Book Description
Discrete Mathematics and its Applications is a focused introduction to the primary themes in a discrete mathematics course, as introduced through extensive applications, expansive discussion, and detailed exercise sets. These themes include mathematical reasoning, combinatorial analysis, discrete structures, algorithmic thinking, and enhanced problem-solving skills through modeling. Its intent is to demonstrate the relevance and practicality of discrete mathematics to all students. The Fifth Edition includes a more thorough and linear presentation of logic, proof types and proof writing, and mathematical reasoning. This enhanced coverage will provide students with a solid understanding of the material as it relates to their immediate field of study and other relevant subjects. The inclusion of applications and examples to key topics has been significantly addressed to add clarity to every subject. True to the Fourth Edition, the text-specific web site supplements the subject matter in meaningful ways, offering additional material for students and instructors. Discrete math is an active subject with new discoveries made every year. The continual growth and updates to the web site reflect the active nature of the topics being discussed. The book is appropriate for a one- or two-term introductory discrete mathematics course to be taken by students in a wide variety of majors, including computer science, mathematics, and engineering. College Algebra is the only explicit prerequisite.
Customer Reviews:
Great buy.......2007-03-15
Purchased for computer science course since it was required. Great price, good book.
A good book .......2006-05-20
I like this book and use it for teaching my students. As a teacher, you have the opportunity to select similar problems for class activity and homework assignment. Also, you can find not only simple and middle-level problems but also high-level problems to provide a good source for active students. Unfortunately, the author uses the letters "F" and "T" in the truth tables instead of much easier and perspective "0" and "1", which correspond to the real needs, e.g., in computer science. That's the most essential lack, which I've found by this time.
This book is interesting.......2006-04-11
I like the exclamations and the concepts/problem solving this book teaches. Maybe I am being swayed by the subject because I enjoy it, however, in comparison to other math books, this one livens it up and is a higher then the average mathematics book.
Very, very helpful..........2006-03-17
This manual was the only reason I got through discrete mathematics for computer science! It's awesome. Now if they only had one for the even solutions...
:-)
The best book on Discrete Math.......2006-03-16
This is the best book I have come across in Discrete Mathematics. It is fun to read and very easy to follow. The subject was taught as a very theoretical way in my course in college, but this book gave a whole new practical dimension to the topics, mainly in the chapters on Graph Theory.
It is a must buy for anybody who wants to know about the fun side of mathematics. That is not to say that this book is not suitable for the serious study program. It has everything that you may require in an Introductory Discrete Math course.
The problem sets are big and the answers to odd number questions are really helpful at times, when you are stuck. The author's detailing of almost every aspect in the book and coupled with the applications of the same idea make it easy to understand.
I have thoroughly enjoyed reading this book and I feel anybody who goes through this book, too will enjoy it immensely.
Book Description
Discrete Mathematics and its Applications, Sixth Edition, is intended for one- or two-term introductory discrete mathematics courses taken by students from a wide variety of majors, including computer science, mathematics, and engineering. This renowned best-selling text, which has been used at over 500 institutions around the world, gives a focused introduction to the primary themes in a discrete mathematics course and demonstrates the relevance and practicality of discrete mathematics to a wide variety of real-world applications…from computer science to data networking, to psychology, to chemistry, to engineering, to linguistics, to biology, to business, and to many other important fields.
Customer Reviews:
The most confusing and worst representated math book I've ever had.......2007-10-01
This book does an excellent job in confusing and misleading student, where as does it a bad job in explanation the concepts. Several of questions ask about the problems that the author never mentions in his text. What is it for? To stimulate students' mind? No, not at all, you confuse your readers instead, Dr. Rosen. I found myself in much easier and helpful situation w/ the help of wikipedia and some other sources than your text itself. I daren't say that your intellectual product is trash, but I definitely can say that your book is a piece of disaster for STUDENTS. You should have better noted in the front cover of the book "MUST BE WITH THE SOLUTION GUIDE." Period.
better than the rest.......2007-09-17
many of the complaints that you read have some validity. i still like it more than most other texts available, because it nicely uses computer science examples (as is clearly stated by the author) AND has a VER through study guide.
