Combinatorial Optimization

Book Description

A complete, highly accessible introduction to one of today's most exciting areas of applied mathematics

One of the youngest, most vital areas of applied mathematics, combinatorial optimization integrates techniques from combinatorics, linear programming, and the theory of algorithms. Because of its success in solving difficult problems in areas from telecommunications to VLSI, from product distribution to airline crew scheduling, the field has seen a ground swell of activity over the past decade.

Combinatorial Optimization is an ideal introduction to this mathematical discipline for advanced undergraduates and graduate students of discrete mathematics, computer science, and operations research. Written by a team of recognized experts, the text offers a thorough, highly accessible treatment of both classical concepts and recent results. The topics include:
* Network flow problems
* Optimal matching
* Integrality of polyhedra
* Matroids
* NP-completeness

Featuring logical and consistent exposition, clear explanations of basic and advanced concepts, many real-world examples, and helpful, skill-building exercises, Combinatorial Optimization is certain to become the standard text in the field for many years to come.

Customer Reviews:

A Classic in Combinatorial Optimization.......2003-03-19

Combinaorial Optimization is one of those rare books that is an instant classic. The authors weave a readable fabric of intuition and theory that is unmatched in this exciting discipline. The choice of topics covered begins with two fundamental optimization problems, namely, the minimum spanning tree and shortest path problems. Next, maximum flow and minimum cost flow problems are discussed, followed by matching problems, polyhedral issues arising in combinatorial optimization, and the famous traveling salesman problem. The text concludes with chapters on matroids and NP-Completeness. The exposition on these topics is very well written and the proofs are rigorous. There is a terrific blend of theory, algorithms and applications without overwhelming the reader with computational details. The authors also do a good job of developing an accurate historical perspective of the material, most of which evolved during the time period 1955 to 1995. The book is suitable for an upper-level undergraduate, or a graduate course. The exercises are very well thought out and are at an appropriate level. I have taught undergraduate courses in combinatorial optimization for over 10 years and have always struggled to find an appropriate text. My problem has now been solved.

Elegant one, but not a lot of details........1999-09-30

This book was thoroughly written by great-minded Masters. It is well-organized in their topics and presentation. However, the book details is unbalnced, some chapters are overwhelm the data, and some others are insufficient. By the way, I graded this book a Very Good one. Worth Reading !!

A superb introduction to Combinatorial Optimisation.......1999-07-17

A good introduction to Combinatorial optimisation and integer programming.

Book Description

Updated with new material, this Fifth Edition of the most widely used book in combinatorial problems explains how to reason and model combinatorically. It also stresses the systematic analysis of different possibilities, exploration of the logical structure of a problem, and ingenuity. Combinatorical reasoning underlies all analysis of computer systems. It plays a similar role in discrete operations research problems and in finite probability. This book seeks to develop proficiency in basic discrete math problem solving in the way that a calculus text develops proficiency in basic analysis problem solving.

Customer Reviews:

Haphazard Applied Combinatorics is more like it..........2007-04-20

Mr. Tucker has, in his mundane brilliance, decided that all college professors would be able enough to fill in the gaps that he has blatantly left out in this book. The examples assume that the reader has actually been well versed on the subject prior to picking up this bound misfit, nor do they offer very detailed explanations on how he gets from point A to point B. It's the cut and dry "Here's the start, then you do this, and here's the answer" approach is very annoying, especially when your instructor is not the greatest. Furthermore, the exercises listed in some chapters have little or no relevance to the examples the author presents prior. This is a poor author and a poor choice of colleges to choose this book to teach from.

Do not be mislead by the positive reviews; this book is mediocre.......2005-11-04

This book covers basically two topics: Graph Theory and Enumeration.

The things I liked about this book were challenging problems. This book will certainly be a great SOURCE of problems for an upper-level undergraduate course in graph theory or combinatorics.

However, there are too many shortcomings. The book does not cover topics in depth, and the definitions and theorems it gives are stated very precisely and not explained. Unless you have had an introductory course in graph theory or combinatorics, these definitions will take a lot of time to sink in and make intuitive sense. Several useful theorems are not presented at all, are subtly stated in the text, or are presented in some problem.

