Proceedings of the NATO Advanced Study Institute, Banff Centre, Canada, April 27-May 5, 1988

From the Rosetta Stone to public-key cryptography, the art and science of cryptology has been used to unlock the vivid history of ancient cultures, to turn the tide of warfare, and to thwart potential hackers from attacking computer systems. Codes: The Guide to Secrecy from Ancient to Modern Times explores the depth and breadth of the field, remaining accessible to the uninitiated while retaining enough rigor for the seasoned cryptologist. The book begins by tracing the development of cryptology from that of an arcane practice used, for example, to conceal alchemic recipes, to the modern scientific method that is studied and employed today. The remainder of the book explores the modern aspects and applications of cryptography, covering symmetric- and public-key cryptography, cryptographic protocols, key management, message authentication, e-mail and Internet security, and advanced applications such as wireless security, smart cards, biometrics, and quantum cryptography. The author also includes non-cryptographic security issues and a chapter devoted to information theory and coding. Nearly 200 diagrams, examples, figures, and tables along with abundant references and exercises complement the discussion. Written by leading authority and best-selling author on the subject Richard A. Mollin, Codes: The Guide to Secrecy from Ancient to Modern Times is the essential reference for anyone interested in this exciting and fascinating field, from novice to veteran practitioner.

Although much literature exists on the subject of RSA and public-key cryptography, until now there has been no single source that reveals recent developments in the area at an accessible level. Acclaimed author Richard A. Mollin brings together all of the relevant information available on public-key cryptography (PKC), from RSA to the latest applications of PKC, including electronic cash, secret broadcasting, secret balloting systems, various banking and payment protocols, high security logins, smart cards, and biometrics. Moreover, he covers public-key infrastructure (PKI) and its various security applications. Throughout the book, Mollin gives a human face to cryptography by including nearly 40 biographies of the individuals who helped develop cryptographic concepts. He includes a number of illustrative and motivating examples, as well as optional topics that go beyond the basics, such as Lenstra's elliptic curve method and the number field sieve. From history and basic concepts to future trends and emerging applications, this book provides a rigorous and detailed treatment of public-key cryptography. Accessible to anyone from the senior undergraduate to the research scientist, RSA and Public-Key Cryptography offers challenging and inspirational material for all readers.

The first thing you will find out about this book is that it is fun to read. It is meant for the browser, as well as for the student and for the specialist wanting to know about the area. The footnotes give an historical background to the text, in addition to providing deeper applications of the concept that is being cited. This allows the browser to look more deeply into the history or to pursue a given sideline. Those who are only marginally interested in the area will be able to read the text, pick up information easily, and be entertained at the same time by the historical and philosophical digressions. It is rich in structure and motivation in its concentration upon quadratic orders. This is not a book that is primarily about tables, although there are 80 pages of appendices that contain extensive tabular material (class numbers of real and complex quadratic fields up to 104; class group structures; fundamental units of real quadratic fields; and more!). This book is primarily a reference book and graduate student text with more than 200 exercises and a great deal of hints! The motivation for the text is best given by a quote from the Preface of Quadratics: "There can be no stronger motivation in mathematical inquiry than the search for truth and beauty. It is this author's long-standing conviction that number theory has the best of both of these worlds. In particular, algebraic and computational number theory have reached a stage where the current state of affairs richly deserves a proper elucidation. It is this author's goal to attempt to shine the best possible light on the subject."

An update of the most accessible introductory number theory text available, Fundamental Number Theory with Applications, Second Edition presents a mathematically rigorous yet easy-to-follow treatment of the fundamentals and applications of the subject. The substantial amount of reorganizing makes this edition clearer and more elementary in its coverage. New to the Second Edition * Removal of all advanced material to be even more accessible in scope * New fundamental material, including partition theory, generating functions, and combinatorial number theory * Expanded coverage of random number generation, Diophantine analysis, and additive number theory * More applications to cryptography, primality testing, and factoring * An appendix on the recently discovered unconditional deterministic polynomial-time algorithm for primality testing Taking a truly elementary approach to number theory, this text supplies the essential material for a first course on the subject. Placed in highlighted boxes to reduce distraction from the main text, nearly 70 biographies focus on major contributors to the field. The presentation of over 1,300 entries in the index maximizes cross-referencing so students can find data with ease.

