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Developed from the authors’ courses at Syracuse University and the U.S. Air Force Research Laboratory, Access Control, Security, and Trust: A Logical Approach equips readers with an access control logic they can use to specify and verify their security designs. Throughout the text, the authors use a single access control logic based on a simple propositional modal logic.

The first part of the book presents the syntax and semantics of access control logic, basic access control concepts, and an introduction to confidentiality and integrity policies. The second section covers access control in networks, delegation, protocols, and the use of cryptography. In the third section, the authors focus on hardware and virtual machines. The final part discusses confidentiality, integrity, and role-based access control.

Taking a logical, rigorous approach to access control, this book shows how logic is a useful tool for analyzing security designs and spelling out the conditions upon which access control decisions depend. It is designed for computer engineers and computer scientists who are responsible for designing, implementing, and verifying secure computer and information systems.

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Information is an important feature of the modern world. Mathematical techniques underlie the devices that we use to handle it, for example, mobile phones, digital cameras, and personal computers.

This book is an integrated introduction to the mathematics of coding, that is, replacing information expressed in symbols, such as a natural language or a sequence of bits, by another message using (possibly) different symbols. There are three main reasons for doing this: economy, reliability, and security, and each is covered in detail. Only a modest mathematical background is assumed, the mathematical theory being introduced at a level that enables the basic problems to be stated carefully, but without unnecessary abstraction. Other features include:

• clear and careful exposition of fundamental concepts, including optimal coding, data compression, and public-key cryptography;
• concise but complete proofs of results;
• coverage of recent advances of practical interest, for example in encryption standards, authentication schemes, and elliptic curve cryptography;
• numerous examples and exercises, and a full solutions manual available to lecturers from www.springer.com

This modern introduction to all aspects of coding is suitable for advanced undergraduate or postgraduate courses in mathematics, computer science, electrical engineering, or informatics. It is also useful for researchers and practitioners in related areas of science, engineering and economics.

Customer Reviews (1)

excellent basic text
I'll be teaching an undergraduate course out of this text next semester. Sections are short, with a few exercises after each one. Proofs are included. Wide selection of basic selected topics in information theory, coding theory (including Huffman codes as well as some error-correcting codes such as Hamming codes), and cryptography. This book is useful for someone who is interested in those topics and wants to have a carefully written but basic introduction. In my opinion, Biggs is a terrific writer, the same one who wrote the Discrete Mathematics book published by Oxford Univ Press. ... Read more

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Juraj Hromkovic takes the reader on an elegant route through the theoretical fundamentals of computer science. The author shows that theoretical computer science is a fascinating discipline, full of spectacular contributions and miracles. The book also presents the development of the computer scientist's way of thinking as well as fundamental concepts such as approximation and randomization in algorithmics, and the basic ideas of cryptography and interconnection network design.

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Nigel Smart's "Cryptography" provides the rigorous detail required for advanced cryptographic studies, yet approaches the subject matter in an accessible style in order to gently guide new students through difficult mathematical topics. ... Read more

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Number theory is one of the oldest and most appealing areas of mathematics. Computation has always played a role in number theory, a role which has increased dramatically in the last 20 or 30 years, both because of the advent of modern computers, and because of the discovery of surprising and powerful algorithms. As a consequence, algorithmic number theory has gradually emerged as an important and distinct field with connections to computer science and cryptography as well as other areas of mathematics. This text provides a comprehensive introduction to algorithmic number theory for beginning graduate students, written by the leading experts in the field. It includes several articles that cover the essential topics in this area, such as the fundamental algorithms of elementary number theory, lattice basis reduction, elliptic curves, algebraic number fields, and methods for factoring and primality proving.In addition, there are contributions pointing in broader directions, including cryptography, computational class field theory, zeta functions and L-series, discrete logarithm algorithms, and quantum computing. ... Read more

