| cryptography in linear algebra: An Introduction to Mathematical Cryptography Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman, 2014-09-11 This self-contained introduction to modern cryptography emphasizes the mathematics behind the theory of public key cryptosystems and digital signature schemes. The book focuses on these key topics while developing the mathematical tools needed for the construction and security analysis of diverse cryptosystems. Only basic linear algebra is required of the reader; techniques from algebra, number theory, and probability are introduced and developed as required. This text provides an ideal introduction for mathematics and computer science students to the mathematical foundations of modern cryptography. The book includes an extensive bibliography and index; supplementary materials are available online. The book covers a variety of topics that are considered central to mathematical cryptography. Key topics include: classical cryptographic constructions, such as Diffie–Hellmann key exchange, discrete logarithm-based cryptosystems, the RSA cryptosystem, and digital signatures; fundamental mathematical tools for cryptography, including primality testing, factorization algorithms, probability theory, information theory, and collision algorithms; an in-depth treatment of important cryptographic innovations, such as elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem. The second edition of An Introduction to Mathematical Cryptography includes a significant revision of the material on digital signatures, including an earlier introduction to RSA, Elgamal, and DSA signatures, and new material on lattice-based signatures and rejection sampling. Many sections have been rewritten or expanded for clarity, especially in the chapters on information theory, elliptic curves, and lattices, and the chapter of additional topics has been expanded to include sections on digital cash and homomorphic encryption. Numerous new exercises have been included. |
| cryptography in linear algebra: Algebraic Aspects of Cryptography Neal Koblitz, 2012-12-06 From the reviews: This is a textbook in cryptography with emphasis on algebraic methods. It is supported by many exercises (with answers) making it appropriate for a course in mathematics or computer science. [...] Overall, this is an excellent expository text, and will be very useful to both the student and researcher. Mathematical Reviews |
| cryptography in linear algebra: Algebraic Cryptanalysis Gregory Bard, 2009-08-14 Algebraic Cryptanalysis bridges the gap between a course in cryptography, and being able to read the cryptanalytic literature. This book is divided into three parts: Part One covers the process of turning a cipher into a system of equations; Part Two covers finite field linear algebra; Part Three covers the solution of Polynomial Systems of Equations, with a survey of the methods used in practice, including SAT-solvers and the methods of Nicolas Courtois. Topics include: Analytic Combinatorics, and its application to cryptanalysis The equicomplexity of linear algebra operations Graph coloring Factoring integers via the quadratic sieve, with its applications to the cryptanalysis of RSA Algebraic Cryptanalysis is designed for advanced-level students in computer science and mathematics as a secondary text or reference book for self-guided study. This book is suitable for researchers in Applied Abstract Algebra or Algebraic Geometry who wish to find more applied topics or practitioners working for security and communications companies. |
| cryptography in linear algebra: A Course in Number Theory and Cryptography Neal Koblitz, 2012-09-05 This is a substantially revised and updated introduction to arithmetic topics, both ancient and modern, that have been at the centre of interest in applications of number theory, particularly in cryptography. As such, no background in algebra or number theory is assumed, and the book begins with a discussion of the basic number theory that is needed. The approach taken is algorithmic, emphasising estimates of the efficiency of the techniques that arise from the theory, and one special feature is the inclusion of recent applications of the theory of elliptic curves. Extensive exercises and careful answers are an integral part all of the chapters. |
| cryptography in linear algebra: Understanding Cryptography Christof Paar, Jan Pelzl, 2009-11-27 Cryptography is now ubiquitous – moving beyond the traditional environments, such as government communications and banking systems, we see cryptographic techniques realized in Web browsers, e-mail programs, cell phones, manufacturing systems, embedded software, smart buildings, cars, and even medical implants. Today's designers need a comprehensive understanding of applied cryptography. After an introduction to cryptography and data security, the authors explain the main techniques in modern cryptography, with chapters addressing stream ciphers, the Data Encryption Standard (DES) and 3DES, the Advanced Encryption Standard (AES), block ciphers, the RSA cryptosystem, public-key cryptosystems based on the discrete logarithm problem, elliptic-curve cryptography (ECC), digital signatures, hash functions, Message Authentication Codes (MACs), and methods for key establishment, including certificates and public-key infrastructure (PKI). Throughout the book, the authors focus on communicating the essentials and keeping the mathematics to a minimum, and they move quickly from explaining the foundations to describing practical implementations, including recent topics such as lightweight ciphers for RFIDs and mobile devices, and current key-length recommendations. The authors have considerable experience teaching applied cryptography to engineering and computer science students and to professionals, and they make extensive use of examples, problems, and chapter reviews, while the book’s website offers slides, projects and links to further resources. This is a suitable textbook for graduate and advanced undergraduate courses and also for self-study by engineers. |
| cryptography in linear algebra: Mathematics of Public Key Cryptography Steven D. Galbraith, 2012-03-15 This advanced graduate textbook gives an authoritative and insightful description of the major ideas and techniques of public key cryptography. |
| cryptography in linear algebra: Group Theoretic Cryptography Maria Isabel Gonzalez Vasco, Rainer Steinwandt, 2015-04-01 Group theory appears to be a promising source of hard computational problems for deploying new cryptographic constructions. This reference focuses on the specifics of using groups, including in particular non-Abelian groups, in the field of cryptography. It provides an introduction to cryptography with emphasis on the group theoretic perspective, making it one of the first books to use this approach. The authors provide the needed cryptographic and group theoretic concepts, full proofs of essential theorems, and formal security evaluations of the cryptographic schemes presented. They also provide references for further reading and exercises at the end of each chapter. |