Rosen's study guide is thicker than most texts. VERY detailed answers so that you can actually figure out why you got the wrong answer. i would pay more for the study guide than i would for 90% of the krapp being sold today. too many authors take the cheap cop-out of "letting the student explore on their own" = lazy author.
get the study guide, you'll be happy you did.
I liked it.......2007-08-09
Having had many, many math classes, this is one of the better-written books I've used. The subject is, as some have implied or explicitly stated, confusing and kind of a melange of topics. This book does a good job of explaining most topics very clearly and has copius examples for each sub-topic that make learning easier. Overall I was very happy with this book.
Rosen's book: Profs love it, students hate it.......2007-07-09
I have used Rosen's book half a dozen or more times for classes I have taught to undergraduates and graduate students (using editions 3, 4, 5, and 6). My universal experience has been that students find it hard to follow and incomplete. However, because it is so broad in its coverage of topics, has lots of excellent problem sets, and treats the subject seriously, I find it useful as a resource in the class, and a reference outside of class. When I use this book, I know that the students will have to get the concepts from me (won't get them from the text)... but that's what I'm there for. The depth of the text pulls the more advanced students along, and is a sufficient review of a well-planned lecture that, overall, it works.
Could be much better..........2007-05-20
I'm going to have to disagree with previous review and say this book takes a rather easy subject (basic logic..probability...graph theory) and turns it into a mess! I was so confused for the first week or two because i was just reading book and going to lecture. So I stopped reading the book altogether and just went to lecture and it was much better. It seemed like the book was just confusing me more than it was really worth.
Amazon.com
Douglas R. Stinson's Cryptography: Theory and Practice is a mathematically intensive examination of cryptography, including ciphers, the Data Encryption Standard (DES), public key cryptography, one-way hash functions, and digital signatures. Stinson's explication of "zero-sum proofs"--a process by which one person lets another person know that he or she has a password without actually revealing any information--is especially good.
If you are new to the math behind cryptography but want to tackle it, the author covers all of the required background to understand the real mathematics here. Cryptography includes extensive exercises with each chapter and makes an ideal introduction for any math-literate person willing to get acquainted with this material.
Book Description
THE LEGACY… First introduced in 1995, Cryptography: Theory and Practice garnered enormous praise and popularity, and soon became the standard textbook for cryptography courses around the world. The second edition was equally embraced, and enjoys status as a perennial bestseller. Now in its third edition, this authoritative text continues to provide a solid foundation for future breakthroughs in cryptography. WHY A THIRD EDITION? The art and science of cryptography has been evolving for thousands of years. Now, with unprecedented amounts of information circling the globe, we must be prepared to face new threats and employ new encryption schemes on an ongoing basis. This edition updates relevant chapters with the latest advances and includes seven additional chapters covering: · Pseudorandom bit generation in cryptography · Entity authentication, including schemes built from primitives and special purpose "zero-knowledge" schemes · Key establishment including key distribution and protocols for key agreement, both with a greater emphasis on security models and proofs · Public key infrastructure, including identity-based cryptography · Secret sharing schemes · Multicast security, including broadcast encryption and copyright protection THE RESULT… Providing mathematical background in a "just-in-time" fashion, informal descriptions of cryptosystems along with more precise pseudocode, and a host of numerical examples and exercises, Cryptography: Theory and Practice, Third Edition offers comprehensive, in-depth treatment of the methods and protocols that are vital to safeguarding the mind-boggling amount of information circulating around the world.
Customer Reviews:
It lacks theory.......2007-06-15
The book is good about protocols and methods but it lacks the real theory underlying modern cryptography.
