induction), the book then proceeds to the core topics: the elements of the theory of groups and fields (Lagrange's Theorem, cosets, the complex numbers and the prime fields), matrix theory and matrix groups, determinants, vector spaces, linear mappings, eigentheory and diagonalization, Jordan decomposition and normal form, normal matrices, and quadratic forms. The final two chapters consist of a more intensive look at group theory, emphasizing orbit stabilizer methods, and an introduction to linear algebraic groups, which enriches the notion of a matrix group. Applications involving symm etry groups, determinants, linear coding theory and cryptography are interwoven throughout. Each section ends with ample practice problems assisting the reader to better understand the material. Some of the applications are illustrated in the chapter appendices. The author's unique melding of topics evolved from a two semester course that he taught at the University of British Columbia consisting of an undergraduate honors course on abstract linear algebra and a similar course on the theory of groups. The combined content from both makes this rare text ideal for a year-long course, covering more material than most linear algebra texts. It is also optimal for independent study and as a supplementary text for various professional applications. Advanced undergraduate or graduate students in mathematics, physics, computer science and engineering will find this book both useful and enjoyable.

Author: Samuel S. Wagstaff, Jr.Publish On: 2002-12-10

First, the book provides the mathematical background needed in cryptography as well as definitions and simple examples from cryptography.

Author: Samuel S. Wagstaff, Jr.

Publisher: CRC Press

ISBN: 1584881534

Category: Mathematics

Page: 336

View: 450

At the heart of modern cryptographic algorithms lies computational number theory. Whether you're encrypting or decrypting ciphers, a solid background in number theory is essential for success. Written by a number theorist and practicing cryptographer, Cryptanalysis of Number Theoretic Ciphers takes you from basic number theory to the inner workings of ciphers and protocols. First, the book provides the mathematical background needed in cryptography as well as definitions and simple examples from cryptography. It includes summaries of elementary number theory and group theory, as well as common methods of finding or constructing large random primes, factoring large integers, and computing discrete logarithms. Next, it describes a selection of cryptographic algorithms, most of which use number theory. Finally, the book presents methods of attack on the cryptographic algorithms and assesses their effectiveness. For each attack method the author lists the systems it applies to and tells how they may be broken with it. Computational number theorists are some of the most successful cryptanalysts against public key systems. Cryptanalysis of Number Theoretic Ciphers builds a solid foundation in number theory and shows you how to apply it not only when breaking ciphers, but also when designing ones that are difficult to break.

Essentially , such primitives are derived from hard subgroup membership
problems of suitable abelian groups . On the other hand , group theory has lately
attracted a lot of attention as a potential source of cryptographic primitives .
Having in ...

The main tool used in algorithms is Fourier sampling, i.e. computing the Fourier
transform and measuring, and its nice group theoretic properties lead to the
solution of the HSP when the underlying group is finite and abelian. However ...

Author: Daniel J. Bernstein

Publisher: Springer Science & Business Media

ISBN: 9783540887027

Category: Mathematics

Page: 246

View: 732

Quantum computers will break today's most popular public-key cryptographic systems, including RSA, DSA, and ECDSA. This book introduces the reader to the next generation of cryptographic algorithms, the systems that resist quantum-computer attacks: in particular, post-quantum public-key encryption systems and post-quantum public-key signature systems. Leading experts have joined forces for the first time to explain the state of the art in quantum computing, hash-based cryptography, code-based cryptography, lattice-based cryptography, and multivariate cryptography. Mathematical foundations and implementation issues are included. This book is an essential resource for students and researchers who want to contribute to the field of post-quantum cryptography.

INFORMATION THEORY WITHOUT THE FINITENESS ASSUMPTION, III: DATA
COMPRESSION AND CODES WHOSE RATES EXCEED UNITY G. R. Blakley ...
The group-theoretic viewpoint helps explain the nature of Gray codes. This paper
...

Author: Henry Beker

Publisher: Oxford University Press, USA

ISBN: UCSC:32106008644418

Category: Literary Collections

Page: 297

View: 712

This new title is based on papers presented at The Conference on Cryptography and Coding, the first such conference to bring together mathematicians working in both cryptography and coding theory. It is a "hands-on" volume that provides a state-of-the-art account of mathematical research in these closely inter-related areas, one that will enable readers to produce and store information efficiently, securely, and cost-effectively.

Note that extractors for this family have been used for other applications than
privacy amplification ( LLTT05 ) . We give two new constructions for strong
extractors for the family Th ( k ) . The first construction is based on special group - theoretic ...

Assuming the covering X0 (p") —> X0 (pf) is Galois Of order p"”', we also see that
its Galois group is F0 (pT)/AT However, the ... All group theoretic arguments used
before are then still valid for the reductions, as long as the field Of definition ...

Author: Gilles Lachaud

Publisher: American Mathematical Soc.

ISBN: 9780821847169

Category: Mathematics

Page: 206

View: 199

This volume contains the proceedings of the 11th conference on $\mathrm{AGC^{2}T}$, held in Marseille, France in November 2007. There are 12 original research articles covering asymptotic properties of global fields, arithmetic properties of curves and higher dimensional varieties, and applications to codes and cryptography. This volume also contains a survey article on applications of finite fields by J.-P. Serre. $\mathrm{AGC^{2}T}$ conferences take place in Marseille, France every 2 years. These international conferences have been a major event in the area of applied arithmetic geometry for more than 20 years.

The discrete logarithm problem involves an area of abstract algebra known as group theory . A group is actually a sophisticated and somewhat generalized
concept that transcends familiar elementary school arithmetic . However , for our
...