| cryptography in linear algebra: A Course in Cryptography Heiko Knospe, 2019-09-27 This book provides a compact course in modern cryptography. The mathematical foundations in algebra, number theory and probability are presented with a focus on their cryptographic applications. The text provides rigorous definitions and follows the provable security approach. The most relevant cryptographic schemes are covered, including block ciphers, stream ciphers, hash functions, message authentication codes, public-key encryption, key establishment, digital signatures and elliptic curves. The current developments in post-quantum cryptography are also explored, with separate chapters on quantum computing, lattice-based and code-based cryptosystems. Many examples, figures and exercises, as well as SageMath (Python) computer code, help the reader to understand the concepts and applications of modern cryptography. A special focus is on algebraic structures, which are used in many cryptographic constructions and also in post-quantum systems. The essential mathematics and the modern approach to cryptography and security prepare the reader for more advanced studies. The text requires only a first-year course in mathematics (calculus and linear algebra) and is also accessible to computer scientists and engineers. This book is suitable as a textbook for undergraduate and graduate courses in cryptography as well as for self-study. |
| cryptography in linear algebra: Cryptology and Error Correction Lindsay N. Childs, 2019-04-18 This text presents a careful introduction to methods of cryptology and error correction in wide use throughout the world and the concepts of abstract algebra and number theory that are essential for understanding these methods. The objective is to provide a thorough understanding of RSA, Diffie–Hellman, and Blum–Goldwasser cryptosystems and Hamming and Reed–Solomon error correction: how they are constructed, how they are made to work efficiently, and also how they can be attacked. To reach that level of understanding requires and motivates many ideas found in a first course in abstract algebra—rings, fields, finite abelian groups, basic theory of numbers, computational number theory, homomorphisms, ideals, and cosets. Those who complete this book will have gained a solid mathematical foundation for more specialized applied courses on cryptology or error correction, and should also be well prepared, both in concepts and in motivation, to pursue more advanced study in algebra and number theory. This text is suitable for classroom or online use or for independent study. Aimed at students in mathematics, computer science, and engineering, the prerequisite includes one or two years of a standard calculus sequence. Ideally the reader will also take a concurrent course in linear algebra or elementary matrix theory. A solutions manual for the 400 exercises in the book is available to instructors who adopt the text for their course. |
| cryptography in linear algebra: Introduction to Cryptography Wade Trappe, Lawrence C. Washington, 2006 This text is for a course in cryptography for advanced undergraduate and graduate students. Material is accessible to mathematically mature students having little background in number theory and computer programming. Core material is treated in the first eight chapters on areas such as classical cryptosystems, basic number theory, the RSA algorithm, and digital signatures. The remaining nine chapters cover optional topics including secret sharing schemes, games, and information theory. Appendices contain computer examples in Mathematica, Maple, and MATLAB. The text can be taught without computers. |
| cryptography in linear algebra: Elliptic Curves Lawrence C. Washington, 2008-04-03 Like its bestselling predecessor, Elliptic Curves: Number Theory and Cryptography, Second Edition develops the theory of elliptic curves to provide a basis for both number theoretic and cryptographic applications. With additional exercises, this edition offers more comprehensive coverage of the fundamental theory, techniques, and application |
| cryptography in linear algebra: Introduction to Modern Cryptography Jonathan Katz, Yehuda Lindell, 2020-12-21 Now the most used texbook for introductory cryptography courses in both mathematics and computer science, the Third Edition builds upon previous editions by offering several new sections, topics, and exercises. The authors present the core principles of modern cryptography, with emphasis on formal definitions, rigorous proofs of security. |
| cryptography in linear algebra: Making, Breaking Codes Paul B. Garrett, 2001 This unique book explains the basic issues of classical and modern cryptography, and provides a self contained essential mathematical background in number theory, abstract algebra, and probability--with surveys of relevant parts of complexity theory and other things. A user-friendly, down-to-earth tone presents concretely motivated introductions to these topics. More detailed chapter topics include simple ciphers; applying ideas from probability; substitutions, transpositions, permutations; modern symmetric ciphers; the integers; prime numbers; powers and roots modulo primes; powers and roots for composite moduli; weakly multiplicative functions; quadratic symbols, quadratic reciprocity; pseudoprimes; groups; sketches of protocols; rings, fields, polynomials; cyclotomic polynomials, primitive roots; pseudo-random number generators; proofs concerning pseudoprimality; factorization attacks finite fields; and elliptic curves. For personnel in computer security, system administration, and information systems. |
