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cryptografie --- cryptologie --- Information systems --- 621.391.7 --- 681.3*E3 --- Secret systems of electrical communication --- Data encryption: data encryption standard; DES; public key cryptosystems --- 681.3*E3 Data encryption: data encryption standard; DES; public key cryptosystems --- 621.391.7 Secret systems of electrical communication --- Data transmission systems --- Cryptography --- Computers --- Données --- Cryptographie --- Ordinateurs --- Security measures --- Congresses --- Access control --- Transmissions --- Sécurité --- Mesures --- Congrès --- Accès --- Contrôle --- Information theory. --- Data transmission systems. --- Telecommunication. --- Theory of Computation. --- Input/Output and Data Communications. --- Communications Engineering, Networks. --- Electric communication --- Mass communication --- Telecom --- Telecommunication industry --- Telecommunications --- Communication --- Information theory --- Telecommuting --- Data communication systems --- Transmission of data --- Digital communications --- Electronic data processing --- Electronic systems --- Telecommunication systems --- Communication theory --- Cybernetics

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The objective of this book is to provide tools for solving problems which involve cubic number fields. Many such problems can be considered geometrically; both in terms of the geometry of numbers and geometry of the associated cubic Diophantine equations that are similar in many ways to the Pell equation. With over 50 geometric diagrams, this book includes illustrations of many of these topics. The book may be thought of as a companion reference for those students of algebraic number theory who wish to find more examples, a collection of recent research results on cubic fields, an easy-to-understand source for learning about Voronoi’s unit algorithm and several classical results which are still relevant to the field, and a book which helps bridge a gap in understanding connections between algebraic geometry and number theory. The exposition includes numerous discussions on calculating with cubic fields including simple continued fractions of cubic irrational numbers, arithmetic using integer matrices, ideal class group computations, lattices over cubic fields, construction of cubic fields with a given discriminant, the search for elements of norm 1 of a cubic field with rational parametrization, and Voronoi's algorithm for finding a system of fundamental units. Throughout, the discussions are framed in terms of a binary cubic form that may be used to describe a given cubic field. This unifies the chapters of this book despite the diversity of their number theoretic topics. .

Geometry, Algebraic. --- Equations, Cubic. --- Cubic equations --- Algebraic geometry --- Geometry --- Geometry, algebraic. --- Number theory. --- Algorithms. --- Algebraic Geometry. --- Number Theory. --- Algorism --- Algebra --- Arithmetic --- Number study --- Numbers, Theory of --- Foundations --- Algebraic geometry.

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The objective of this book is to provide tools for solving problems which involve cubic number fields. Many such problems can be considered geometrically; both in terms of the geometry of numbers and geometry of the associated cubic Diophantine equations that are similar in many ways to the Pell equation. With over 50 geometric diagrams, this book includes illustrations of many of these topics. The book may be thought of as a companion reference for those students of algebraic number theory who wish to find more examples, a collection of recent research results on cubic fields, an easy-to-understand source for learning about Voronoi’s unit algorithm and several classical results which are still relevant to the field, and a book which helps bridge a gap in understanding connections between algebraic geometry and number theory. The exposition includes numerous discussions on calculating with cubic fields including simple continued fractions of cubic irrational numbers, arithmetic using integer matrices, ideal class group computations, lattices over cubic fields, construction of cubic fields with a given discriminant, the search for elements of norm 1 of a cubic field with rational parametrization, and Voronoi's algorithm for finding a system of fundamental units. Throughout, the discussions are framed in terms of a binary cubic form that may be used to describe a given cubic field. This unifies the chapters of this book despite the diversity of their number theoretic topics. .

Number theory --- Algebraic geometry --- Geometry --- Computer science --- landmeetkunde --- algoritmen --- getallenleer --- geometrie

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Pell's equation is a very simple, yet fundamental Diophantine equation which is believed to have been known to mathematicians for over 2000 years. Because of its popularity, the Pell equation is often discussed in textbooks and recreational books concerning elementary number theory, but usually not in much depth. This book provides a modern and deeper approach to the problem of solving the Pell equation. The main component of this will be computational techniques, but in the process of deriving these it will be necessary to develop the corresponding theory. One objective of this book is to provide a less intimidating introduction for senior undergraduates and others with the same level of preparedness to the delights of algebraic number theory through the medium of a mathematical object that has fascinated people since the time of Archimedes. To achieve this, this work is made accessible to anyone with some knowledge of elementary number theory and abstract algebra. Many references and notes are provided for those who wish to follow up on various topics, and the authors also describe some rather surprising applications to cryptography. The intended audience is number theorists, both professional and amateur, and students, but we wish to emphasize that this is not intended to be a textbook; its focus is much too narrow for that. It could, however be used as supplementary reading for students enrolled in a second course in number theory.

Number theory --- getallenleer

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Number theory --- getallenleer