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Since the appearance of the authors' first volume on elliptic curve cryptography in 1999 there has been tremendous progress in the field. In some topics, particularly point counting, the progress has been spectacular. Other topics such as the Weil and Tate pairings have been applied in new and important ways to cryptographic protocols that hold great promise. Notions such as provable security, side channel analysis and the Weil descent technique have also grown in importance. This second volume addresses these advances and brings the reader up to date. Prominent contributors to the research literature in these areas have provided articles that reflect the current state of these important topics. They are divided into the areas of protocols, implementation techniques, mathematical foundations and pairing based cryptography. Each of the topics is presented in an accessible, coherent and consistent manner for a wide audience that will include mathematicians, computer scientists and engineers. ... Read more

Customer Reviews (3)

The latest cutting edge research on Elliptic Curve Cryptography
First, the reviews dated below (July 25, 2002, July 29, 2000 [Lee Carlson] and January 31, 2000) are refering to Blake, Seroussi and Smart's first book: Elliptic Curves in Cryptography: London Mathematical Society Lecture Note Series 265, not the new book Advances in Elliptic Curve Cryptography, London Mathematical Society Lecture Note Series 317.

Contents of Advances in Elliptic Curve Cryptography, London Mathematical Society Lecture Note Series 317 (ISBN-10: 052160415X).

Chapter I: covers Elliptic Curve Based Protocols in the IEEE 1363 standard, ECDSA (EC Digital Signature Algorithm), ECDH (EC Diffie-Hellman) /ECMQV (EC MQV protocol of Law, Menezes, QU, Solinas and Vanstone) and ECIES (EC Integrated Encryption Scheme).

Chapter II: on the provable security of ECDSA.

Chapter III: proofs of security for ECIES,

Chapter IV: side-channel analysis.

Chapter V: defenses against side-analysis.

Chapter VI: advances in point counting.(This is an advanced chapter covering Takakazu Satoh's fast p-adic algorithm.Note, a very brief introduction to p-adic fields and extensions is given at the start of this chapter.)

Chapter VII: hyperelliptic curves and HCDLP.

Chapter VIII: weil descent attacks.

Chapter IX: pairings.

Chapter X: cryptography from pairings.(Highlight: coversBoneh and Franklin's identity based encryption (IBE) using Weil pairings.)

This book, published in April, 2005, brings the reader up to date with much of the latest research on Elliptic Curve Cryptography.

The algorithms are in the same format as in Elliptic Curves in Cryptography.Also, like in their first book, this book also does not always give proofs.

Highly recommended for advanced graduate students, applied mathematicians and computer scientists in the field of public key cryptography.The mathematics is more advanced than in their first book on Elliptic Curve Cryptography.

Good compact book on elliptic curves in cryptography
This book gives a good summary of the current algorithms and methodologies employed in elliptic curve cryptography. The book is short (less than 200 pages), so most of the mathematical proofs of the main results are omitted.The authors instead concentrate on the mathematics needed to implementelliptic curve cryptography. The book is written for the reader with someexperience in cryptography and one who has some background in the theory ofelliptic curves.A reader coming to the field for the first time mightfind the reading difficult. The authors do give a brief summary in Chapter1 on the idea of doing cryptography based on group theory. They then moveon to discuss finite field arithmetic in Chapter 2. The reader is expectedto know some of the basic notions of multiprecision arithmetic forintegers. The authors choose to work with 2^16. Psuedocode is given fordoing modular arithmetic with Montgomery arithmetic given specialattention. The last section of the chapter gives a good summary ofarithmetic in fields of characteristic 2. Chapter 3 discusses verycompactly arithmetic in elliptic curves. This is where the reader shouldalready have the background in the theory of elliptic curves, since thereading is very fast and formal. The authors do a good job of summarizinghow modular polynomials come into play in elliptic curve cryptography andgive some explicit examples of these polynomials. The most importantchapter of the book is Chapter 4, where the authors give a discussion ofhow to implement elliptic curves efficiently in cryptosystems. This chapteris nicely written and pseudocode appears many times with lots of niceexamples. This chapter serves as background for the next one on thediscrete logarithm problem using elliptic curves over finite fields. TheMOV attack, the anomalous attack, and the baby step/giant step methods arediscussed very nicely. Random methods, such as the tame and wild kangarooare discussed at the end of the chapter.