The next problem is that this book is riddled with errors. And these are more than just errors in the Answers section, of which there are many, but errors in the actual problems! Sometimes even errors in the proofs. Usually these are typographical errors or sometimes just flat out wrong answers. You can find an errata list on the author's site, but it is far from complete.

I assume the other reviewers did not thoroughly work through this book and did not notice the errors. It is inexcusable for a math textbook to have this many errors. It almost seems as though this book wasn't edited at all. It is truly poor.

Excellent for applications.......2004-06-28

The book covers the fundamentals of graph theory and combinatorics (enumeration) and is designed for first courses for undergraduates.

The material is presented in a clear, friendly manner. The sections are short and specific and the emphasis is on problem-solving. Many examples are provided and constitute the majority of the book's volume. Each section ends with 20-30 exercises with answers (not full solutions) at the end of the book.

The book is excellent for computer science and applied math majors looking for a clear, application-based introduction to combinatorics and graph theory. It is also excellent for self-study.

The book's main flaw is that the proofs are not rigorous and are sometimes more intuitive than mathematical. For pure math students looking to explore graph theory and combinatorics in a more rigorous manner, other books (e.g. Diestel, "Graph Theory") will serve that purpose better.

An almost ideal introduction book to combinatorics.......2002-05-23

There have been wonderfully written reviews of this book, but since this is really an excellent textbook, I am urged to praise again. Fully recommended.

This book is easily and clearly written; covers almost every important basic concept and technic in graph theory and enumerative combinatorics, with neatly selected and wonderfully organised exercises.

And I highly suggest the author give the references to those last exercises in every section, since each of them does lead into a theory.

An almost ideal introduction book to combinatorics.......2002-05-23

There have been wonderfully written reviews of this book, but since this is really an excellent textbook, I am urged to praise again. Fully recommended.

And I highly suggested the author give the references to those last exercises in every section, since each of them does lead into a theory.

Small Worlds: The Dynamics of Networks between Order and Randomness (Princeton Studies in Complexity)
Average customer rating:

All the details you need to know to understand Watts' and Strogatz' famous article

Not enough contents to be a good book

Good, but some typos

Inspiring

Great scientific synthesis

Small Worlds: The Dynamics of Networks between Order and Randomness (Princeton Studies in Complexity)
Duncan J. Watts
Manufacturer: Princeton University Press
ProductGroup: Book
Binding: Paperback

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The Structure and Dynamics of Networks: (Princeton Studies in Complexity)

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Sync: The Emerging Science of Spontaneous Order

ASIN: 0691117047

Book Description

Everyone knows the small-world phenomenon: soon after meeting a stranger, we are surprised to discover that we have a mutual friend, or we are connected through a short chain of acquaintances. In his book, Duncan Watts uses this intriguing phenomenon--colloquially called "six degrees of separation"--as a prelude to a more general exploration: under what conditions can a small world arise in any kind of network?

The networks of this story are everywhere: the brain is a network of neurons; organisations are people networks; the global economy is a network of national economies, which are networks of markets, which are in turn networks of interacting producers and consumers. Food webs, ecosystems, and the Internet can all be represented as networks, as can strategies for solving a problem, topics in a conversation, and even words in a language. Many of these networks, the author claims, will turn out to be small worlds.

How do such networks matter? Simply put, local actions can have global consequences, and the relationship between local and global dynamics depends critically on the network's structure. Watts illustrates the subtleties of this relationship using a variety of simple models---the spread of infectious disease through a structured population; the evolution of cooperation in game theory; the computational capacity of cellular automata; and the sychronisation of coupled phase-oscillators.

Watts's novel approach is relevant to many problems that deal with network connectivity and complex systems' behaviour in general: How do diseases (or rumours) spread through social networks? How does cooperation evolve in large groups? How do cascading failures propagate through large power grids, or financial systems? What is the most efficient architecture for an organisation, or for a communications network? This fascinating exploration will be fruitful in a remarkable variety of fields, including physics and mathematics, as well as sociology, economics, and biology.

