Chinese Remainder Theorem

Author : C Ding,D Pei,A Salomaa
Publisher : World Scientific
Page : 224 pages
File Size : 46,6 Mb
Release : 1996-10-25
Category : Computers
ISBN : 9789814498364

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Chinese Remainder Theorem by C Ding,D Pei,A Salomaa Pdf

Chinese Remainder Theorem, CRT, is one of the jewels of mathematics. It is a perfect combination of beauty and utility or, in the words of Horace, omne tulit punctum qui miscuit utile dulci. Known already for ages, CRT continues to present itself in new contexts and open vistas for new types of applications. So far, its usefulness has been obvious within the realm of “three C's”. Computing was its original field of application, and continues to be important as regards various aspects of algorithmics and modular computations. Theory of codes and cryptography are two more recent fields of application. This book tells about CRT, its background and philosophy, history, generalizations and, most importantly, its applications. The book is self-contained. This means that no factual knowledge is assumed on the part of the reader. We even provide brief tutorials on relevant subjects, algebra and information theory. However, some mathematical maturity is surely a prerequisite, as our presentation is at an advanced undergraduate or beginning graduate level. We have tried to make the exposition innovative, many of the individual results being new. We will return to this matter, as well as to the interdependence of the various parts of the book, at the end of the Introduction. A special course about CRT can be based on the book. The individual chapters are largely independent and, consequently, the book can be used as supplementary material for courses in algorithmics, coding theory, cryptography or theory of computing. Of course, the book is also a reference for matters dealing with CRT. Contents:Introduction and PhilosophyChinese Remainder AlgorithmIn Modular ComputationsIn AlgorithmicsIn Bridging ComputationsIn Coding TheoryIn CryptographyTutorial in Information TheoryTutorial in AlgebraList of Mathematical SymbolsBibliography Readership: Postgraduate students, researchers and scientists of theoretical foundations of computer science, numerical and computational methods. keywords: “It is a good book about the basic principles of trellis decoding for block codes, existing open problems, some recent solutions, and different applications of this technique.” Computing Reviews

Author : Gareth A. Jones,Josephine M. Jones
Publisher : Springer Science & Business Media
Page : 305 pages
File Size : 53,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781447106135

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Elementary Number Theory by Gareth A. Jones,Josephine M. Jones Pdf

An undergraduate-level introduction to number theory, with the emphasis on fully explained proofs and examples. Exercises, together with their solutions are integrated into the text, and the first few chapters assume only basic school algebra. Elementary ideas about groups and rings are then used to study groups of units, quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part, suitable for third-year students, uses ideas from algebra, analysis, calculus and geometry to study Dirichlet series and sums of squares. In particular, the last chapter gives a concise account of Fermat's Last Theorem, from its origin in the ancient Babylonian and Greek study of Pythagorean triples to its recent proof by Andrew Wiles.

Author : Ulrich Oberst,Martin Scheicher,Ingrid Scheicher
Publisher : Springer Nature
Page : 757 pages
File Size : 44,7 Mb
Release : 2020-06-27
Category : Mathematics
ISBN : 9783030439361

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Linear Time-Invariant Systems, Behaviors and Modules by Ulrich Oberst,Martin Scheicher,Ingrid Scheicher Pdf

This book comprehensively examines various significant aspects of linear time-invariant systems theory, both for continuous-time and discrete-time. Using a number of new mathematical methods it provides complete and exact proofs of all the systems theoretic and electrical engineering results, as well as important results and algorithms demonstrated with nontrivial computer examples. The book is intended for readers who have completed the first two years of a university mathematics course. All further mathematical results required are proven in the book.

Author : Ulrich Libbrecht
Publisher : Courier Corporation
Page : 594 pages
File Size : 51,6 Mb
Release : 2005-01-01
Category : Mathematics
ISBN : 9780486446196

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Chinese Mathematics in the Thirteenth Century by Ulrich Libbrecht Pdf

An exploration of the life and work of the thirteenth-century mathematician Ch'in, this fascinating book examines a range of mathematical issues that reflect Chinese life of a millennium ago. Its first part consists of four closely related studies of Ch'in and his work. The first study brings together what is known of the mathematician's life and of the history of his only extant work, the Shu-shu chiu-chang. Subsequent studies examine the entire range of mathematical techniques and problems found within Ch'in's book. The core of this book consists of an in-depth study of what modern mathematicians still refer to as the Chinese remainder theorem for the solution of indeterminate equations of the first degree. This was Ch'in's most original contribution to mathematics--so original that no one could correctly explain Ch'in's procedure until the early nineteenth century. This volume's concluding study unites information on artisanal, economic, administrative, and military affairs dispersed throughout Ch'in's writings, providing rare insights into thirteenth-century China.