For a more serious treatment of these topics i recommend "Introduction to modern cryptography" by J. Katz and Y. Lindell (which i read in form of draft) and "Foundations of Cryptography Vol. 1" by O. Goldreich.
A good reference.......2007-04-24
I may not intend to read this book from cover to cover but would rather use it as a reference. As an engineer I like chapter 2 Shannon's Theory which gives an answer to why a cryptosytem is secure.
Personally I am doing the job related to network security and perfer to recommend the book by C. Kaufman, R. Perlman, and M. Speciner:Network Security: Private Communication in a Public World, Second Edition.
Good book after Schneier's Non-Mathematical Treatment.......2005-05-28
If you are an engineer trying to learn crypto, maybe get a book on number theory to go with this book. It'd be nice if there were fewer errors and more worked out problems, as well.
Overall a good effort but written by a mathematican so you need a book like Schenier's that explains how to use the tools. Maybe Scheier is the one I'd read first. Then read Stinson's to understand how the tools work because Scheier's book is mathematically barren.
Volume III of the Definitive Work.......2005-04-17
This book takes a fairly rigorous mathematical approach to cryptography. It is intended for upper level undergraduate and graduate students in mathematics, computer science and engineering. I suspect only the quite mathematically inclined computer science and engineering students will find this book helpful. This is not a Boy Scout how to do secret messages book, but a book that will give the professional the data needed to implement cryptographic software, and the mathematician hints on both code breaking and creating.
This is the third edition of this book. With the second edition, the author got rid of several several subjects that were not right at the core of cryptography, with the intend of doing a second volume. Instead, the art and scienct of cryptography has changed so fast during the past few years that a two volume approach isn't practical. Instead, he has produced this third edition that picks back up many of the subjects from the first edition. All of the material in this edition has been extensively re-written to incorporate the latest theories and practices.
In recent years the use of cryptography has increased by several orders of magnitude. Every time we buy something with a credit card, use on line banking, send a password to access e-mail, we use cryptography. With this growth, the interest at software companies, universities, and other places has grown accordingly and this text has become the standard by which others are compared.
Highly recommended for the serious student.
Could be a great book .... but it falls short.......2004-05-02
As other people have pointed out, this is not a mathematics book, and it is not an algorithm (recipies) book. It could be a great book for people that are interested in learning these tools to actually use them, either in a research or product development context (something besides homework). Unfortunately, the number of typos, in key mathematical expressions AND PORTIONS OF THE EXPLANATIONS is staggering. Go to the author's web page and you will find that some chapters, like 4 for example, average more than one typo per page (and some of these 'typos' are full sentences, or math expressions that do not look like anything that is actually printed on the page). If you do not have that errata sheet handy, you will waste a lot of time trying to understand the text, or trying to solve the exercises. If you are trying to learn from this book, without attending a class and without the errata, you will simply give up. It is a real shame because it has all the makings of a great book.
Average customer rating:
- A very handy collection
- Printout of Java programs
- Useful problem-solving tool
- A mere compendium of poorly written algorithms
|
A Java Library of Graph Algorithms and Optimization (Discrete Mathematics and Its Applications)
Hang T. Lau
Manufacturer: Chapman & Hall/CRC
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An Inconvenient Truth
ASIN: 1584887184 |
Book Description
Because of its portability and platform-independence, Java is the ideal computer programming language to use when working on graph algorithms and other mathematical programming problems. Collecting some of the most popular graph algorithms and optimization procedures, A Java Library of Graph Algorithms and Optimization provides the source code for a library of Java programs that can be used to solve problems in graph theory and combinatorial optimization. Self-contained and largely independent, each topic starts with a problem description and an outline of the solution procedure, followed by its parameter list specification, source code, and a test example that illustrates the usage of the code. The book begins with a chapter on random graph generation that examines bipartite, regular, connected, Hamilton, and isomorphic graphs as well as spanning, labeled, and unlabeled rooted trees. It then discusses connectivity procedures, followed by a paths and cycles chapter that contains the Chinese postman and traveling salesman problems, Euler and Hamilton cycles, and shortest paths. The author proceeds to describe two test procedures involving planarity and graph isomorphism. Subsequent chapters deal with graph coloring, graph matching, network flow, and packing and covering, including the assignment, bottleneck assignment, quadratic assignment, multiple knapsack, set covering, and set partitioning problems. The final chapters explore linear, integer, and quadratic programming. The appendices provide references that offer further details of the algorithms and include the definitions of many graph theory terms used in the book.