| cryptography in linear algebra: Introduction to Cryptography Hans Delfs, Helmut Knebl, 2007-05-31 Due to the rapid growth of digital communication and electronic data exchange, information security has become a crucial issue in industry, business, and administration. Modern cryptography provides essential techniques for securing information and protecting data. In the first part, this book covers the key concepts of cryptography on an undergraduate level, from encryption and digital signatures to cryptographic protocols. Essential techniques are demonstrated in protocols for key exchange, user identification, electronic elections and digital cash. In the second part, more advanced topics are addressed, such as the bit security of one-way functions and computationally perfect pseudorandom bit generators. The security of cryptographic schemes is a central topic. Typical examples of provably secure encryption and signature schemes and their security proofs are given. Though particular attention is given to the mathematical foundations, no special background in mathematics is presumed. The necessary algebra, number theory and probability theory are included in the appendix. Each chapter closes with a collection of exercises. The second edition contains corrections, revisions and new material, including a complete description of the AES, an extended section on cryptographic hash functions, a new section on random oracle proofs, and a new section on public-key encryption schemes that are provably secure against adaptively-chosen-ciphertext attacks. |
| cryptography in linear algebra: Algorithmic Cryptanalysis Antoine Joux, 2009-06-15 Illustrating the power of algorithms, Algorithmic Cryptanalysis describes algorithmic methods with cryptographically relevant examples. Focusing on both private- and public-key cryptographic algorithms, it presents each algorithm either as a textual description, in pseudo-code, or in a C code program.Divided into three parts, the book begins with a |
| cryptography in linear algebra: Applied Algebra Darel W. Hardy, Fred Richman, Carol L. Walker, 2009-02-17 Using mathematical tools from number theory and finite fields, Applied Algebra: Codes, Ciphers, and Discrete Algorithms, Second Edition presents practical methods for solving problems in data security and data integrity. It is designed for an applied algebra course for students who have had prior classes in abstract or linear algebra. While the con |
| cryptography in linear algebra: Codes, Cryptology and Curves with Computer Algebra Ruud Pellikaan, Xin-Wen Wu, Stanislav Bulygin, Relinde Jurrius, 2017-11-02 This well-balanced text touches on theoretical and applied aspects of protecting digital data. The reader is provided with the basic theory and is then shown deeper fascinating detail, including the current state of the art. Readers will soon become familiar with methods of protecting digital data while it is transmitted, as well as while the data is being stored. Both basic and advanced error-correcting codes are introduced together with numerous results on their parameters and properties. The authors explain how to apply these codes to symmetric and public key cryptosystems and secret sharing. Interesting approaches based on polynomial systems solving are applied to cryptography and decoding codes. Computer algebra systems are also used to provide an understanding of how objects introduced in the book are constructed, and how their properties can be examined. This book is designed for Masters-level students studying mathematics, computer science, electrical engineering or physics. |
| cryptography in linear algebra: Coding the Matrix Philip N. Klein, 2013-07 An engaging introduction to vectors and matrices and the algorithms that operate on them, intended for the student who knows how to program. Mathematical concepts and computational problems are motivated by applications in computer science. The reader learns by doing, writing programs to implement the mathematical concepts and using them to carry out tasks and explore the applications. Examples include: error-correcting codes, transformations in graphics, face detection, encryption and secret-sharing, integer factoring, removing perspective from an image, PageRank (Google's ranking algorithm), and cancer detection from cell features. A companion web site, codingthematrix.com provides data and support code. Most of the assignments can be auto-graded online. Over two hundred illustrations, including a selection of relevant xkcd comics. Chapters: The Function, The Field, The Vector, The Vector Space, The Matrix, The Basis, Dimension, Gaussian Elimination, The Inner Product, Special Bases, The Singular Value Decomposition, The Eigenvector, The Linear Program A new edition of this text, incorporating corrections and an expanded index, has been issued as of September 4, 2013, and will soon be available on Amazon. |
| cryptography in linear algebra: Mastering Linear Algebra Cybellium Ltd, Unlock the Language of Vectors and Matrices for Enhanced Problem Solving In the realm of mathematics and science, linear algebra stands as a powerful language that underlies numerous disciplines. Mastering Linear Algebra is your definitive guide to understanding and harnessing the potential of this essential mathematical framework, empowering you to solve complex problems with clarity and precision. About the Book: As mathematical concepts become more integral to various fields, a strong grasp of linear algebra becomes increasingly valuable. Mastering Linear Algebra offers a comprehensive exploration of this foundational subject—a cornerstone of mathematics and its applications. This book caters to both newcomers and experienced learners aiming to excel in linear algebra concepts, computations, and applications. Key Features: Linear Algebra Fundamentals: Begin by understanding the core principles of linear algebra. Learn about vectors, matrices, and linear transformations—the fundamental building blocks of the subject. Matrix Operations: Dive into matrix operations. Explore techniques for matrix addition, multiplication, inversion, and determinant computation. Vector Spaces: Grasp the art of vector spaces and subspaces. Understand how to define, visualize, and analyze vector spaces for various applications. Eigenvalues and Eigenvectors: Explore the significance of eigenvalues and eigenvectors. Learn how they enable the analysis of dynamic systems and transformations. Linear Systems: Understand how linear algebra solves systems of linear equations. Explore techniques for Gaussian elimination, LU decomposition, and matrix factorization. Applications in Science and Engineering: Delve into real-world applications of linear algebra. Discover how it's applied in physics, computer graphics, data analysis, and more. Inner Product Spaces: Grasp the concepts of inner product spaces and orthogonality. Explore applications in geometric interpretations and least-squares solutions. Singular Value Decomposition: Explore the power of singular value decomposition. Understand how it enables data compression, noise reduction, and dimensionality reduction. Why This Book Matters: In a world driven by data and technological advancement, mastering linear algebra offers a competitive edge. Mastering Linear Algebra empowers students, researchers, scientists, and technology enthusiasts to leverage this fundamental mathematical language, enabling them to analyze and solve problems across diverse fields. Unlock the Power of Mathematical Insight: In the landscape of mathematics and science, linear algebra is the key to understanding complex relationships and transformations. Mastering Linear Algebra equips you with the knowledge needed to leverage linear algebra concepts, enabling you to solve intricate problems with clarity and precision. Whether you're a seasoned learner or new to the world of linear algebra, this book will guide you in building a solid foundation for effective mathematical analysis and application. Your journey to mastering linear algebra starts here. © 2023 Cybellium Ltd. All rights reserved. www.cybellium.com |