Author : Leonard Eugene Dickson
Publisher : Legare Street Press
Page : 0 pages
File Size : 45,5 Mb
Release : 2023-07-22
Category : Electronic
ISBN : 1022888676

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History Of The Theory Of Numbers - I by Leonard Eugene Dickson Pdf

A landmark work in the field of mathematics, History of the Theory of Numbers - I traces the development of number theory from ancient civilizations to the early 20th century. Written by mathematician Leonard Eugene Dickson, this book is a comprehensive and accessible introduction to the history of one of the most fundamental branches of mathematics. This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

Author : Lindsay N. Childs
Publisher : Springer Science & Business Media
Page : 540 pages
File Size : 54,7 Mb
Release : 2012-12-04
Category : Mathematics
ISBN : 9781441987020

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A Concrete Introduction to Higher Algebra by Lindsay N. Childs Pdf

An informal and readable introduction to higher algebra at the post-calculus level. The concepts of ring and field are introduced through study of the familiar examples of the integers and polynomials, with much emphasis placed on congruence classes leading the way to finite groups and finite fields. New examples and theory are integrated in a well-motivated fashion and made relevant by many applications -- to cryptography, coding, integration, history of mathematics, and especially to elementary and computational number theory. The later chapters include expositions of Rabiin's probabilistic primality test, quadratic reciprocity, and the classification of finite fields. Over 900 exercises, ranging from routine examples to extensions of theory, are scattered throughout the book, with hints and answers for many of them included in an appendix.

Author : Pascal Paillier,David Naccache
Publisher : Springer
Page : 384 pages
File Size : 50,5 Mb
Release : 2003-07-31
Category : Computers
ISBN : 9783540456643

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Public Key Cryptography by Pascal Paillier,David Naccache Pdf

This book constitutes the thoroughly refereed proceedings of the PKC Public Key Cryptography, PKC 2002, held in Paris, France in February 2002. This book presents 26 carefully reviewed papers selected from 69 submissions plus one invited talk. Among the topics addressed are encryption schemes, signature schemes, protocols, cryptanalysis, elliptic curve cryptography, and side channels.

Author : Yoshio Mikami
Publisher : Unknown
Page : 392 pages
File Size : 42,9 Mb
Release : 1913
Category : Mathematics, Chinese
ISBN : PSU:000028269007

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The Development of Mathematics in China and Japan by Yoshio Mikami Pdf

Author : Oystein Ore
Publisher : Courier Corporation
Page : 400 pages
File Size : 55,8 Mb
Release : 2012-07-06
Category : Mathematics
ISBN : 9780486136431

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Number Theory and Its History by Oystein Ore Pdf

Unusually clear, accessible introduction covers counting, properties of numbers, prime numbers, Aliquot parts, Diophantine problems, congruences, much more. Bibliography.

Author : Hong-Bing Yu
Publisher : World Scientific
Page : 115 pages
File Size : 42,8 Mb
Release : 2010
Category : Mathematics
ISBN : 9789814271141

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Problems of Number Theory in Mathematical Competitions by Hong-Bing Yu Pdf

Number theory is an important research field of mathematics. In mathematical competitions, problems of elementary number theory occur frequently. These problems use little knowledge and have many variations. They are flexible and diverse. In this book, the author introduces some basic concepts and methods in elementary number theory via problems in mathematical competitions. Readers are encouraged to try to solve the problems by themselves before they read the given solutions of examples. Only in this way can they truly appreciate the tricks of problem-solving.