Exploring one of the most dynamic areas of mathematics, Advanced Number Theory with Applications covers a wide range of algebraic, analytic, combinatorial, cryptographic, and geometric aspects of number theory. Written by a recognized leader in algebra and number theory, the book includes a page reference for every citing in the bibliography and more than 1,500 entries in the index so that students can easily cross-reference and find the appropriate data. With numerous examples throughout, the text begins with coverage of algebraic number theory, binary quadratic forms, Diophantine approximation, arithmetic functions, p-adic analysis, Dirichlet characters, density, and primes in arithmetic progression. It then applies these tools to Diophantine equations, before developing elliptic curves and modular forms. The text also presents an overview of Fermat's Last Theorem (FLT) and numerous consequences of the ABC conjecture, including Thue-Siegel-Roth theorem, Hall's conjecture, the Erdoes-Mollin--Walsh conjecture, and the Granville-Langevin Conjecture. In the appendix, the author reviews sieve methods, such as Eratothesenes', Selberg's, Linnik's, and Bombieri's sieves. He also discusses recent results on gaps between primes and the use of sieves in factoring. By focusing on salient techniques in number theory, this textbook provides the most up-to-date and comprehensive material for a second course in this field. It prepares students for future study at the graduate level.

Exploring one of the most dynamic areas of mathematics, Advanced Number Theory with Applications covers a wide range of algebraic, analytic, combinatorial, cryptographic, and geometric aspects of number theory. Written by a recognized leader in algebra and number theory, the book includes a page reference for every citing in the bibliography and more than 1,500 entries in the index so that students can easily cross-reference and find the appropriate data.

With numerous examples throughout, the text begins with coverage of algebraic number theory, binary quadratic forms, Diophantine approximation, arithmetic functions, p-adic analysis, Dirichlet characters, density, and primes in arithmetic progression. It then applies these tools to Diophantine equations, before developing elliptic curves and modular forms. The text also presents an overview of Fermat's Last Theorem (FLT) and numerous consequences of the ABC conjecture, including Thue-Siegel-Roth theorem, Hall's conjecture, the Erdos-Mollin--Walsh conjecture, and the Granville-Langevin Conjecture. In the appendix, the author reviews sieve methods, such as Eratothesenes', Selberg's, Linnik's, and Bombieri's sieves. He also discusses recent results on gaps between primes and the use of sieves in factoring.

By focusing on salient techniques in number theory, this textbook provides the most up-to-date and comprehensive material for a second course in this field. It prepares students for future study at the graduate level."

From the Rosetta Stone to public-key cryptography, the art and science of cryptology has been used to unlock the vivid history of ancient cultures, to turn the tide of warfare, and to thwart potential hackers from attacking computer systems. Codes: The Guide to Secrecy from Ancient to Modern Times explores the depth and breadth of the field, remaining accessible to the uninitiated while retaining enough rigor for the seasoned cryptologist. The book begins by tracing the development of cryptology from that of an arcane practice used, for example, to conceal alchemic recipes, to the modern scientific method that is studied and employed today. The remainder of the book explores the modern aspects and applications of cryptography, covering symmetric- and public-key cryptography, cryptographic protocols, key management, message authentication, e-mail and Internet security, and advanced applications such as wireless security, smart cards, biometrics, and quantum cryptography. The author also includes non-cryptographic security issues and a chapter devoted to information theory and coding. Nearly 200 diagrams, examples, figures, and tables along with abundant references and exercises complement the discussion. Written by leading authority and best-selling author on the subject Richard A. Mollin, Codes: The Guide to Secrecy from Ancient to Modern Times is the essential reference for anyone interested in this exciting and fascinating field, from novice to veteran practitioner.

The first thing you will find out about this book is that it is fun to read. It is meant for the browser, as well as for the student and for the specialist wanting to know about the area. The footnotes give an historical background to the text, in addition to providing deeper applications of the concept that is being cited. This allows the browser to look more deeply into the history or to pursue a given sideline. Those who are only marginally interested in the area will be able to read the text, pick up information easily, and be entertained at the same time by the historical and philosophical digressions. It is rich in structure and motivation in its concentration upon quadratic orders.
This is not a book that is primarily about tables, although there are 80 pages of appendices that contain extensive tabular material (class numbers of real and complex quadratic fields up to 104; class group structures; fundamental units of real quadratic fields; and more!). This book is primarily a reference book and graduate student text with more than 200 exercises and a great deal of hints!
The motivation for the text is best given by a quote from the Preface of Quadratics: "There can be no stronger motivation in mathematical inquiry than the search for truth and beauty. It is this author's long-standing conviction that number theory has the best of both of these worlds. In particular, algebraic and computational number theory have reached a stage where the current state of affairs richly deserves a proper elucidation. It is this author's goal to attempt to shine the best possible light on the subject."

