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

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.

Average customer rating:

fabulous introduction to implementing ECC

Detailed and practical

A very nice introduction to the field

Guide to Elliptic Curve Cryptography (Springer Professional Computing)
Darrel Hankerson , Alfred J. Menezes , and Scott Vanstone
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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

Implementing Elliptic Curve Cryptography

Advances in Elliptic Curve Cryptography (London Mathematical Society Lecture Note Series)

A Course in Number Theory and Cryptography (Graduate Texts in Mathematics)

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ASIN: 038795273X

Book Description

After two decades of research and development, elliptic curve cryptography now has widespread exposure and acceptance. Industry, banking, and government standards are in place to facilitate extensive deployment of this efficient public-key mechanism.

Anchored by a comprehensive treatment of the practical aspects of elliptic curve cryptography (ECC), this guide explains the basic mathematics, describes state-of-the-art implementation methods, and presents standardized protocols for public-key encryption, digital signatures, and key establishment. In addition, the book addresses some issues that arise in software and hardware implementation, as well as side-channel attacks and countermeasures. Readers receive the theoretical fundamentals as an underpinning for a wealth of practical and accessible knowledge about efficient application.

Features & Benefits:

* Breadth of coverage and unified, integrated approach to elliptic curve cryptosystems

* Describes important industry and government protocols, such as the FIPS 186-2 standard from the U.S. National Institute for Standards and Technology

* Provides full exposition on techniques for efficiently implementing finite-field and elliptic curve arithmetic

* Distills complex mathematics and algorithms for easy understanding

* Includes useful literature references, a list of algorithms, and appendices on sample parameters, ECC standards, and software tools

This comprehensive, highly focused reference is a useful and indispensable resource for practitioners, professionals, or researchers in computer science, computer engineering, network design, and network data security.

Customer Reviews:

fabulous introduction to implementing ECC.......2005-08-11

I bought this book because I was designing a cryptographic protocol, and wanted to know if I could use ECC in my design. It begins with an explanation of "traditional" public key cryptography (i.e., cryptography over prime fields), introduces binary fields and elliptic curves, shows how to perform computations over elliptic curves, puts this together into ECC protocols, and then includes very useful implementaiton information. This book does a good job explaining not only how to use ECC algorithms, but why they work.

As advertised, this book doesn't go into too much mathematical depth, omitting most proofs. This doesn't mean that there is no math in this book; if you don't have a decent background in algebra (no, not the stuff you learned in seventh grade), you're likely to get confused. However, if you have a little background in theoretical math and cryptography, you'll find this a very readable and easy to understand book.

The one thing that's left out of this book are intellectual property issues. Certicom owns a lot of patents on ECC, and it's not clear which ideas in this book are covered by Certicom patents. This is a minor complaint though; overall, the book is excellent. It's rare to find a book that is so exactly on target. Highly recommeneded.

Detailed and practical.......2005-05-17

This is the only source I've found that goes into the nuts and bolts of elliptic curve (EC) cryptography. The mathematical content is rich, although proofs are generally in references rather than in the text itself. The real value is in its many and detailed algorithm examples, and in the way it builds up to them.

Before it even gets into the text, Hankerson et al have created a model of clarity. In addition to the usual, front matter includes a list of abbreviations. If you've ever choked on the alphabet soup in other books, you'll appreciate how this makes the discussion much easier to absorb. There's also a list of the algorithms presented - what the practitioner wanted in the first place.

After an introductory chapter, the authors present finite field arithmetic in a thorough but readable way. First they present prime fields over the integers, then optimal extension fields and (most importantly) binary fields. There's nothing here for the cut&paste programmer, but dozens of algorithms help the thoughtful developer work through material that is immensely complicated in other presentations. Other goodies, like Karatsuba-Ofman fast multiplication appear here as well.

The third chapter is the book's real payload: EC techniques. I've been looking for years for a book that was so explicit in the how-to, without watering down the technical content. This is practical stuff - not just the theory of EC operations, but the techniques that make EC calculations practical for high-speed implementations.

The rest of the book - about half - discusses what to do with EC codes. That includes protocols for choosing parameters, public-key and signature algorithms, and standard kinds of attacks. It also includes hardware-level description of possible implementations, down to specific instruction sets and cache structures and different kinds of chip implementations. That leads to another set of discussions on attacks, the kind that go in through the power supply or RF emissions. Appendices provide or point to pragmatic details such as parameters to use or software support available.

The only thing that could be improved in this book is the index - it's just too brief, and lacks the thoroughness the rest of the book led me to expect. I hope you realize just how small a complaint that is. In all other ways, this book meets the highest expectations.