The next three chaptersconcentrate on how to actually generate elliptic curves for cryptosystems,with particular attention payed to the Schoof Algorithm. The chapter onSchoof's algorithm is more detailed than the rest of the chapters and thismakes for better reading. The authors do discuss how to generate curvesusing complex multiplication although the discussion is somewhat hurried. The next chapter discusses how elliptic curves have been applied to otherareas in cryptography, such as factoring, etc. A good discussion of theECPP algorithm on proving primality ends the chapter.The authors endthe chapter with a discussion of hyperelliptic cryptography. Anyonefamiliar with the theory of elliptic curves and how they are applied tocryptography will naturually ask if hyperelliptic curves have anyadvantages over the elliptic case. The authors never really address thisexplicity but do give examples on just what is involved in implementinghyperelliptic curves in cryptography. Overall a fine addition to theliterature on elliptic curves in cryptography. One would hope that theauthors would write a follow-up book on hyperelliptic curves and maybe ongeneral algebraic curves and their possible use in this area.

Good book
I think this is one of the best introductions to elliptic curve cryptosystems. This book have all the last algorithms in the field. ... Read more

Product Description
In today’s world, data must be sent around the world cheaply and securely, and that requires origin authentication, integrity protection, and confidentiality – the recipient of a message should be able to ascertain who sent the message, be sure that the message has not been changed en route, and be sure that the data arrives without having been read by anyone else.The second editor invented signcryption, an area of cryptography that studies systems that simultaneously provide origin authentication, integrity protection and confidentiality for data. Signcryption schemes combine the features of digital signature schemes with those of public-key encryption schemes and aim to provide security guarantees in a way that is provably correct and significantly less computationally expensive than the “encrypt-then-sign” method most commonly adopted in public-key cryptography. This is the first comprehensive book on signcryption, and brings together leading authors from the field of cryptography in a discussion of the different methods for building efficient and secure signcryption schemes, and the ways in which these schemes can be used in practical systems. Chapters deal with the theory of signcryption, methods for constructing practical signcryption schemes, and the advantages of using such schemes in practical situations. The book will be of benefit to cryptography researchers, graduate students and practitioners. ... Read more

Product Description

This book constitutes the refereed proceedings of the 8th International Workshop on Theory and Practice in Public Key Cryptography, PKC 2005, held in Les Diablerets, Switzerland in January 2005.

The 28 revised full papers presented were carefully reviewed and selected from 126 submissions. The papers are organized in topical sections on cryptanalysis, key establishment, optimization, building blocks, RSA cryptography, multivariate asymmetric cryptography, signature schemes, and identity-based cryptography.

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From the reviews: "This is a textbook in cryptography with emphasis on algebraic methods. It is supported by many exercises (with answers) making it appropriate for a course in mathematics or computer science. [...] Overall, this is an excellent expository text, and will be very useful to both the student and researcher." Mathematical Reviews

Customer Reviews (2)

Excellent, but only if you have extensive knowledge of math
The textbook is intended for students with extensive knowledge of number theory, field theory, and algebraic techniques. It is terse and runs through the material using the definition-lemma-proof-theorem-proof method, with almost no examples. For students without the aforementioned prerequisites, the book is almost impossible to understand.

The book can be great for students with the mathematical prerequisites and with sufficient mathematical maturity to understand elaborate definitions, theorems and proofs and who want to learn the material efficiently and quickly.

The beginning student is advised to look for other, more elementary textbooks.

The book contains many exercises with solutions at the end of the book.