Customer Reviews:

All the details you need to know to understand Watts' and Strogatz' famous article.......2007-03-12

The book basically gives all the details needed to understand Watts and Strogatz famous Nature article 'Collective Dynamics of Complex Networks' in 1998. I think that it is basically Watts PhD-thesis and as such it is of course nicely written, but nothing for the laymen who is rather referred to Watts other, more story-telling book 'Six Degrees', Barabasi's book 'Linked', or to another book that I would recommend most, namely the one by Mark Buchanan titled 'Small Worlds'. Mark is a skillful scientific writer and his book has a broader scope that makes it more interesting than each of the two monographs that are a bit more focused on the scientists own contribution.

Not enough contents to be a good book.......2005-07-08

Networks are since a couple of years object of intense research in several different disciplines. One reason therefore is certainly the outstanding article by Watts and Strogatz, Collective dynamics of small world networks, Nature, 393:440--442, 1998. Unfortunatelly, this book can not continue the high level of this article. Actually, it does not really provide much more information than the article itself. I would suggest to read the article cited above and either decide for another book or to look directly in the literature and read the origninal articles.

To summarize, this book is not terribly weak, but one can clearly sees that it swims on the current 'complex networks' wave without providing enough justification for its existence. Of course, if you do not have access to the original literature and just what to have a general overview of complex networks and what be done with them, you may consider buying this book.

Good, but some typos.......2005-06-02

Mathematical level: Moderate; there's no calculus, and little high level math, but the book is quite mathematical in tone, and some of the arguments may be difficult to follow without a good "math sense". There are MANY equations and graphs.

Good points: Watts covers an area that will interest those who deal with mathematical models of social networks e.g. models of disease-spread, especially HIV. It might, however, cover other things that can spread through networks as well. He presents analysis of graphs (or networks) that are neither random nor highly structured; and begins to examine ways that the degree of structure v. randomness can be measured.

Bad points: There are more than the usual number of typos. The models presented are a "first step", only.

Inspiring.......2001-07-24

The author believes that human thought might be a small world, in the sense that one could reach any idea if he/she finds the right associations and "short-cut"s. The small-world theory is indeed one of those short cuts itself. It links many different domains and uncovers some interesting common behavior.

The theory is developed in a scientific manner with extensive numerical support. Rich literature reviews and many open questions make this book a good research reference. Complex observations are generally followed by qualitative explanations. However, some of the simpler derivations are not fully clear. I believe that adding a few lines here and there can turn this book into a textbook.

The book spans many different areas of science and a deep understanding of the related results may require some background. However, each chapter ends with a brief summary, allowing the reader to move forward if he/she finds the chapter difficult. In summary, as the author puts it, the book is simply the "end of the beginning" in an exciting new field.

Great scientific synthesis.......2000-07-12

The book takes a systematic look at the 'small world' graphs. These natural graphs have been discovered by graph theoretist as erly as 60's, but were not properly understood. The graphs are remarkable in their ability to cluster and scale lengths. There are fundumental connections between these graphs and complex systems, discrete dynamical systems, computation and information processing. Duncan has done a tremendous job in building experimetal and theoretical models trying to understand how these graphs come about and sustain themselves. Read this book.

Handbook of Applied Cryptography (Crc Press Series on Discrete Mathematics and Its Applications)

Average customer rating:

Advanced Crypto for the college mind.
Fantastic traditional reference
A very detailed book, but not for everyone.
Complete and satisfying
Very depthful yet readable

Handbook of Applied Cryptography (Crc Press Series on Discrete Mathematics and Its Applications)
Alfred J. Menezes , Paul C. van Oorschot , and Scott A. Vanstone
Manufacturer: CRC
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Binding: Hardcover

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Applied Cryptography: Protocols, Algorithms, and Source Code in C, Second Edition
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ASIN: 0849385237

Book Description

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.