Customer Reviews:
A very handy collection.......2007-04-15
There are many well-written textbooks that cover the theory
and algorithms on graphs and combinatorial optimization.
Very few provide the computer code for the methods. This
book offers an extensive collection of Java programs in
this area. Each program is self-contained and can be used
independently through parameter passing. The drawback of
the book is that the coding style is not object oriented,
and the programs would be difficult to maintain. The
description of the methods and their implementations is
terse. Hence the book is not intended as a learning text.
But the library of programs is a very convenient handy
device for students and researchers in locating solutions
to classroom didactic problems in graphs and optimization,
which apparently is the main objective of the book.
Printout of Java programs.......2007-03-27
This is my third review; my two previous reviews have been removed. I have already notified Amazon about this fact
As I have stated, book is just a printout of Java program, without any explanation how program is doing what is doing, what are program limitations in terms of memory, time and complexity. Programming style is mostly Fortran IV like. Programs are without single line of comment and with non-intuitive variable names, what makes modificatios difficult or impossible. Book can be useful for somebody who needs "black box" library, doesn't need to understand programs and trusts the author that programs fave no flaws
Useful problem-solving tool.......2007-02-09
This library of ready-to-use programs is extremely useful. I have used the programs with very minimal effort in obtaining solutions to some graph optimization problems. Unfortunately the programs are not well documented; it would be a challenge to make modifications to the code. However, the library serves as an ideal black box tool in solving most of the pedagogical graph theory and optimization problems, especially well suited for users who are not of much concern for the underlying methodology and implementation.
A mere compendium of poorly written algorithms.......2007-02-07
There are so many problems with this book, it's hard to know where to begin. So I don't come across as all and only negative, I will first give it credit for gathering together, at least in name a large number of graph processing algorithms.
That said, here are the problems:
The book is just a catalog of graph algorithms with poorly done documentation and even worse actual code. To wit:
*Each algorithm is preceded by a very brief explanation of what it does and some of the issues involved. Suffice it to say that it's the sparsest and most minimal explanation imaginable; if you don't already understand the issues involved, you probably won't after reading the short paragraph or two that precedes each algorithm / method.
*There is but ONE class and every bit of functionality is contained in its own individual, single static method. This "design" causes not a few of the methods to literally run to a thousand and more lines and contain dozens and dozens of (cryptically named) member variables.
So for instance, if you are interested in planarity testing, there's a "method" called planarityTesting that takes four parameters and returns true or false.
All well and good until you actually look at that method and see declared 51 , that's fifty-one, member variables. Each of these variables has poorly chosen names like, "wkpathfind2" and "store2" and "store3" and of course "store4" and "sortptr1" and "sortptr2". I thought this tactic of vowel-conserving naming of variables went out with the 8 + 3 DOS naming convention. At any rate, the cryptic naming scheme combined with the lack of javadoc combine to render each variable's purpose completely opaque. This makes it all but impossible to relate the code to the underlying graph theory.
Then comes the code.
Imagine a thousand and more lines, literally page after page after page of streaming code, all one single method, manipulating these cryptic variables in virtually uncommented ways.
That is pretty much what you get with this book. One algorithm after another after another.
I would say the following:
1) the author codes as if from another time. There is NO object-oriented design to this code whatsoever. None. Zero. Zip.
2)The methods are hundreds or thousands of lines of what amounts to undocumented symbol manipulation. There is small chance to learn anything from this book with respect to relating the code to graph theory.