Excelent
It's a very good book about cryptography and all the "stuff" about it. Neal Koblick is one of the invertor on ECC, so he knows about he talks. ... Read more

Customer Reviews (1)

A book mismatched to its subject
Cryptography is a fascinating and important subject; so is computer security. But as all of us know who have suffered from breaches of computer security, cryptography, although a useful tool, is only one small aspect of computer security. This book is quite a nice book on cryptography, although not the best; I recommend Bruce Schneier's book and Alan Konheim's book as better texts on cryptography for anyone seriously interested in cryptography. However, the relationship of this book to the subject of computer security is sort of like that of a book on padlocks to the care and protection of racehorses.

Unfortunately, there is no book I know of that gives adequate treatment even to the role of cryptography in computer security, let alone to the entire subject of computer security, so we learn the hard way, from our mistakes. The only book I know that places this topic in proper perspective is "Cryptography's Role In Securing The Information Society" created in 1996 by the Computer Science and Technology Board of the National Research Council, and published by the National Academy Press. This latter book is kind of interminably boring, and it raises far more questions than it answers, but it's the best we've got; it's well balanced, and was prepared by a group of hard-bitten scientists and engineers with lots of practical experience (plenty of it bad.) So if you want to know how cryptography relates to computer security (maybe), read this latter book, not Seberry and Pieprzyck. ... Read more

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Illustrating the power of algorithms, Algorithmic Cryptanalysis describes algorithmic methods with cryptographically relevant examples. Focusing on both private- and public-key cryptographic algorithms, it presents each algorithm either as a textual description, in pseudo-code, or in a C code program.

Divided into three parts, the book begins with a short introduction to cryptography and a background chapter on elementary number theory and algebra. It then moves on to algorithms, with each chapter in this section dedicated to a single topic and often illustrated with simple cryptographic applications. The final part addresses more sophisticated cryptographic applications, including LFSR-based stream ciphers and index calculus methods.

Accounting for the impact of current computer architectures, this book explores the algorithmic and implementation aspects of cryptanalysis methods. It can serve as a handbook of algorithmic methods for cryptographers as well as a textbook for undergraduate and graduate courses on cryptanalysis and cryptography.

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Like its bestselling predecessor, Elliptic Curves: Number Theory and Cryptography, Second Edition develops the theory of elliptic curves to provide a basis for both number theoretic and cryptographic applications. With additional exercises, this edition offers more comprehensive coverage of the fundamental theory, techniques, and applications of elliptic curves.

New to the Second Edition

Taking a basic approach to elliptic curves, this accessible book prepares readers to tackle more advanced problems in the field. It introduces elliptic curves over finite fields early in the text, before moving on to interesting applications, such as cryptography, factoring, and primality testing. The book also discusses the use of elliptic curves in Fermat’s Last Theorem. Relevant abstract algebra material on group theory and fields can be found in the appendices. ... Read more

Customer Reviews (1)

review of "Elliptic Curves" by Washington
I own both the first and second editions of this book. I am an amateur mathetician; I don't think there is a siginicant difference in the two editions, if you are a non-professional like me. They are both excellent books, and almost exponentially inrease in difficulty as one gets further into them. The writing is less dense, and more amateur-friendly, than Washington's other famous book on cyclotomtic theory. There is some surprising humor in "Elliptic Curves" too. Washington makes a clever pun on Fermat's marginalia. I would gladly recommend either addition---unless you are a Faltings to begin with.
charlie sanders ... Read more

Product Description

This comprehensive encyclopedia provides easy access to information on all aspects of cryptography and security. With an A–Z format of over 460 entries, 100+ international experts provide an accessible reference for those seeking entry into any aspect of the broad fields of cryptography and information security. Most entries in this preeminent work include useful literature references, providing more than 2500 references in total.