Customer Reviews:

Advanced Crypto for the college mind........2004-04-26

This very detailed work is not for the light hearted. It's an in depth look at the mathmatics behind cryptography. If you're looking for a book to help you program then look for Applied Cryptography by Bruce the crypto king instead. If you're looking for something to help you learn cryptoanalysis and how to break codes then this is the first step.

Fantastic traditional reference.......2004-01-03

The Chapter 14 - Efficient Implementation - shows several multiple precision algorithms. They are very easy to understand and implement under any microprocessor. It is a very good complement to the book set written by Donald Knuth (The Art of Computer Programming, Volumes 1-3 Boxed Set), another fantastic traditional reference.

A very detailed book, but not for everyone........2003-10-13

This is a fairly strong book on crypto, with heavy detail on the math involved. The upside is that the second chapter is devoted to most of the important mathematical theory you'll need to understand for the rest of the book. The downside? That chapter tries to cover just about the same breadth of information as a semester long course in Number Theory.

If you don't have a ton of mathematical background and are scared of having to take a crash course in number theory, or are looking for a higher level view of things, I'd suggest something more along the lines of Bruce Schneier's 'Applied Cryptography' (ASIN 0471117099). If you have some mathematical background, but want to get into things in detail, this is probably for you.

If you're not sure whether you'll like the book, you should definitely take a look at it. While Amazon currently doesn't have sample pages, if you do a Web Search on "Handbook of Applied Cryptography", you can find Sample Chapters hosted online to give you a good feel for the book's style.

Complete and satisfying.......2003-07-06

This book is a deep detailed analysis of
modern cryptography. It is light on
cryptanalysis.
The mathematical background information
and explanations are complete and clear.
It is very satisfying to be able to read
the prose and implement the ideas in
a computer program with ease.

Very depthful yet readable.......2003-02-22

I read 4 other books before picking this one. It is the most detailed and readable book. Covers all aspect of the Cryptography. Worth the money.

Introduction to Coding Theory (Graduate Texts in Mathematics)

Average customer rating:

Excellent book from mathematical standpoint

Introduction to Coding Theory (Graduate Texts in Mathematics)
J.H. van Lint
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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Accessories:

Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Algebraic Cobordism (Springer Monographs in Mathematics)
Introduction to Singularities and Deformations (Springer Monographs in Mathematics)

ASIN: 3540641335

Book Description

From the reviews: "The 2nd (slightly enlarged) edition of the van Lint's book is a short, concise, mathematically rigorous introduction to the subject. Basic notions and ideas are clearly presented from the mathematician's point of view and illustrated on various special classes of codes...This nice book is a must for every mathematician wishing to introduce himself to the algebraic theory of coding." European Mathematical Society Newsletter, 1993 "Despite the existence of so many other books on coding theory, this present volume will continue to hold its place as one of the standard texts...." The Mathematical Gazette, 1993

Customer Reviews:

Excellent book from mathematical standpoint.......2005-02-20

Very good intro textbook. It gives short, detailed preps to various coding areas (linear, cyclic, convolutional). The biggest advantage this book has is that it does not throw at You tonnes of unnecessary info (like many other thick books do). That is, it assumes reader has some basic understanding of algebra and probability theory. Let's say, it gives good theoretical presentation such that the reader gets good theoretical understanding, it is not example-based.

Average customer rating:

Good Text
Hope this is a 1st edition error...
Superb textbook
Excellent Book for a Graph Theory and Combinatorics Course
One of the best math books written

Applied Combinatorics
Fred Roberts
Manufacturer: Prentice Hall
ProductGroup: Book
Binding: Paperback

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Introductory Combinatorics (4th Edition)
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ASIN: 0130393134

Customer Reviews:

Good Text.......2005-03-08

This text is an excellent introduction to the subject, as it will give the reader a solid base of the main ideas in combinatorics. However, I rated it with four stars for two reasons. One is that the authors sometimes use nonconventional notation, another is that there are occasional typos. Other than that I highly recommend this book.

Hope this is a 1st edition error..........2004-10-21

looking at the sample pages, on page 3, it said 5! = 60... I may not know anything about combinatorics, but I do know my factorials...

