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2015 | Book

A Review of Number Theory and Algebra Read chapter Read first chapter

Applied Number Theory

Authors: Harald Niederreiter, Arne Winterhof

Publisher: Springer International Publishing

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This textbook effectively builds a bridge from basic number theory to recent advances in applied number theory. It presents the first unified account of the four major areas of application where number theory plays a fundamental role, namely cryptography, coding theory, quasi-Monte Carlo methods, and pseudorandom number generation, allowing the authors to delineate the manifold links and interrelations between these areas.

Number theory, which Carl-Friedrich Gauss famously dubbed the queen of mathematics, has always been considered a very beautiful field of mathematics, producing lovely results and elegant proofs. While only very few real-life applications were known in the past, today number theory can be found in everyday life: in supermarket bar code scanners, in our cars’ GPS systems, in online banking, etc.

Starting with a brief introductory course on number theory in Chapter 1, which makes the book more accessible for undergraduates, the authors describe the four main application areas in Chapters 2-5 and offer a glimpse of advanced results that are presented without proofs and require more advanced mathematical skills. In the last chapter they review several further applications of number theory, ranging from check-digit systems to quantum computation and the organization of raster-graphics memory.

Upper-level undergraduates, graduates and researchers in the field of number theory will find this book to be a valuable resource.

Table of Contents

Frontmatter
Chapter 1. A Review of Number Theory and Algebra
Abstract
Elementary number theory may be regarded as a prerequisite for this book, but since we, the authors, want to be nice to you, the readers, we provide a brief review of this theory for those who already have some background on number theory and a crash course on elementary number theory for those who have not. Apart from trying to be friendly, we also follow good practice when we prepare the ground for the coming attractions by collecting some basic notation, terminology, and facts in an introductory chapter, like a playwright who presents the main characters of the play in the first few scenes. Basically, we cover only those results from elementary number theory that are actually needed in this book. For more information, there is an extensive expository literature on number theory, and if you want to read the modern classics, then we recommend the books of Hardy and Wright [61] and of Niven, Zuckerman, and Montgomery [151].
Harald Niederreiter, Arne Winterhof
Chapter 2. Cryptography
Abstract
Cryptology in the modern sense is the theory of data security and data integrity. Cryptology as a practical craft can be traced back several thousand years as it was already used in one form or other in the ancient civilizations of Egypt, Mesopotamia, China, Greece, and Rome. It would lead us too far astray if we were to delineate the colorful history of cryptology here, but we will occasionally mention some tidbits. A systematic account of the history of cryptology up to 1967 is given in the book of Kahn [74]. The more recent treatment by Singh [186] offers very stimulating reading.
Harald Niederreiter, Arne Winterhof
Chapter 3. Coding Theory
Abstract
Life is a comedy of errors, at least in the opinion of William Shakespeare, but you can make a concentrated effort to reduce the number of errors that you commit and thus increase the quality of your life. There is probably no panacea for all human errors and mishaps, but in the setting of communication technology, number theory and finite fields can help to prevent errors and ensure the quality of communication. The aim of this chapter is to explain in sufficient detail how this is achieved.
Harald Niederreiter, Arne Winterhof
Chapter 4. Quasi-Monte Carlo Methods
Abstract
There are many scientific as well as real-world applications where we run into the problem of computing a definite integral. In calculus courses you are taught that a definite integral \(\int _{a}^{b}f(u)du\) is evaluated by the fundamental theorem of integral calculus which says that
$$\displaystyle{ \int _{a}^{b}f(u)du = F(b) - F(a), }$$
(4.1)
where the function F is an antiderivative of the integrand f. What you are often not told is that there are many cases where F cannot be expressed in finite terms by means of elementary functions, and in such situations the formula (4.1) is useless for computational purposes. Examples are \(\int _{0}^{1}e^{-u^{2} }du\) and \(\int _{0}^{1}(\sin u)(u + 1)^{-1}du\). We then have to settle for numerical approximations of \(\int _{a}^{b}f(u)du\). The process of approximately computing definite integrals with a sufficient degree of precision is called numerical integration.
Harald Niederreiter, Arne Winterhof
Chapter 5. Pseudorandom Numbers
Abstract
We pointed out in Sect. 4.1.2 that the Monte Carlo method for numerical integration uses random samples, but we did not say anything about how to produce random samples. Maybe we were wise to keep quiet, because nobody really knows how to generate honest-to-goodness random samples in practice. On the other hand, the purely theoretical framework for random sampling is clear: we have a set with at least two elements, we are given a probability distribution (or a probability measure) on this set, and we want to pick elements from this set that fairly represent the probability distribution. But how do we decide whether the sampling is fair? Normally in mathematics there is a definition to which we refer for the verification of a property, but here no generally accepted definition of a fair random sample is codified, unless we deceive ourselves and tolerate tautologies like “a random sample is a collection of elements chosen at random”.
Harald Niederreiter, Arne Winterhof
Chapter 6. Further Applications
Abstract
Check-digit systems and error-correcting codes (see Chap. 3 for the latter) are birds of a feather, but it must be conceded that error-correcting codes are the more colorful birds. Just like error-correcting codes, check-digit systems help to eliminate errors in data, but their aims are more modest than those of error-correcting codes. In a check-digit system we extend an identification number, as for example a bank account number, by a control symbol primarily to detect any single error. A check-digit system can be formally defined over any finite abelian group.
Harald Niederreiter, Arne Winterhof
Backmatter
Metadata
Title
Applied Number Theory
Authors
Harald Niederreiter
Arne Winterhof
Copyright Year
2015
Electronic ISBN
978-3-319-22321-6
Print ISBN
978-3-319-22320-9
DOI
https://doi.org/10.1007/978-3-319-22321-6

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