| cryptography in linear algebra: Algebra for Cryptologists Alko R. Meijer, 2016-09-01 This textbook provides an introduction to the mathematics on which modern cryptology is based. It covers not only public key cryptography, the glamorous component of modern cryptology, but also pays considerable attention to secret key cryptography, its workhorse in practice. Modern cryptology has been described as the science of the integrity of information, covering all aspects like confidentiality, authenticity and non-repudiation and also including the protocols required for achieving these aims. In both theory and practice it requires notions and constructions from three major disciplines: computer science, electronic engineering and mathematics. Within mathematics, group theory, the theory of finite fields, and elementary number theory as well as some topics not normally covered in courses in algebra, such as the theory of Boolean functions and Shannon theory, are involved. Although essentially self-contained, a degree of mathematical maturity on the part of the reader is assumed, corresponding to his or her background in computer science or engineering. Algebra for Cryptologists is a textbook for an introductory course in cryptography or an upper undergraduate course in algebra, or for self-study in preparation for postgraduate study in cryptology. |
| cryptography in linear algebra: Handbook of Applied Cryptography Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone, 2018-12-07 Cryptography, in particular public-key cryptography, has emerged in the last 20 years as an important discipline that is not only the subject of an enormous amount of research, but provides the foundation for information security in many applications. Standards are emerging to meet the demands for cryptographic protection in most areas of data communications. Public-key cryptographic techniques are now in widespread use, especially in the financial services industry, in the public sector, and by individuals for their personal privacy, such as in electronic mail. This Handbook will serve as a valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography. It is a necessary and timely guide for professionals who practice the art of cryptography. The Handbook of Applied Cryptography provides a treatment that is multifunctional: It serves as an introduction to the more practical aspects of both conventional and public-key cryptography It is a valuable source of the latest techniques and algorithms for the serious practitioner It provides an integrated treatment of the field, while still presenting each major topic as a self-contained unit It provides a mathematical treatment to accompany practical discussions It contains enough abstraction to be a valuable reference for theoreticians while containing enough detail to actually allow implementation of the algorithms discussed Now in its third printing, this is the definitive cryptography reference that the novice as well as experienced developers, designers, researchers, engineers, computer scientists, and mathematicians alike will use. |
| cryptography in linear algebra: Cryptography, Information Theory, and Error-Correction Aiden A. Bruen, Mario A. Forcinito, James M. McQuillan, 2021-10-08 CRYPTOGRAPHY, INFORMATION THEORY, AND ERROR-CORRECTION A rich examination of the technologies supporting secure digital information transfers from respected leaders in the field As technology continues to evolve Cryptography, Information Theory, and Error-Correction: A Handbook for the 21ST Century is an indispensable resource for anyone interested in the secure exchange of financial information. Identity theft, cybercrime, and other security issues have taken center stage as information becomes easier to access. Three disciplines offer solutions to these digital challenges: cryptography, information theory, and error-correction, all of which are addressed in this book. This book is geared toward a broad audience. It is an excellent reference for both graduate and undergraduate students of mathematics, computer science, cybersecurity, and engineering. It is also an authoritative overview for professionals working at financial institutions, law firms, and governments who need up-to-date information to make critical decisions. The book’s discussions will be of interest to those involved in blockchains as well as those working in companies developing and applying security for new products, like self-driving cars. With its reader-friendly style and interdisciplinary emphasis this book serves as both an ideal teaching text and a tool for self-learning for IT professionals, statisticians, mathematicians, computer scientists, electrical engineers, and entrepreneurs. Six new chapters cover current topics like Internet of Things security, new identities in information theory, blockchains, cryptocurrency, compression, cloud computing and storage. Increased security and applicable research in elliptic curve cryptography are also featured. The book also: Shares vital, new research in the field of information theory Provides quantum cryptography updates Includes over 350 worked examples and problems for greater understanding of ideas. Cryptography, Information Theory, and Error-Correction guides readers in their understanding of reliable tools that can be used to store or transmit digital information safely. |
| cryptography in linear algebra: Algebra for Applications Arkadii Slinko, 2015-08-19 This book examines the relationship between mathematics and data in the modern world. Indeed, modern societies are awash with data which must be manipulated in many different ways: encrypted, compressed, shared between users in a prescribed manner, protected from an unauthorised access and transmitted over unreliable channels. All of these operations can be understood only by a person with knowledge of basics in algebra and number theory. This book provides the necessary background in arithmetic, polynomials, groups, fields and elliptic curves that is sufficient to understand such real-life applications as cryptography, secret sharing, error-correcting, fingerprinting and compression of information. It is the first to cover many recent developments in these topics. Based on a lecture course given to third-year undergraduates, it is self-contained with numerous worked examples and exercises provided to test understanding. It can additionally be used for self-study. |