Superb textbook.......2002-04-23

We used this text for our introductory class in Combinatorics in graduate Mathematics at Louisville. Very well written with fantastic examples.

Excellent Book for a Graph Theory and Combinatorics Course.......2000-04-13

Simply stated, this is an excellent book. I used this book to teach an upper division undergraduate course in Graph Theory and Combinatorics. This book is by far the best book I have come across for teaching these topics. Incredibly the book contains very few errors (I've only found two errors (typos) ). In addition, the book has applications of the concepts for a wide range of disciplines.

The author has a wide range of problems at the end of each section. Almost all of the problems are well written with clear directions.

Every Computer Science/Mathematics major should have this book in their library. It's great!

One of the best math books written.......1999-07-21

This is one of the best mathematics books I have ever read. I used this in a discrete modeling course. Unfortunately, we didn't cover some of the more interesting topics like coding theory. I've since graduated and when I get a penchant to study a little math, this is the book I go to first because there is such a diverse array of topics presented in a clear, concise manner. I only wish there was a new edition.

A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems (Nonconvex Optimization and Its Applications)

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A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems (Nonconvex Optimization and Its Applications)
Hanif D. Sherali , and W.P. Adams
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 0792354877

Book Description

This book addresses a new method for generating tight linear or convex programming relaxations for discrete and continuous nonconvex programming problems. Problems of this type arise in many economics, location-allocation, scheduling and routing, and process control and engineering design applications. The principal thrust is to commence with a model that affords a useful representation and structure, and then to further strengthen this representation through an automatic reformulation and constraint generation technique. The contents of this book comprise the original work of the authors compiled from several journal publications, and not covered in any other book on this subject. The outstanding feature of this book is that it offers for the first time a unified treatment of discrete and continuous nonconvex programming problems. In essence, the bridge between these two types of nonconvexities is made via a polynomial representation of discrete constraints. The book lays the foundation of an idea that is stimulating and that has served to enhance the solubility of many challenging problems in the field.
Audience: This book is intended for researchers and practitioners who work in the area of discrete or continuous nonlinear, nonconvex optimization problems, as well as for students who are interested in learning about techniques for solving such problems.

Average customer rating:

Abstract Regular Polytopes
Peter McMullen , and Egon Schulte
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Hardcover

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ASIN: 0521814960

Book Description

Abstract regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra. The rapid development of the subject in the past twenty years has resulted in a rich new theory featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology. This is the first comprehensive, up-to-date account of the subject and its ramifications. It meets a critical need for such a text, because no book has been published in this area since Coxeter's "Regular Polytopes" (1948) and "Regular Complex Polytopes" (1974).

Download Description

Abstract regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra. They are highly symmetric combinatorial structures with distinctive geometric, algebraic or topological properties; in many ways more fascinating than traditional regular polytopes and tessellations. The rapid development of the subject in the past 20 years has resulted in a rich new theory, featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology. Abstract regular polytopes and their groups provide an appealing new approach to understanding geometric and combinatorial symmetry. This is the first comprehensive up-to-date account of the subject and its ramifications, and meets a critical need for such a text, because no book has been published in this area of classical and modern discrete geometry since Coxeter's Regular Polytopes (1948) and Regular Complex Polytopes (1974). The book should be of interest to researchers and graduate students in discrete geometry, combinatorics and group theory.

Elliptic Curves: Number Theory and Cryptography (Discrete Mathematics and Its Applications)

Average customer rating:

Washington Elliptic Curves
Solid intermediate introduction to elliptic curves
A clear, concise introduction to elliptic curves
It might be a good book for a mathematic student but not a good one for an engineering student.
Excellent

Elliptic Curves: Number Theory and Cryptography (Discrete Mathematics and Its Applications)
Lawrence C. Washington
Manufacturer: Chapman & Hall/CRC
ProductGroup: Book
Binding: Hardcover

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Guide to Elliptic Curve Cryptography (Springer Professional Computing)
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ASIN: 1584883650