Highly recommended for anyone who needs to understand exactly how EC cryptography works, right down to the bit level.

//wiredweird

A very nice introduction to the field.......2004-08-07

This book is a must have if you are interested in implementing elliptic curve cryptography. It does not have any of the juicy ellpitic curve mathematics, but that is okay as this book is directed towards engineers and others who want to learn about how elliptic curve cryptosystems are being deployed.

Average customer rating:

Implementing Elliptic Curve
Lack of clarity. Hard to read and follow.
Good for engineers, as the title says
Full of good, helpful information
rated 1 star as -5 isn't an option

Implementing Elliptic Curve Cryptography
Michael Rosing
Manufacturer: Manning Publications
ProductGroup: Book
Binding: Paperback

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ASIN: 1884777694

Book Description

How to develop cryptosystems that utilize minimal resources to get maximum security.

Customer Reviews:

Implementing Elliptic Curve .......2007-07-14

There is a very good description on various codes on BIG INTEGER arithmatic. It is very helpful for the developers . The Book also gives a very good description of the various types of algorithms used in ECC.

Lack of clarity. Hard to read and follow........2002-04-21

Unfortunately the book is written without clarity. The author needs to write better to communicate with his reader more clearly.

Good for engineers, as the title says.......2001-03-05

The book allowed me to gain fair understanding of ECC principles in a matter of hours. It would be difficult to understand without having taken a previous course in cryptography, but if you already have some idea of numbers theory, and you need to get a quick feel of ECC this would be a good place to start. The continued focus on implementation is important to me (being someone who would eventually have to do it). I would add a summary to each section, describing what EXACTLY needs to be done for each operation - less words, more math.

Full of good, helpful information.......2001-02-23

This book is the first I have read on elliptic curves that actually attempts to explain just how they are used in cryptography from a practical standpoint. It does not attempt to prove the many interesting properties of elliptic curves but instead concentrates on the computer code that one might use to put in place an elliptic curve cryptosystem. The code the author admits could be done in many other ways, but the one he chose I think does its job in instructing the reader just how to implement elliptic curves in cryptography. Indeed, his implementation of large integer math routines is very clear and points out the difference in using a (high level) language like C versus doing the same in Assembly. The only minus to the book from a didactic standpoint are the subroutine schematics that permeate the book. These could have been omitted without any serious damage to understanding what is going on. Readers who need a more rigorous introduction to the mathematics can go to the (immense) literature on elliptic curves. A fine book, and definitely worth reading to gain a practial understanding of elliptic curve cryptosystems.

rated 1 star as -5 isn't an option.......2000-11-21

"Lack of clear definitions?" Not a single definition in this book is correct! The author so badly mangles the terminology of basic undergraduate algebra, that it's hard to believe he understands anything of this advanced subject. This book is an abomination that was printed over the strong objections of an expert reviewer by a company more interested in publishing quantity than quality. I mourn the trees that have been sacrificed.

Book Description

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.

Download Description

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.

Customer Reviews:

The latest cutting edge research on Elliptic Curve Cryptography.......2005-09-26

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: covers Boneh 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.

too much math.......2002-07-25

This is a fairly complete treatment of elliptic curve cryptography. It suffers from a very uneven treatment. The chapters on implementation are well written and easy to read. The material on the logarithm problem, however, is much too advanced and will only be accessible to research mathematicians. A big omission in the book are protocols such as signatures and encryption.

Good compact book on elliptic curves in cryptography.......2000-07-29

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 implement elliptic curve cryptography. The book is written for the reader with some experience in cryptography and one who has some background in the theory of elliptic curves. A reader coming to the field for the first time might find the reading difficult. The authors do give a brief summary in Chapter 1 on the idea of doing cryptography based on group theory. They then move on to discuss finite field arithmetic in Chapter 2. The reader is expected to know some of the basic notions of multiprecision arithmetic for integers. The authors choose to work with 2^16. Psuedocode is given for doing modular arithmetic with Montgomery arithmetic given special attention. The last section of the chapter gives a good summary of arithmetic in fields of characteristic 2. Chapter 3 discusses very compactly arithmetic in elliptic curves. This is where the reader should already have the background in the theory of elliptic curves, since the reading is very fast and formal. The authors do a good job of summarizing how modular polynomials come into play in elliptic curve cryptography and give some explicit examples of these polynomials. The most important chapter of the book is Chapter 4, where the authors give a discussion of how to implement elliptic curves efficiently in cryptosystems. This chapter is nicely written and pseudocode appears many times with lots of nice examples. This chapter serves as background for the next one on the discrete logarithm problem using elliptic curves over finite fields. The MOV attack, the anomalous attack, and the baby step/giant step methods are discussed very nicely. Random methods, such as the tame and wild kangaroo are discussed at the end of the chapter.