| cryptography in linear algebra: Algebraic Aspects of the Advanced Encryption Standard Carlos Cid, Sean Murphy, Matthew Robshaw, 2006-11-24 The Belgian block cipher Rijndael was chosen in 2000 by the U.S. government’s National Institute of Standards and Technology (NIST) to be the successor to the Data Encryption Standard. Rijndael was subsequently standardized as the Advanced Encryption Standard (AES), which is potentially the world’s most important block cipher. In 2002, some new analytical techniques were suggested that may have a dramatic effect on the security of the AES. Existing analytical techniques for block ciphers depend heavily on a statistical approach, whereas these new techniques are algebraic in nature. Algebraic Aspects of the Advanced Encryption Standard, appearing five years after publication of the AES, presents the state of the art for the use of such algebraic techniques in analyzing the AES. The primary audience for this work includes academic and industry researchers in cryptology; the book is also suitable for advanced-level students. |
| cryptography in linear algebra: Introduction to Cryptography with Maple José Luis Gómez Pardo, 2012-12-19 This introduction to cryptography employs a programming-oriented approach to study the most important cryptographic schemes in current use and the main cryptanalytic attacks against them. Discussion of the theoretical aspects, emphasizing precise security definitions based on methodological tools such as complexity and randomness, and of the mathematical aspects, with emphasis on number-theoretic algorithms and their applications to cryptography and cryptanalysis, is integrated with the programming approach, thus providing implementations of the algorithms and schemes as well as examples of realistic size. A distinctive feature of the author's approach is the use of Maple as a programming environment in which not just the cryptographic primitives but also the most important cryptographic schemes are implemented following the recommendations of standards bodies such as NIST, with many of the known cryptanalytic attacks implemented as well. The purpose of the Maple implementations is to let the reader experiment and learn, and for this reason the author includes numerous examples. The book discusses important recent subjects such as homomorphic encryption, identity-based cryptography and elliptic curve cryptography. The algorithms and schemes which are treated in detail and implemented in Maple include AES and modes of operation, CMAC, GCM/GMAC, SHA-256, HMAC, RSA, Rabin, Elgamal, Paillier, Cocks IBE, DSA and ECDSA. In addition, some recently introduced schemes enjoying strong security properties, such as RSA-OAEP, Rabin-SAEP, Cramer--Shoup, and PSS, are also discussed and implemented. On the cryptanalysis side, Maple implementations and examples are used to discuss many important algorithms, including birthday and man-in-the-middle attacks, integer factorization algorithms such as Pollard's rho and the quadratic sieve, and discrete log algorithms such as baby-step giant-step, Pollard's rho, Pohlig--Hellman and the index calculus method. This textbook is suitable for advanced undergraduate and graduate students of computer science, engineering and mathematics, satisfying the requirements of various types of courses: a basic introductory course; a theoretically oriented course whose focus is on the precise definition of security concepts and on cryptographic schemes with reductionist security proofs; a practice-oriented course requiring little mathematical background and with an emphasis on applications; or a mathematically advanced course addressed to students with a stronger mathematical background. The main prerequisite is a basic knowledge of linear algebra and elementary calculus, and while some knowledge of probability and abstract algebra would be helpful, it is not essential because the book includes the necessary background from these subjects and, furthermore, explores the number-theoretic material in detail. The book is also a comprehensive reference and is suitable for self-study by practitioners and programmers. |
| cryptography in linear algebra: Groups, Matrices, and Vector Spaces James B. Carrell, 2017-09-02 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. |
| cryptography in linear algebra: Algebraic Geometry in Coding Theory and Cryptography Harald Niederreiter, Chaoping Xing, 2009-09-21 This textbook equips graduate students and advanced undergraduates with the necessary theoretical tools for applying algebraic geometry to information theory, and it covers primary applications in coding theory and cryptography. Harald Niederreiter and Chaoping Xing provide the first detailed discussion of the interplay between nonsingular projective curves and algebraic function fields over finite fields. This interplay is fundamental to research in the field today, yet until now no other textbook has featured complete proofs of it. Niederreiter and Xing cover classical applications like algebraic-geometry codes and elliptic-curve cryptosystems as well as material not treated by other books, including function-field codes, digital nets, code-based public-key cryptosystems, and frameproof codes. Combining a systematic development of theory with a broad selection of real-world applications, this is the most comprehensive yet accessible introduction to the field available. Introduces graduate students and advanced undergraduates to the foundations of algebraic geometry for applications to information theory Provides the first detailed discussion of the interplay between projective curves and algebraic function fields over finite fields Includes applications to coding theory and cryptography Covers the latest advances in algebraic-geometry codes Features applications to cryptography not treated in other books |
