Interactions between Group Theory, Symmetry and Cryptology

Interactions between Group Theory, Symmetry and Cryptology PDF Author: María Isabel González Vasco
Publisher: MDPI
ISBN: 3039288024
Category : Mathematics
Languages : en
Pages : 164

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Book Description
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.

Interactions between Group Theory, Symmetry and Cryptology

Interactions between Group Theory, Symmetry and Cryptology PDF Author: María Isabel González Vasco
Publisher: MDPI
ISBN: 3039288024
Category : Mathematics
Languages : en
Pages : 164

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Book Description
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.

Exploring the Infinite Possibilities

Exploring the Infinite Possibilities PDF Author: Barrett Williams
Publisher: Barrett Williams
ISBN:
Category : Mathematics
Languages : en
Pages : 113

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Book Description
**Exploring the Infinite Possibilities Unlock the Mysteries of Mathematics** Dive into a world where numbers weave patterns of breathtaking beauty and equations reveal the secrets of the universe. "Exploring the Infinite Possibilities" is a captivating journey through the landscape of mathematics, offering a fresh and insightful perspective on a subject often shrouded in mystery and complexity. Begin your exploration with the aesthetic allure of mathematics, where the harmony of numbers and the elegance of patterns and symmetry spark a sense of wonder. Venture into the rich history of mathematical thought, tracing its evolution from ancient civilizations, through the intellectual fervor of the Renaissance, to the innovations that define modern mathematics today. Discover the boundless nature of infinity, uncover the mysteries of fractals and chaos theory, and delve into the intriguing realm of transfinite numbers. Wander through the natural world, where the Fibonacci sequence and the Golden Ratio manifest in mesmerizing forms and patterns, and explore the symmetrical beauty inherent in biological structures. Unravel the intricacies of mathematical proofs, from historical breakthroughs to contemporary challenges that drive mathematical discovery. Appreciate mathematics as a universal language, bridging the gap between the abstract and the tangible, and see its unifying power in science. From the elegance of Euclidean geometry to the peculiarities of non-Euclidean spaces, geometric concepts open the door to endless possibilities. Explore the hidden symmetries in abstract algebra, the enigmatic nature of prime numbers, and the profound impacts of calculus—the mathematics of change. Venture into the realms of mathematical analysis, probability, and statistics, uncovering the profound insights these fields offer into our world. Engage with the foundations of mathematical logic and embark on a journey through the digital age, where algorithms and machine learning reshape our lives. "Exploring the Infinite Possibilities" is not just a book—it's an inspiring odyssey into a vibrant mathematical universe. Whether you're a curious enthusiast or a seasoned mathematician, this book invites you to continue the great journey of mathematical exploration, inspiring future generations and highlighting the global impact of mathematics.

Abstract Algebra: Group Theory

Abstract Algebra: Group Theory PDF Author: N.B. Singh
Publisher: N.B. Singh
ISBN:
Category : Mathematics
Languages : en
Pages : 130

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Book Description


Geometric Methods in Signal and Image Analysis

Geometric Methods in Signal and Image Analysis PDF Author: Hamid Krim
Publisher: Cambridge University Press
ISBN: 110703390X
Category : Computers
Languages : en
Pages : 299

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Book Description
A comprehensive guide to modern geometric methods for signal and image analysis, from basic principles to state-of-the-art concepts and applications.

Theory of Quantum Computation, Communication, and Cryptography

Theory of Quantum Computation, Communication, and Cryptography PDF Author: Kazuo Iwama
Publisher: Springer
ISBN: 3642356567
Category : Computers
Languages : en
Pages : 161

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Book Description
This book constitutes revised selected papers from the 7th Conference on Theory of Quantum Computation, Communication, and Cryptography, TQC 2012, held in Tokyo, Japan, in May 2012. The 12 papers presented were carefully reviewed and selected for inclusion in this book. They contain original research on the rapidly growing, interdisciplinary field of quantum computation, communication and cryptography. Topics addressed are such as quantum algorithms, quantum computation models, quantum complexity theory, simulation of quantum systems, quantum programming languages, quantum cryptography, quantum communication, quantum estimation, quantum measurement, quantum tomography, completely positive maps, decoherence, quantum noise, quantum coding theory, fault-tolerant quantum computing, entanglement theory, and quantum teleportation.