The next three chapters concentrate on how to actually generate elliptic curves for cryptosystems, with particular attention payed to the Schoof Algorithm. The chapter on Schoof's algorithm is more detailed than the rest of the chapters and this makes for better reading. The authors do discuss how to generate curves using complex multiplication although the discussion is somewhat hurried. The next chapter discusses how elliptic curves have been applied to other areas in cryptography, such as factoring, etc. A good discussion of the ECPP algorithm on proving primality ends the chapter. The authors end the chapter with a discussion of hyperelliptic cryptography. Anyone familiar with the theory of elliptic curves and how they are applied to cryptography will naturually ask if hyperelliptic curves have any advantages over the elliptic case. The authors never really address this explicity but do give examples on just what is involved in implementing hyperelliptic curves in cryptography. Overall a fine addition to the literature on elliptic curves in cryptography. One would hope that the authors would write a follow-up book on hyperelliptic curves and maybe on general algebraic curves and their possible use in this area.

Good book.......2000-02-01

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

Handbook of Elliptic and Hyperelliptic Curve Cryptography (Discrete Mathematics and Its Applications)

Average customer rating:

A seminal work in its field
One of the best books on this domain
Very understandable overview of modern developments

Handbook of Elliptic and Hyperelliptic Curve Cryptography (Discrete Mathematics and Its Applications)

Manufacturer: Chapman & Hall/CRC
ProductGroup: Book
Binding: Hardcover

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ASIN: 1584885181

Book Description

The discrete logarithm problem based on elliptic and hyperelliptic curves has gained a lot of popularity as a cryptographic primitive. The main reason is that no subexponential algorithm for computing discrete logarithms on small genus curves is currently available, except in very special cases. Therefore curve-based cryptosystems require much smaller key sizes than RSA to attain the same security level. This makes them particularly attractive for implementations on memory-restricted devices like smart cards and in high-security applications. The Handbook of Elliptic and Hyperelliptic Curve Cryptography introduces the theory and algorithms involved in curve-based cryptography. After a very detailed exposition of the mathematical background, it provides ready-to-implement algorithms for the group operations and computation of pairings. It explores methods for point counting and constructing curves with the complex multiplication method and provides the algorithms in an explicit manner. It also surveys generic methods to compute discrete logarithms and details index calculus methods for hyperelliptic curves. For some special curves the discrete logarithm problem can be transferred to an easier one; the consequences are explained and suggestions for good choices are given. The authors present applications to protocols for discrete-logarithm-based systems (including bilinear structures) and explain the use of elliptic and hyperelliptic curves in factorization and primality proving. Two chapters explore their design and efficient implementations in smart cards. Practical and theoretical aspects of side-channel attacks and countermeasures and a chapter devoted to (pseudo-)random number generation round off the exposition. The broad coverage of all- important areas makes this book a complete handbook of elliptic and hyperelliptic curve cryptography and an invaluable reference to anyone interested in this exciting field.

Customer Reviews:

A seminal work in its field.......2007-05-15

When I first opened this book, a tear fell from my eye. Never in my life have I seen such mathematical beauty as summarized from this book. Elliptic curves, isogenies, complex multiplication, higher order abelian varieties, finite fields, point counting, Teichmuller modulus, p-adic numbers, and applications to cryptography: it's all there, and in one amazing book. And the algorithms are written so perfectly that it is easy to translate to the computer language of your choice. Ladies and gentlemen, I promise you, you will not be disappointed by this masterpiece.

One of the best books on this domain.......2007-04-05

It is one of the best books about elliptic curve cryptography, taking the reader from the basics of number theory to the elaborate and tricky field of elliptic curves.
It takes into discussion both theoretical and practical aspects of the domain.

Very understandable overview of modern developments.......2005-09-17

Elliptic curve cryptography is now an entrenched field and has been subjected to an enormous amount of research in the last fifteen years. As soon as encryption schemes based on arithmetic in elliptic curves were proposed, it was natural to speculate on whether these schemes could be generalized to hyperelliptic curves or even general abelian varieties. This book gives an overview of what has been done, and even though most of the proofs are omitted, it does serve a need for those interested in the latest developments in the subject. This reviewer did not read the entire book, but concentrated instead on only a few parts that discussed developments in the last few years. Just skimming the book though will reveal that the authors have been very thorough in giving the reader the necessary mathematical background for a study of ECC and HECC cryptography. Readers needing more detailed background can consult the many references.