| cryptography in linear algebra: Modern Cryptography William Easttom, 2022-10-29 This expanded textbook, now in its second edition, is a practical yet in depth guide to cryptography and its principles and practices. Now featuring a new section on quantum resistant cryptography in addition to expanded and revised content throughout, the book continues to place cryptography in real-world security situations using the hands-on information contained throughout the chapters. Prolific author Dr. Chuck Easttom lays out essential math skills and fully explains how to implement cryptographic algorithms in today's data protection landscape. Readers learn and test out how to use ciphers and hashes, generate random keys, handle VPN and Wi-Fi security, and encrypt VoIP, Email, and Web communications. The book also covers cryptanalysis, steganography, and cryptographic backdoors and includes a description of quantum computing and its impact on cryptography. This book is meant for those without a strong mathematics background with only just enough math to understand the algorithms given. The book contains a slide presentation, questions and answers, and exercises throughout. Presents new and updated coverage of cryptography including new content on quantum resistant cryptography; Covers the basic math needed for cryptography - number theory, discrete math, and algebra (abstract and linear); Includes a full suite of classroom materials including exercises, Q&A, and examples. |
| cryptography in linear algebra: Elementary Linear Algebra Howard Anton, 2010-03-15 When it comes to learning linear algebra, engineers trust Anton. The tenth edition presents the key concepts and topics along with engaging and contemporary applications. The chapters have been reorganized to bring up some of the more abstract topics and make the material more accessible. More theoretical exercises at all levels of difficulty are integrated throughout the pages, including true/false questions that address conceptual ideas. New marginal notes provide a fuller explanation when new methods and complex logical steps are included in proofs. Small-scale applications also show how concepts are applied to help engineers develop their mathematical reasoning. |
| cryptography in linear algebra: Applied Cryptography Bruce Schneier, 2017-05-25 From the world's most renowned security technologist, Bruce Schneier, this 20th Anniversary Edition is the most definitive reference on cryptography ever published and is the seminal work on cryptography. Cryptographic techniques have applications far beyond the obvious uses of encoding and decoding information. For developers who need to know about capabilities, such as digital signatures, that depend on cryptographic techniques, there's no better overview than Applied Cryptography, the definitive book on the subject. Bruce Schneier covers general classes of cryptographic protocols and then specific techniques, detailing the inner workings of real-world cryptographic algorithms including the Data Encryption Standard and RSA public-key cryptosystems. The book includes source-code listings and extensive advice on the practical aspects of cryptography implementation, such as the importance of generating truly random numbers and of keeping keys secure. . . .the best introduction to cryptography I've ever seen. . . .The book the National Security Agency wanted never to be published. . . . -Wired Magazine . . .monumental . . . fascinating . . . comprehensive . . . the definitive work on cryptography for computer programmers . . . -Dr. Dobb's Journal . . .easily ranks as one of the most authoritative in its field. -PC Magazine The book details how programmers and electronic communications professionals can use cryptography-the technique of enciphering and deciphering messages-to maintain the privacy of computer data. It describes dozens of cryptography algorithms, gives practical advice on how to implement them into cryptographic software, and shows how they can be used to solve security problems. The book shows programmers who design computer applications, networks, and storage systems how they can build security into their software and systems. With a new Introduction by the author, this premium edition will be a keepsake for all those committed to computer and cyber security. |
| cryptography in linear algebra: A Classical Introduction to Cryptography Serge Vaudenay, 2005-09-16 A Classical Introduction to Cryptography: Applications for Communications Security introduces fundamentals of information and communication security by providing appropriate mathematical concepts to prove or break the security of cryptographic schemes. This advanced-level textbook covers conventional cryptographic primitives and cryptanalysis of these primitives; basic algebra and number theory for cryptologists; public key cryptography and cryptanalysis of these schemes; and other cryptographic protocols, e.g. secret sharing, zero-knowledge proofs and undeniable signature schemes. A Classical Introduction to Cryptography: Applications for Communications Security is designed for upper-level undergraduate and graduate-level students in computer science. This book is also suitable for researchers and practitioners in industry. A separate exercise/solution booklet is available as well, please go to www.springeronline.com under author: Vaudenay for additional details on how to purchase this booklet. |
| cryptography in linear algebra: Matrices and Linear Algebra Hans Schneider, George Phillip Barker, 2012-06-08 Linear algebra is one of the central disciplines in mathematics. A student of pure mathematics must know linear algebra if he is to continue with modern algebra or functional analysis. Much of the mathematics now taught to engineers and physicists requires it. This well-known and highly regarded text makes the subject accessible to undergraduates with little mathematical experience. Written mainly for students in physics, engineering, economics, and other fields outside mathematics, the book gives the theory of matrices and applications to systems of linear equations, as well as many related topics such as determinants, eigenvalues, and differential equations. Table of Contents: l. The Algebra of Matrices 2. Linear Equations 3. Vector Spaces 4. Determinants 5. Linear Transformations 6. Eigenvalues and Eigenvectors 7. Inner Product Spaces 8. Applications to Differential Equations For the second edition, the authors added several exercises in each chapter and a brand new section in Chapter 7. The exercises, which are both true-false and multiple-choice, will enable the student to test his grasp of the definitions and theorems in the chapter. The new section in Chapter 7 illustrates the geometric content of Sylvester's Theorem by means of conic sections and quadric surfaces. 6 line drawings. lndex. Two prefaces. Answer section. |