Topics for the encyclopedia were selected by a distinguished advisory board consisting of 18 of the world’s leading scholars and practitioners. Main subject areas include:

Authentication and identification

Block ciphers and stream ciphers

Computational issues

Copy protection

Cryptanalysis and security

Cryptographic protocols

Electronic payment and digital certificates

Elliptic curve cryptography

Factorization algorithms and primality tests

Hash functions and MACs

Historical systems

Identity-based cryptography

Implementation aspects for smart cards and standards

Key management

Multiparty computations like voting schemes

Public key cryptography

Quantum cryptography

Secret sharing schemes

Sequences

Web security

The style of the entries in the Encyclopedia of Cryptography and Security is expository and tutorial rather than detailed and technical, making the book a practical resource for information security experts as well as professionals in other fields who need to access this vital information but who may not have time to work their way through an entire text on their topic of interest.

The underlying concepts in information security can be difficult to understand and may even be counter-intuitive. The Encyclopedia of Cryptography and Security will become the premier reference work on this complex subject.

Customer Reviews (1)

excellent coverage of crypto!
For the acutely interested reader, this encylopedia is well worth perusing from cover to cover. It is not that long that this is an arduous task. Nor is it that highly mathematical that you will take months doing so. The level of mathematical treatment is moderate. By that I mean it is less than in journal papers. To be sure, you need a strong maths background going in. And previous exposure to various crypto ideas, notably public key infrastructure, would be good.

For Web usage, people worried about the rising levels of malware might see what the book offers as possible countermeasures. As ecommerce and other activities become more frequent, the possibility of man in the middle attacks increases. Plus, the incredible recent surge in phishing and pharming has led some to suggest that strong cryptographic methods for authentication of messages and web pages might be useful. Keep in mind that this is not necessarily a given. Other methods are possible which avoid cryptography. (I am the co-inventor of several of these methods.) This is something that an astute reader should keep in mind when going through the text. Step up from the details of the various crypto procedures and ask yourself if there are ways to avoid using these, and still prevent attacks. The entire mindset of this book is about using crypto. While it is a dreadful cliche to say 'think outside the box', perhaps you should try to do so here.

There is even a discussion of quantum cryptography. Which may be an ultimate, unbreakable method, if it ever becomes feasible. The quantum systems are so delicate, especially if coherence needs to be maintained over macroscopic distances, that one might reasonably wonder about the eventual efficacies. There are two main postulated usages. One is to break an existing encryption. The other is to prevent a man in the middle attack. It is in the latter that coherence may be more of a problem. For the former, one can imagine a spatially limited, microscopic or mesoscopic system, in which the decryption is occurring. ... Read more

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This book constitutes the refereed proceedings of the Third International Workshop on Practice and Theory in Public Key Cryptography, PKC 2000, held in Melbourne, Victoria, Australia in January 2000. The 31 revised full papers presented were carefully reviewed and selected from 70 submissions. Among the topics addressed are cryptographic protocols, digital signature schemes, elliptic curve cryptography, discrete logarithm, authentication, encryption protocols, key recovery, time stamping, shared cryptography, certification, zero-knowledge proofs, auction protocols, mobile communications security, etc. ... Read more

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Thirty years after RSA was first publicized, it remains an active research area. Although several good surveys exist, they are either slightly outdated or only focus on one type of attack. Offering an updated look at this field, Cryptanalysis of RSA and Its Variants presents the best known mathematical attacks on RSA and its main variants, including CRT-RSA, multi-prime RSA, and multi-power RSA.

Divided into three parts, the book first introduces RSA and reviews the mathematical background needed for the majority of attacks described in the remainder of the text. It then brings together all of the most popular mathematical attacks on RSA and its variants. For each attack presented, the author includes a mathematical proof if possible or a mathematical justification for attacks that rely on assumptions. For the attacks that cannot be proven, he gives experimental evidence to illustrate their practical effectiveness.

Focusing on mathematical attacks that exploit the structure of RSA and specific parameter choices, this book provides an up-to-date collection of the most well-known attacks, along with details of the attacks. It facilitates an understanding of the cryptanalysis of public-key cryptosystems, applications of lattice basis reduction, and the security of RSA and its variants.