Proceedings of the NATO Advanced Study Institute, Banff Centre, Canada, April 27-May 5, 1988

Bringing the material up to date to reflect modern applications, Algebraic Number Theory, Second Edition has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. This edition focuses on integral domains, ideals, and unique factorization in the first chapter; field extensions in the second chapter; and class groups in the third chapter. Applications are now collected in chapter four and at the end of chapter five, where primality testing is highlighted as an application of the Kronecker-Weber theorem. In chapter five, the sections on ideal decomposition in number fields have been more evenly distributed. The final chapter continues to cover reciprocity laws. New to the Second Edition * Reorganization of all chapters * More complete and involved treatment of Galois theory * A study of binary quadratic forms and a comparison of the ideal and form class groups * More comprehensive section on Pollard's cubic factoring algorithm * More detailed explanations of proofs, with less reliance on exercises, to provide a sound understanding of challenging material The book includes mini-biographies of notable mathematicians, convenient cross-referencing, a comprehensive index, and numerous exercises. The appendices present an overview of all the concepts used in the main text, an overview of sequences and series, the Greek alphabet with English transliteration, and a table of Latin phrases and their English equivalents. Suitable for a one-semester course, this accessible, self-contained text offers broad, in-depth coverage of numerous applications. Readers are lead at a measured pace through the topics to enable a clear understanding of the pinnacles of algebraic number theory.

An update of the most accessible introductory number theory text available, Fundamental Number Theory with Applications, Second Edition presents a mathematically rigorous yet easy-to-follow treatment of the fundamentals and applications of the subject. The substantial amount of reorganizing makes this edition clearer and more elementary in its coverage. New to the Second Edition * Removal of all advanced material to be even more accessible in scope * New fundamental material, including partition theory, generating functions, and combinatorial number theory * Expanded coverage of random number generation, Diophantine analysis, and additive number theory * More applications to cryptography, primality testing, and factoring * An appendix on the recently discovered unconditional deterministic polynomial-time algorithm for primality testing Taking a truly elementary approach to number theory, this text supplies the essential material for a first course on the subject. Placed in highlighted boxes to reduce distraction from the main text, nearly 70 biographies focus on major contributors to the field. The presentation of over 1,300 entries in the index maximizes cross-referencing so students can find data with ease.

Although a vast literature exists on the subject of RSA and public-key cryptography, until now there has been no single source that reveals recent developments in the area at an accessible level. Acclaimed author Richard A. Mollin brings together all of the relevant information available on public-key cryptography (PKC), from RSA to the latest applications of PKC, including electronic cash, secret broadcasting, secret balloting systems, various banking and payment protocols, high security logins, smart cards, and biometrics. Moreover, he covers public-key infrastructure (PKI) and its various security applications.

Throughout the book, Mollin gives a human face to cryptography by including nearly 40 biographies of the individuals who helped develop cryptographic concepts. He includes a number of illustrative and motivating examples, as well as optional topics that go beyond the basics, such as Lenstra's elliptic curve method and the number field sieve. From history and basic concepts to future trends and emerging applications, this book provides a rigorous and detailed treatment of public-key cryptography. Accessible to anyone from the senior undergraduate to the research scientist, RSA and Public-Key Cryptography offers challenging and inspirational material for all readers.

Continuing a bestselling tradition, An Introduction to Cryptography, Second Edition provides a solid foundation in cryptographic concepts that features all of the requisite background material on number theory and algorithmic complexity as well as a historical look at the field. With numerous additions and restructured material, this edition presents the ideas behind cryptography and the applications of the subject. The first chapter provides a thorough treatment of the mathematics necessary to understand cryptography, including number theory and complexity, while the second chapter discusses cryptographic fundamentals, such as ciphers, linear feedback shift registers, modes of operation, and attacks. The next several chapters discuss DES, AES, public-key cryptography, primality testing, and various factoring methods, from classical to elliptical curves. The final chapters are comprised of issues pertaining to the Internet, such as pretty good privacy (PGP), protocol layers, firewalls, and cookies, as well as applications, including login and network security, viruses, smart cards, and biometrics. The book concludes with appendices on mathematical data, computer arithmetic, the Rijndael S-Box, knapsack ciphers, the Silver-Pohlig-Hellman algorithm, the SHA-1 algorithm, radix-64 encoding, and quantum cryptography. New to the Second Edition: -An introductory chapter that provides more information on mathematical facts and complexity theory -Expanded and updated exercises sets, including some routine exercises -More information on primality testing and cryptanalysis Accessible and logically organized, An Introduction to Cryptography, Second Edition is the essential book on the fundamentalsof cryptography.