| cryptography in linear algebra: Advances in Cryptology — ASIACRYPT 2001 Colin Boyd, 2003-06-30 The origins of the Asiacrypt series of conferences can be traced back to 1990, when the ?rst Auscrypt conference was held, although the name Asiacrypt was ?rst used for the 1991 conference in Japan. Starting with Asiacrypt 2000, the conference is now one of three annual conferences organized by the Inter- tional Association for Cryptologic Research (IACR). The continuing success of Asiacrypt is in no small part due to the e?orts of the Asiacrypt Steering C- mittee (ASC) and the strong support of the IACR Board of Directors. There were 153 papers submitted to Asiacrypt 2001 and 33 of these were accepted for inclusion in these proceedings. The authors of every paper, whether accepted or not, made a valued contribution to the success of the conference. Sending out rejection noti?cations to so many hard working authors is one of the most unpleasant tasks of the Program Chair. The review process lasted some 10 weeks and consisted of an initial refe- eing phase followed by an extensive discussion period. My heartfelt thanks go to all members of the Program Committee who put in extreme amounts of time to give their expert analysis and opinions on the submissions. All papers were reviewed by at least three committee members; in many cases, particularly for those papers submitted by committee members, additional reviews were obt- ned. Specialist reviews were provided by an army of external reviewers without whom our decisions would have been much more di?cult. |
| cryptography in linear algebra: Cryptographic Boolean Functions and Applications Thomas W. Cusick, Pantelimon Stanica, 2009-03-04 Boolean functions are the building blocks of symmetric cryptographic systems. Symmetrical cryptographic algorithms are fundamental tools in the design of all types of digital security systems (i.e. communications, financial and e-commerce).Cryptographic Boolean Functions and Applications is a concise reference that shows how Boolean functions are used in cryptography. Currently, practitioners who need to apply Boolean functions in the design of cryptographic algorithms and protocols need to patch together needed information from a variety of resources (books, journal articles and other sources). This book compiles the key essential information in one easy to use, step-by-step reference. Beginning with the basics of the necessary theory the book goes on to examine more technical topics, some of which are at the frontier of current research. Serves as a complete resource for the successful design or implementation of cryptographic algorithms or protocols using Boolean functions Provides engineers and scientists with a needed reference for the use of Boolean functions in cryptography Addresses the issues of cryptographic Boolean functions theory and applications in one concentrated resource Organized logically to help the reader easily understand the topic |
| cryptography in linear algebra: Boolean Functions for Cryptography and Coding Theory Claude Carlet, 2021-01-07 Boolean functions are essential to systems for secure and reliable communication. This comprehensive survey of Boolean functions for cryptography and coding covers the whole domain and all important results, building on the author's influential articles with additional topics and recent results. A useful resource for researchers and graduate students, the book balances detailed discussions of properties and parameters with examples of various types of cryptographic attacks that motivate the consideration of these parameters. It provides all the necessary background on mathematics, cryptography, and coding, and an overview on recent applications, such as side channel attacks on smart cards, cloud computing through fully homomorphic encryption, and local pseudo-random generators. The result is a complete and accessible text on the state of the art in single and multiple output Boolean functions that illustrates the interaction between mathematics, computer science, and telecommunications. |
| cryptography in linear algebra: Cryptology Richard Klima, Richard E. Klima, Neil Sigmon, Neil P. Sigmon, 2018-12-07 Cryptology: Classical and Modern, Second Edition proficiently introduces readers to the fascinating field of cryptology. The book covers classical methods including substitution, transposition, Playfair, ADFGVX, Alberti, Vigene re, and Hill ciphers. It also includes coverage of the Enigma machine, Turing bombe, and Navajo code. Additionally, the book presents modern methods like RSA, ElGamal, and stream ciphers, as well as the Diffie-Hellman key exchange and Advanced Encryption Standard. When possible, the book details methods for breaking both classical and modern methods. The new edition expands upon the material from the first edition which was oriented for students in non-technical fields. At the same time, the second edition supplements this material with new content that serves students in more technical fields as well. Thus, the second edition can be fully utilized by both technical and non-technical students at all levels of study. The authors include a wealth of material for a one-semester cryptology course, and research exercises that can be used for supplemental projects. Hints and answers to selected exercises are found at the end of the book. |
| cryptography in linear algebra: Cryptology and Error Correction Lindsay Childs, 2019 This text presents a careful introduction to methods of cryptology and error correction in wide use throughout the world and the concepts of abstract algebra and number theory that are essential for understanding these methods. The objective is to provide a thorough understanding of RSA, Diffie-Hellman, and Blum-Goldwasser cryptosystems and Hamming and Reed-Solomon error correction: how they are constructed, how they are made to work efficiently, and also how they can be attacked. To reach that level of understanding requires and motivates many ideas found in a first course in abstract algebra-rings, fields, finite abelian groups, basic theory of numbers, computational number theory, homomorphisms, ideals, and cosets. Those who complete this book will have gained a solid mathematical foundation for more specialized applied courses on cryptology or error correction, and should also be well prepared, both in concepts and in motivation, to pursue more advanced study in algebra and number theory. This text is suitable for classroom or online use or for independent study. Aimed at students in mathematics, computer science, and engineering, the prerequisite includes one or two years of a standard calculus sequence. Ideally the reader will also take a concurrent course in linear algebra or elementary matrix theory. A solutions manual for the 400 exercises in the book is available to instructors who adopt the text for their course. |