Author : Lindsay Childs
Publisher : Springer Science & Business Media
Page : 348 pages
File Size : 54,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781468400656

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A Concrete Introduction to Higher Algebra by Lindsay Childs Pdf

This book is written as an introduction to higher algebra for students with a background of a year of calculus. The book developed out of a set of notes for a sophomore-junior level course at the State University of New York at Albany entitled Classical Algebra. In the 1950s and before, it was customary for the first course in algebra to be a course in the theory of equations, consisting of a study of polynomials over the complex, real, and rational numbers, and, to a lesser extent, linear algebra from the point of view of systems of equations. Abstract algebra, that is, the study of groups, rings, and fields, usually followed such a course. In recent years the theory of equations course has disappeared. Without it, students entering abstract algebra courses tend to lack the experience in the algebraic theory of the basic classical examples of the integers and polynomials necessary for understanding, and more importantly, for ap preciating the formalism. To meet this problem, several texts have recently appeared introducing algebra through number theory.

Author : John Stillwell
Publisher : Springer Science & Business Media
Page : 576 pages
File Size : 48,5 Mb
Release : 2004-01-27
Category : Mathematics
ISBN : 0387953361

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Mathematics and Its History by John Stillwell Pdf

This book offers a collection of historical essays detailing a large variety of mathematical disciplines and issues; it’s accessible to a broad audience. This second edition includes new chapters on Chinese and Indian number theory, on hypercomplex numbers, and on algebraic number theory. Many more exercises have been added as well as commentary that helps place the exercises in context.

Author : Tianxin Cai
Publisher : World Scientific
Page : 430 pages
File Size : 44,5 Mb
Release : 2021-07-21
Category : Mathematics
ISBN : 9789811218316

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A Modern Introduction To Classical Number Theory by Tianxin Cai Pdf

Natural numbers are the oldest human invention. This book describes their nature, laws, history and current status. It has seven chapters. The first five chapters contain not only the basics of elementary number theory for the convenience of teaching and continuity of reading, but also many latest research results. The first time in history, the traditional name of the Chinese Remainder Theorem is replaced with the Qin Jiushao Theorem in the book to give him a full credit for his establishment of this famous theorem in number theory. Chapter 6 is about the fascinating congruence modulo an integer power, and Chapter 7 introduces a new problem extracted by the author from the classical problems of number theory, which is out of the combination of additive number theory and multiplicative number theory.One feature of the book is the supplementary material after each section, there by broadening the reader's knowledge and imagination. These contents either discuss the rudiments of some aspects or introduce new problems or conjectures and their extensions, such as perfect number problem, Egyptian fraction problem, Goldbach's conjecture, the twin prime conjecture, the 3x + 1 problem, Hilbert Waring problem, Euler's conjecture, Fermat's Last Theorem, Laudau's problem and etc.This book is written for anyone who loves natural numbers, and it can also be read by mathematics majors, graduate students, and researchers. The book contains many illustrations and tables. Readers can appreciate the author's sensitivity of history, broad range of knowledge, and elegant writing style, while benefiting from the classical works and great achievements of masters in number theory.

Author : H. E. Rose
Publisher : Oxford University Press
Page : 420 pages
File Size : 49,7 Mb
Release : 1995
Category : Mathematics
ISBN : 0198523769

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A Course in Number Theory by H. E. Rose Pdf

This textbook covers the main topics in number theory as taught in universities throughout the world. Number theory deals mainly with properties of integers and rational numbers; it is not an organized theory in the usual sense but a vast collection of individual topics and results, with some coherent sub-theories and a long list of unsolved problems. This book excludes topics relying heavily on complex analysis and advanced algebraic number theory. The increased use of computers in number theory is reflected in many sections (with much greater emphasis in this edition). Some results of a more advanced nature are also given, including the Gelfond-Schneider theorem, the prime number theorem, and the Mordell-Weil theorem. The latest work on Fermat's last theorem is also briefly discussed. Each chapter ends with a collection of problems; hints or sketch solutions are given at the end of the book, together with various useful tables.

Author : Jin Li,Zheli Liu,Hao Peng
Publisher : Springer
Page : 751 pages
File Size : 40,9 Mb
Release : 2019-06-07
Category : Computers
ISBN : 9783030213732

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Security and Privacy in New Computing Environments by Jin Li,Zheli Liu,Hao Peng Pdf

This book constitutes the refereed proceedings of the 2nd EAI International Conference on Security and Privacy in New Computing Environments, SPNCE 2019, held in Tianjin, China, in April 2019. The 62 full papers were selected from 112 submissions and are grouped into topics on privacy and security analysis, Internet of Things and cloud computing, system building, scheme, model and application for data, mechanism and method in new computing.