Author: Peter Thorsteinson

Publisher: Prentice Hall Professional

ISBN: 013100851X

Category: Computers

Page: 466

View: 385

Learn how to make your .NET applications secure! Security and cryptography, while always an essential part of the computing industry, have seen their importance increase greatly in the last several years. Microsoft's .NET Framework provides developers with a powerful new set of tools to make their applications secure. NET Security and Cryptography is a practical and comprehensive guide to implementing both the security and the cryptography features found in the .NET platform. The authors provide numerous clear and focused examples in both C# and Visual Basic .NET, as well as detailed commentary on how the code works. They cover topics in a logical sequence and context, where they are most relevant and most easily understood. All of the sample code is available online at . This book will allow developers to: Develop a solid basis in the theory of cryptography, so they can understand how the security tools in the .NET Framework function Learn to use symmetric algorithms, asymmetric algorithms, and digital signatures Master both traditional encryption programming as well as the new techniques of XML encryption and XML signatures Learn how these tools apply to ASP.NET and Web Services security

Supplementary. Algorithmic. Number. Theory. For the cryptographic constructions
given in this book to be efficient (i.e., to run in ... themselves are quite clever, and
an analysis of their performance may rely on non-trivial group-theoretic results.

Author: Jonathan Katz

Publisher: CRC Press

ISBN: 9781420010756

Category: Computers

Page: 552

View: 181

Cryptography plays a key role in ensuring the privacy and integrity of data and the security of computer networks. Introduction to Modern Cryptography provides a rigorous yet accessible treatment of modern cryptography, with a focus on formal definitions, precise assumptions, and rigorous proofs. The authors introduce the core principles of modern cryptography, including the modern, computational approach to security that overcomes the limitations of perfect secrecy. An extensive treatment of private-key encryption and message authentication follows. The authors also illustrate design principles for block ciphers, such as the Data Encryption Standard (DES) and the Advanced Encryption Standard (AES), and present provably secure constructions of block ciphers from lower-level primitives. The second half of the book focuses on public-key cryptography, beginning with a self-contained introduction to the number theory needed to understand the RSA, Diffie-Hellman, El Gamal, and other cryptosystems. After exploring public-key encryption and digital signatures, the book concludes with a discussion of the random oracle model and its applications. Serving as a textbook, a reference, or for self-study, Introduction to Modern Cryptography presents the necessary tools to fully understand this fascinating subject.

This open access book presents selected papers from International Symposium on Mathematics, Quantum Theory, and Cryptography (MQC), which was held on September 25-27, 2019 in Fukuoka, Japan.

Author: Tsuyoshi Takagi

Publisher: Springer Nature

ISBN: 9789811551918

Category: Applied mathematics

Page: 274

View: 777

This open access book presents selected papers from International Symposium on Mathematics, Quantum Theory, and Cryptography (MQC), which was held on September 25-27, 2019 in Fukuoka, Japan. The international symposium MQC addresses the mathematics and quantum theory underlying secure modeling of the post quantum cryptography including e.g. mathematical study of the light-matter interaction models as well as quantum computing. The security of the most widely used RSA cryptosystem is based on the difficulty of factoring large integers. However, in 1994 Shor proposed a quantum polynomial time algorithm for factoring integers, and the RSA cryptosystem is no longer secure in the quantum computing model. This vulnerability has prompted research into post-quantum cryptography using alternative mathematical problems that are secure in the era of quantum computers. In this regard, the National Institute of Standards and Technology (NIST) began to standardize post-quantum cryptography in 2016. This book is suitable for postgraduate students in mathematics and computer science, as well as for experts in industry working on post-quantum cryptography.

The idea behind this book is to provide the mathematical foundations for assessing modern developments in the Information Age.

Author: Volker Diekert

Publisher: Walter de Gruyter GmbH & Co KG

ISBN: 9783110413335

Category: Mathematics

Page: 354

View: 932

The idea behind this book is to provide the mathematical foundations for assessing modern developments in the Information Age. It deepens and complements the basic concepts, but it also considers instructive and more advanced topics. The treatise starts with a general chapter on algebraic structures; this part provides all the necessary knowledge for the rest of the book. The next chapter gives a concise overview of cryptography. Chapter 3 on number theoretic algorithms is important for developping cryptosystems, Chapter 4 presents the deterministic primality test of Agrawal, Kayal, and Saxena. The account to elliptic curves again focuses on cryptographic applications and algorithms. With combinatorics on words and automata theory, the reader is introduced to two areas of theoretical computer science where semigroups play a fundamental role.The last chapter is devoted to combinatorial group theory and its connections to automata. Contents: Algebraic structures Cryptography Number theoretic algorithms Polynomial time primality test Elliptic curves Combinatorics on words Automata Discrete infinite groups

One way of looking at such algorithms is in the context of computational group theory . We have already shown that knowing the order of the group ( Z / NZ ) * ,
for an RSA modulus N , is the same as knowing the prime factors of N . Hence ,
the ...

Author: Nigel Paul Smart

Publisher:

ISBN: UCSD:31822033230921

Category: Computer algorithms

Page: 433

View: 206

Nigel Smartâ¬"s Cryptography provides the rigorous detail required for advanced cryptographic studies, yet approaches the subject matter in an accessible style in order to gently guide new students through difficult mathematical topics.

... Coding Theory, Cryptography and Related Areas, held in Guanajuato, Mexico,
in April 1998 Johannes Buchmann, Tom Hoeholdt, Henning Stichtenoth, Horacio
Tapia-Recillas. On Weierstrass Semigroups and One-point Algebraic Geometry ...

Author: Johannes Buchmann

Publisher: Springer Science & Business Media

ISBN: 3540662480

Category: Computers

Page: 260

View: 777

A series of research papers on various aspects of coding theory, cryptography, and other areas, including new and unpublished results on the subjects. The book will be useful to students, researchers, professionals, and tutors interested in this area of research.