| cryptography in linear algebra: Cryptography Simon Rubinstein-Salzedo, 2018-09-27 This text introduces cryptography, from its earliest roots to cryptosystems used today for secure online communication. Beginning with classical ciphers and their cryptanalysis, this book proceeds to focus on modern public key cryptosystems such as Diffie-Hellman, ElGamal, RSA, and elliptic curve cryptography with an analysis of vulnerabilities of these systems and underlying mathematical issues such as factorization algorithms. Specialized topics such as zero knowledge proofs, cryptographic voting, coding theory, and new research are covered in the final section of this book. Aimed at undergraduate students, this book contains a large selection of problems, ranging from straightforward to difficult, and can be used as a textbook for classes as well as self-study. Requiring only a solid grounding in basic mathematics, this book will also appeal to advanced high school students and amateur mathematicians interested in this fascinating and topical subject. |
| cryptography in linear algebra: Number-Theoretic Methods in Cryptology Jerzy Kaczorowski, Josef Pieprzyk, Jacek Pomykała, 2018-03-09 This book constitutes the refereed post-conference proceedings of the First International Conference on Number-Theoretic Methods in Cryptology, NuTMiC 2017, held in Warsaw, Poland, in September 2017.The 15 revised full papers presented in this book together with 3 invited talks were carefully reviewed and selected from 32 initial submissions. The papers are organized in topical sections on elliptic curves in cryptography; public-key cryptography; lattices in cryptography; number theory; pseudorandomness; and algebraic structures and analysis. |
| cryptography in linear algebra: Advances in Cryptology - CRYPTO '89 Gilles Brassard, 1995-01-01 CRYPTO is a conference devoted to all aspects of cryptologic research. It is held each year at the University of California at Santa Barbara. Annual meetings on this topic also take place in Europe and are regularly published in this Lecture Notes series under the name of EUROCRYPT. This volume presents the proceedings of the ninth CRYPTO meeting. The papers are organized into sections with the following themes: Why is cryptography harder than it looks?, pseudo-randomness and sequences, cryptanalysis and implementation, signature and authentication, threshold schemes and key management, key distribution and network security, fast computation, odds and ends, zero-knowledge and oblivious transfer, multiparty computation. |
Cryptography - Wikipedia
Cryptography, or cryptology (from Ancient Greek: κρυπτός, romanized: kryptós "hidden, secret"; and γράφειν graphein, "to write", or -λογία-logia, "study", respectively [1]), is the practice and …
Cryptography and its Types - GeeksforGeeks
May 8, 2025 · Cryptography is a technique of securing information and communications using codes to ensure confidentiality, integrity and authentication. Thus, preventing unauthorized …
What Is Cryptography? - IBM
In our modern digital age, cryptography has become an essential cybersecurity tool for protecting sensitive information from hackers and other cybercriminals. Derived from the Greek word …
What is Cryptography? Definition, Importance, Types - Fortinet
Cryptography is the process of hiding or coding information so only the intended recipient can read a message. Discover how cryptography works and the potential risks it poses.
ISO - What is cryptography?
Cryptography refers to the techniques and algorithms that are used today for secure communication and data in storage. It incorporates mathematics, computer science, …
What is cryptography? How algorithms keep information secret …
May 22, 2022 · Cryptography is the art of keeping information secret and safe by transforming it into form that unintended recipients cannot understand. It makes secure data transmission …
Cryptography | NIST - National Institute of Standards and …
Cryptography uses mathematical techniques to transform data and prevent it from being read or tampered with by unauthorized parties. That enables exchanging secure messages even in the …
Welcome to the Art of Secrets | Cryptography Guide
Comprehensive guide to cryptography covering basic concepts, advanced topics, CTF challenges, and practical implementations.
What is Cryptography? A Complete Guide - The Knowledge …
May 16, 2025 · Cryptography is the practice of safeguarding information and communications by converting data into a coded format that only authorised individuals can decipher. This process …
What is Cryptography? Types and Examples You Need to Know
Cryptography is a method of protecting information and communications by employing codes, so that only those for whom the information is intended can read and process it.
Cryptography - Wikipedia
Cryptography, or cryptology (from Ancient Greek: κρυπτός, romanized: kryptós "hidden, secret"; and γράφειν graphein, "to write", or -λογία-logia, "study", respectively [1]), is the practice and …
Cryptography and its Types - GeeksforGeeks
May 8, 2025 · Cryptography is a technique of securing information and communications using codes to ensure confidentiality, integrity and authentication. Thus, preventing unauthorized …
What Is Cryptography? - IBM
In our modern digital age, cryptography has become an essential cybersecurity tool for protecting sensitive information from hackers and other cybercriminals. Derived from the Greek word …
What is Cryptography? Definition, Importance, Types - Fortinet
Cryptography is the process of hiding or coding information so only the intended recipient can read a message. Discover how cryptography works and the potential risks it poses.
ISO - What is cryptography?
Cryptography refers to the techniques and algorithms that are used today for secure communication and data in storage. It incorporates mathematics, computer science, …
What is cryptography? How algorithms keep information secret …
May 22, 2022 · Cryptography is the art of keeping information secret and safe by transforming it into form that unintended recipients cannot understand. It makes secure data transmission …
Cryptography | NIST - National Institute of Standards and …
Cryptography uses mathematical techniques to transform data and prevent it from being read or tampered with by unauthorized parties. That enables exchanging secure messages even in …
Welcome to the Art of Secrets | Cryptography Guide
Comprehensive guide to cryptography covering basic concepts, advanced topics, CTF challenges, and practical implementations.
What is Cryptography? A Complete Guide - The Knowledge …
May 16, 2025 · Cryptography is the practice of safeguarding information and communications by converting data into a coded format that only authorised individuals can decipher. This process …
What is Cryptography? Types and Examples You Need to Know
Cryptography is a method of protecting information and communications by employing codes, so that only those for whom the information is intended can read and process it.