Book Description

Elliptic curves have played an increasingly important role in number theory and related fields over the last several decades, most notably in areas such as cryptography, factorization, and the proof of Fermat's Last Theorem. However, most books on the subject assume a rather high level of mathematical sophistication, and few are truly accessible to senior undergraduate or beginning graduate students. Assuming only a modest background in elementary number theory, groups, and fields, Elliptic Curves: Number Theory and Cryptography introduces both the cryptographic and number theoretic sides of elliptic curves, interweaving the theory of elliptic curves with their applications. The author introduces elliptic curves over finite fields early in the treatment, leading readers directly to the intriguing cryptographic applications, but the book is structured so that readers can explore the number theoretic aspects independently if desired. By side-stepping algebraic geometry in favor an approach based on basic formulas, this book clearly demonstrates how elliptic curves are used and opens the doors to higher-level studies. Elliptic Curves offers a solid introduction to the mathematics and applications of elliptic curves that well prepares its readers to tackle more advanced problems in cryptography and number theory.

Customer Reviews:

Washington Elliptic Curves.......2007-01-12

I bought this book as a follow-up to working my way through "Introduction to Cryptography with Coding Theory" (by the same author together Wade Trappe) (which I strongly recommend as well). I was not disappointed - Washington covers a difficult but important topic in a masterly fashion which should be accessible to anyone with a serious interest in elliptic curve cryptography. It successfully follows a middle road between the standard, but rather abstract texts on number theory and those which give details of algorithms but few proofs. There are ample examples and enjoyable exercises. Strongly recommended.

Solid intermediate introduction to elliptic curves.......2006-06-12

I compare this book to Rational Points on Elliptic Curves (RP) by Tate and Silverman, and The Arithmetic of Ellipitic Curves (AEC) by Silverman.

RP is definitely for junior and senior undergraduates interested in elliptic curves. With modest knowledge of real and complex analysis (calculus and some complex calculus), RP introduces the concept of elliptic curves and presents many interesting results. Unfortunately, a lot of hand waving goes on, i.e., many results are merely stated, instead of proved.

AEC is definitely for graduate students who have all ready taken the graduate algebra and geometry sequences. A lot of high powered mathematics is used in this text to get at the heart of elliptic curves.

Washington's book falls right in between these two books. He assumes knowledge of some analysis and algebra (particulary abelian groups), then develops much of what else is needed. Some hand waving exists (mainly for some of the high powered projective geometry needed to fully understand the geometry of elliptic curves) in this book, but this does not detract from the understanding of the additive group on elliptic curves, the primary focus of the book.

For those with a basic handle on real analysis and group theory, this book can easily be used for self-teaching.

A clear, concise introduction to elliptic curves.......2006-02-20

I used this book as my main resource when writing my undergraduate dissertation on elliptic curve group structure. Although once I wanted to have a more in-depth look into any particular subject I had to chase up the references, this book made an excellent starting point. This book is a solid, clear introduction to the subject, which can be easily understood even by maths undergrads in the later years of their study (though if you're not a mathematician you may find it hard going!!) I found it be the clearest textbook on elliptic curves I came across, especially as it doesn't assume any background knowledge of algebraic geometry.

It might be a good book for a mathematic student but not a good one for an engineering student........2005-09-06

It might be a good book for a mathematic student but not a good one for an engineering student. There are too many mathematic jargons with very limited explanations. Many notations just take for granted that the readers have already known them. It is very hard for people who have limited math background. Moreover, there are so many editorial errors in the current version. I would suggest that the author put a mathematical symbol/sign index at the end of the book and make it easier for the readers to look for their meanings.

Excellent.......2003-07-19

Anyone who writes a book on elliptic curves will never do a bad job, for these objects are so beautiful that it would be a sacrilege to do otherwise. Those who study elliptic curves fall under their spell, not only because of their beauty, but also because of their many applications: the spinning top in mechanics, cryptography, exactly solved models in statistical mechanics, precession of the Mercury perihelion in general relativity, the proof of Fermat's Last (Wiles) Theorem, control theory, and string theory, to name a few. This book is an excellent treatment of ECs and would be good for a graduate student starting out in the field. The author gives many concrete examples of the main theorems, and helpful exercises are found at the end of each chapter.