3) I can say that, having implemented many of the algorithms in this book myself prior to buying this book, the book has contributed nothing to my understanding and further, that already understanding the issues surrounding many of these methods, that is being a qualified reader, is NOT sufficient to allow the reader to follow and understand the algorithms.
4) If you only want to use the (static) methods to return a value or ascertain some property of a graph and you don't care to understand how it works or why it works, then perhaps you'll be happy with this book, but then , why not release the object code as blackbox library? If the code was never meant to be read, and there is no attempt at explaining graph theory as it relates to the code, then what of value is left for the reader?
5) Finally, if the purpose of the book is deliver a good "black-box" library, readers should know that the actual implementation of the graph "object" chosen in this book makes will make that problematic. The book uses an adjacency matrix to represent the graph, a well known data structure in graph theory. Unfortunately, this data structure has the following well-known problem: it is only suitable for the rare instance of dense graphs. The runtime performance and memory demands of this data structure make it unsuitable to any but very very small graphs. Most graphs are neither very very small nor very very dense, (as dense is defined in graph theory), and for that reason almost all graph drawing packages opt for a linked-list data structure to represent the graph.
This is solidly the worst book on this subject I have yet encountered. Amazon offers a number of alternative books, including the fine Graph Algorithms, Third Edition by Robert Sedgewick and Michael Schidlowsky, a book I have no connection with whatsoever and two authors who are otherwise unknown to me. Bundle of Algorithms in Java, Third Edition (Parts 1-5): Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms, Third Edition
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
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Applied Cryptography: Protocols, Algorithms, and Source Code in C, Second Edition
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Practical Cryptography
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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.
Book Description
This book is devoted to a rapidly developing branch of the qualitative theory of difference equations with or without delays. It presents the theory of oscillation of difference equations, exhibiting classical as well as very recent results in that area. While there are several books on difference equations and also on oscillation theory for ordinary differential equations, there is until now no book devoted solely to oscillation theory for difference equations. This book is filling the gap, and it can easily be used as an encyclopedia and reference tool for discrete oscillation theory.
In nine chapters, the book covers a wide range of subjects, including oscillation theory for second-order linear difference equations, systems of difference equations, half-linear difference equations, nonlinear difference equations, neutral difference equations, delay difference equations, and differential equations with piecewise constant arguments. This book summarizes almost 300 recent research papers and hence covers all aspects of discrete oscillation theory that have been discussed in recent journal articles. The presented theory is illustrated with 121 examples throughout the book. Each chapter concludes with a section that is devoted to notes and bibliographical and historical remarks.
The book is addressed to a wide audience of specialists such as mathematicians, engineers, biologists, and physicists. Besides serving as a reference tool for researchers in difference equations, this book can also be easily used as a textbook for undergraduate or graduate classes. It is written at a level easy to understand for college students who have had courses in calculus.
Average customer rating:
|
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
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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.
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.
The last chapter of the book is an excellent introduction to the proof of Fermat's Last Theorem. Considering the level of the book, the author captures very well the essential ideas. Readers will be well prepared, after studying more algebraic number theory and the theory of Galois representations (which the author only skims in the book), to tackle the full proof if so desired.
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The Handbook of Linear Algebra provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use handbook format. The esteemed international contributors guide you from the very elementary aspects of the subject to the frontiers of current research. The book features an accessible layout of parts, chapters, and sections, with each section containing definition, fact, and example segments. The five main parts of the book encompass the fundamentals of linear algebra, combinatorial and numerical linear algebra, applications of linear algebra to various mathematical and nonmathematical disciplines, and software packages for linear algebra computations. Within each section, the facts (or theorems) are presented in a list format and include references for each fact to encourage further reading, while the examples illustrate both the definitions and the facts. Linearization often enables difficult problems to be estimated by more manageable linear ones, making the Handbook of Linear Algebra essential reading for professionals who deal with an assortment of mathematical problems.
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