Introduction to Abstract Algebra

Introduction to Abstract Algebra PDF Author: Dr. Kuldeep Singh
Publisher: Navi International Book Publication house
ISBN: 9395745193
Category : Mathematics
Languages : en
Pages : 219

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Book Description
Mathematicians who specialize in abstract algebra study algebraic structures like fields, rings, and groups. Abstract algebra investigates the fundamental ideas and patterns that underpin these procedures, as contrast to elementary algebra, which works with particular equations and operations on numbers. It is a fundamental topic with applications in computer science, cryptography, and physics. It also offers the theoretical basis for many other areas of mathematics. The idea of a group, which is a set with a single operation that meets axioms such as closure, associativity, the presence of an identity element, and the existence of inverses, is one of the fundamental ideas in abstract algebra. A common subject in the study of symmetry and transformations is groups. By adding new operations, including addition and multiplication, and examining their interactions, rings and fields expand on fundamental concepts. By studying abstract algebra, mathematicians may identify patterns and correlations that remain across many systems by moving from concrete numbers to more generalized things. This abstraction makes it possible to comprehend mathematical structures more deeply and inspires the creation of new ideas and instruments. As a field of study, abstract algebra serves as a doorway to more complicated mathematical analysis and as a potent language for characterizing intricate systems across a range of scientific fields. The importance of abstract algebra is not limited to mathematics alone; it also affects other practical disciplines. For example, in computer science, knowledge of abstract algebraic structures is essential to comprehending data structures, algorithms, and cryptographic systems. Group theory and field theory ideas play a major role in cryptography, which protects digital communications, in the creation and cracking of encryption systems. Similar to this, group theory's description of symmetry operations in physics aids in the explanation of key ideas in relativity and quantum mechanics. This field's intrinsic abstraction encourages other ways of thinking. It promotes the development of rigorous yet creative problem-solving abilities since it often calls for identifying patterns and generalizations that are not immediately apparent. This ability to think abstractly is useful not just in mathematics but also in other fields like economics, engineering, and biology that study complex systems. Because of its degree of abstraction and divergence from the arithmetic and algebraic intuition acquired in previous mathematics courses, abstract algebra may be difficult to understand in educational settings

Cryptology and Network Security

Cryptology and Network Security PDF Author: Juan A. Garay
Publisher: Springer
ISBN: 3642104339
Category : Computers
Languages : en
Pages : 551

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Book Description
This book constitutes the refereed proceedings of the 8th International Conference on Cryptology and Network Security, CANS 2009, held in Kanazawa, Japan, in December 2009. The 32 revised full papers presented together with 3 invited talks were carefully reviewed and selected from 109 submissions. The papers are organized in topical sections on cryptographic protocols and schemes; cryptanalysis; wireless and sensor security; network security; privacy and anonymity; functional and searchable encryption; authentication; block cipher design; and algebraic and number-theoretic schemes.

High Energy Physics

High Energy Physics PDF Author: Glenn Cunningham
Publisher: Scientific e-Resources
ISBN: 1839473851
Category :
Languages : en
Pages : 316

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Book Description
Particle physics (also high energy physics) is the branch of physics that studies the nature of the particles that constitute matter and radiation. Although the word "e;particle"e; can refer to various types of very small objects "e;particle physics"e; usually investigates the irreducibly smallest detectable particles and the fundamental interactions necessary to explain their behaviour. By our current understanding, these elementary particles are excitations of the quantum fields that also govern their interactions. The currently dominant theory explaining these fundamental particles and fields, along with their dynamics, is called the Standard Model. Thus, modern particle physics generally investigates the Standard Model and its various possible extensions, e.g. to the newest "e;known"e; particle, the Higgs boson, or even to the oldest known force field, gravity. Written in a clear pedagogic style by active researchers, this book will prepare a beginner to work in the field and at the same time will also provide useful reference material for active researchers.

Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry: Volume 1

Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry: Volume 1 PDF Author: Raf Cluckers
Publisher: Cambridge University Press
ISBN: 1139499793
Category : Mathematics
Languages : en
Pages : 347

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Book Description
Assembles different theories of motivic integration for the first time, providing all of the necessary background for graduate students and researchers from algebraic geometry, model theory and number theory. In a rapidly-evolving area of research, this volume and Volume 2, which unite the several viewpoints and applications, will prove invaluable.

Discrete Algebraic Methods

Discrete Algebraic Methods PDF Author: Volker Diekert
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110413337
Category : Mathematics
Languages : en
Pages : 354

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Book Description
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