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With the scope and frequency of attacks on valuable corporate data growing enormously in recent years, a solid understanding of cryptography is essential for computer/network security practitioners, systems designers, applications developers, and researchers. This timely book delivers the hands-on knowledge professionals need, offering comprehensive coverage on the latest and most-important standardized cryptographic techniques to help them protect their data and computing resources to the fullest. Rather than bogging professionals down with theory like other books on the market, this unique resource describes cryptography from an end-user perspective, presenting in-depth, highly practical comparisons of standards and techniques. Moreover, practitioners learn how to select standardized techniques that are most suitable for their specific needs. ... Read more

Customer Reviews (1)

UNDERPINNING THE SECURITY OF THE ELECTRONICS WORLD
Over the last 31 years, cryptography has grown from being a specialist technology used almost exclusively by governments and the military, to a technology underpinning the security of the electronics world. Authors Alex W. Dent and Chris J. Mitchell have done an outstanding job of bringing you both cryptography and standards for cryptography.

Dent and Mitchell begin this book by reviewing the main relevant standardization bodies and provide a brief introduction to the main standardization processes. Next, the authors define a set of terminology for the use of cryptography. In addition, the authors describe in detail the different types of encryption algorithms and their use for providing data confidentiality. They also describe cryptographic hash functions, message authentication codes, and digital signatures, respectively. Then, they look at non-repudiation and authentication protocols. The authors then describe a standardized framework for key management. Next, protocols for establishing shared secret keys are the main focus. The topic of public key infrastructures is then discussed. Then, they deal with standards governing the use of trusted third parties to support the use and management of cryptographic techniques. The standardization of cryptographic application program interfaces is then considered. The authors then summarize a variety of other standards with relevance to cryptography. Finally, they discuss the future of cryptographic standardization.

With the preceding in mind, the authors have done an excellent job of making this book accessible to the general reader, although it is not really designed for the reader completely unfamiliar with cryptography and its possible applications. Nevertheless, the reader is encouraged to get started on the book, and only refer to mathematical algorithms if concepts arise with which they are not familiar.
... Read more

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Quantum cryptography (or quantum key distribution) is a state-of-the-art technique that exploits properties of quantum mechanics to guarantee the secure exchange of secret keys. This self-contained text introduces the principles and techniques of quantum cryptography, setting it in the wider context of cryptography and security, with specific focus on secret-key distillation.The book starts with an overview chapter, progressing to classical cryptography, information theory (classical and quantum), and applications of quantum cryptography. The discussion moves to secret-key distillation, privacy amplification and reconciliation techniques, concluding with the security principles of quantum cryptography. The author explains the physical implementation and security of these systems, enabling engineers to gauge the suitability of quantum cryptography for securing transmission in their particular application.With its blend of fundamental theory, implementation techniques, and details of recent protocols, this book will be of interest to graduate students, researchers, and practitioners in electrical engineering, physics, and computer science. ... Read more

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The LLL algorithm is a polynomial-time lattice reduction algorithm, named after its inventors, Arjen Lenstra, Hendrik Lenstra and László Lovász. The algorithm has revolutionized computational aspects of the geometry of numbers since its introduction in 1982, leading to breakthroughs in fields as diverse as computer algebra, cryptology and algorithmic number theory.

This book consists of 15 survey chapters on computational aspects of Euclidean lattices and their main applications. Topics covered include polynomial factorization, lattice reduction algorithms, applications in number theory, integer programming, provable security, lattice-based cryptography and complexity.

The authors include many detailed motivations, explanations and examples, and the contributions are largely self-contained. The book will be of value to a wide range of researchers and graduate students working in related fields of theoretical computer science and mathematics.

Product 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. ... Read more

Customer Reviews (5)

Washington Elliptic Curves
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
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
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.
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
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. ... Read more