The author begins the book with two neat problems that motivate well the subject of elliptic curves: the pyramid of cannonballs and the right triangle problem, i.e. which integers can occur as areas of right triangles with integer sides? He then immediately begins the elementary theory of ECs in chapter 2. The treatment is pretty standard, although he proves Pascal's and Pappus's theorems using the associativity of the group operation on ECs, which is not usually done in books on ECs. Also somewhat non-standard this early in the game is the discussion of reduction of ECs modulo various primes, and the subsequent definitions of additive, split multiplicative, and non-split multiplicative reduction.

The study of torsion points is done in chapter 3 with the Weil pairing on the n-torsion of an EC taking center stage. A fairly short chapter, the author delays the proof of the properties of the Weil pairing until chapter 11, where it is done with divisors.

Chapter 4 deals with elliptic curves over finite fields, and is one of the most important in the book from the standpoint of cryptographic applications of ECs. Hasse's theorem, giving the bounds for the group of points on an EC over a finite field, is proven in detail. The Frobenius endomorphism is introduced, and a proof of Schoof's algorithm for computing the number of points on ECs over a finite field is given a detailed treatment. There are many symbolic computational software packages in both the open and commerical realm which will do the counting straightforwardly, and anyone interested in cryptography will need to be familiar with some of these. Supersingular curves in characteristic p are introduced, and the author gives a good discussion of the reason why they are named as such.

The discrete logarithm problem, a topic also very important for cryptographic applications, is discussed in chapter 5. The chapter beings with the index calculus, and, recognizing that it does not apply to general groups, the Pohlig-Hellman, baby step-giant step method, and Pollards rho and lambda methods are discussed in details. The author then shows that for supersingular and "anomalous" curves, that the discrete logarithm problem can be reduced to an easier discrete logarithm problem. Along the way, two important concepts are introduced: the p-adic valuation, and the Tate-Lichtenbaum pairing, the latter of which is related to the Weil pairing, but applies to situations where the Weil pairing does not.

Elliptic curve cryptography is then discussed in chapter 6, and the treatment is fairly thorough. The author shows to what extent the Decision Diffie-Hellman problem can be solved using the Weil pairing. He also shows how to represent a message on an elliptic curve, satisfying early on any reader's curiosity on just how this is done. The El Gamal and ECDSA are compared in terms of their computational efficiency. An EC generalization of RSA is also discussed in some detail, along with a cryptosystem based on the Weil pairing. Chapter 7 then gives other applications of ECs, such as factoring and primality testing.

Chapter 8 marks the beginning of the "heavy artillery" in the theory of ECs, for here the author begins the discussion of elliptic curves over the rational numbers, which can be viewed as an example of Diophantine geometry. The famous Mordell-Weil theorem is proved, and as a sign that one is definitely in the arena of modern mathematics, the proof is given in terms of Galois cohomology, which is an abstraction of the Fermat method of descent. The reader gets a taste of height functions, and via some good examples, gets insight into why the rank of the EC is so difficult to compute. A neat example is given of a nontrivial Shafarevich-Tate group.

I did not read the chapters 9, 10, or 11 on ECs over the complex numbers, complex multiplication, and divisors, so I will omit their review. Chapter 12 introduces the famous zeta functions, and their use in obtaining arithmetic information about an EC. Zeta functions motivate the definition of an L-function of an EC, these being tremendously important in modern developments in the theory of ECs, such as the Swinnerton-Dyer and Birch conjecture, the latter of which is motivated rather nicely in this chapter.

Book Description

This is the first volume of a two-volume text on design theory. Since the first edition, there has been extensive development of the theory. In particular, the growing importance of discrete mathematics to many parts of engineering and science has made designs a useful tool for applications. The authors acknowledge this trend with an additional chapter on applications. It is suitable for advanced courses and as a reference work, not only for researchers in discrete mathematics or finite algebra, but also for those working in computer and communications engineering. The book features exercises throughout and concludes with an extensive and updated bibliography of over 1800 entries.

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