As expected, a substantial portion of the book is devoted to point counting methods. One of the methods discussed is the p-adic approach to counting the number of points on an elliptic curve over a field with a small characteristic, with the three most practical ones given the most attention. One of these, the Satoh algorithm, first computes the p-adic approximation of the canonical lift of an elliptic curve E over a finite field F(q), where q = p^d and p is a small prime. This involves lifting the j-invariants using a multivariate version of Newton's root finding algorithm. The trace of the Frobenius endomorphism must then be recovered, and this is done by using the action of the lift on a holomorphic differential on the lift. The resulting factoring problems are formidable, so instead the q-th Verschiebung, which is the dual isogeny to the Frobenius endomorphism is used. The Verschiebung is a separable morphism and the trace of an endomorphism is the trace of its dual. These facts are used to express the trace of the Frobenius endomorphism as a product (modulo q) of coefficients in Z(q). These coefficients are then calculated using certain polynomials.

Another algorithm using the p-adic approach to counting is the Arithmetic-Geometric-Mean (AGM) algorithm, which is discussed for the 2-adic case. As the name implies, this method is based on the AGM iteration, wherein a sequence of elliptic curves is constructed all of which are 2-isogenous to each other. This sequence is constructed so that it converges to the canonical lift of an ordinary elliptic curve, and then an explicit formula for the trace of the Frobenius map is derived. It is then shown how to extend the AGM algorithm to hyperelliptic curves by interpreting it as a special case of the Riemann duplication formula for theta functions.

The third p-adic algorithm discussed is called the Kedlaya algorithm and involves working with the affine curve associated to a hyperelliptic curve of genus g. Associated with this affine curve is its `dagger algebra,' the latter of which is discussed in the book and has its origins in the Monsky-Washnitzer cohomology for nonsingular affine curves over a finite field. This cohomology, which is currently listed under the classification of `rigid cohomology' is a cohomology for algebraic fields over fields of nonzero characteristic and can be considered to be a version of de Rham cohomology (in positive characteristic). In arises when one attempts to lift the Frobenius endomorphism on the coordinate ring of the curve to the coordinate ring of a lift of the curve. Taking the p-adic completion of the coordinate ring of the lift results in a de Rham cohomology which is even larger than the coordinate ring (the limit of exact differentials may not be exact), and so one works with a subring of the completion, which is called the `dagger ring.' The Frobenius endomorphism on the coordinate ring can then be lifted to a (Z(q)) endomorphism on the dagger ring. One can then define differentials of elements in the dagger ring, yielding a module over the dagger ring. The kernel and cokernel of this differential map can then be used to construct the zeroth and first Monsky-Washnitzer cohomology groups. The lift of the Frobenius endomorphism to the dagger ring induces an endomorphism on the cohomology groups, and this allows a Lefschetz fixed point formula to be proved, thus giving the number of rational points on the curve. The Kedlaya algorithm essentially follows this approach to do the point counting, but outputting the zeta function and working only for p greater than or equal to 3.

The book is not just a discussion on theoretical developments and computational algorithms, as an entire part of the book is devoted to applications. One of the applications discussed is that of `smart cards' which to date have been one of most widely used applications of cryptography. An entire chapter is spent on the hardware of smart cards, followed by one on how to attack the implementations of cryptosystems. One particular method for extracting the keys from inside a tamper-proof device involves the use of `power consumption analysis,' which is discussed in some detail in this chapter. The power consumption curve of the device or smart card is analyzed by the attacker, and this, coupled with an understanding of cryptographic algorithms, allows the keys to be compromised. Countermeasures against these attacks are discussed in the next chapter. The discussion is general enough in these chapters to give the motivated reader enough information to experiment with both attacking and with designing and testing effective countermeasures.

Elliptic Curve Public Key Cryptosystems (The Springer International Series in Engineering and Computer Science)

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Understanding DSA

Elliptic Curve Public Key Cryptosystems (The Springer International Series in Engineering and Computer Science)
Alfred J. Menezes
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 0792393686

Book Description

Elliptic curves have been intensively studied in algebraic geometry and number theory. In recent years they have been used in devising efficient algorithms for factoring integers and primality proving, and in the construction of public key cryptosystems.
Elliptic Curve Public Key Cryptosystems provides an up-to-date and self-contained treatment of elliptic curve-based public key cryptology. Elliptic curve cryptosystems potentially provide equivalent security to the existing public key schemes, but with shorter key lengths. Having short key lengths means smaller bandwidth and memory requirements and can be a crucial factor in some applications, for example the design of smart card systems. The book examines various issues which arise in the secure and efficient implementation of elliptic curve systems.
Elliptic Curve Public Key Cryptosystems is a valuable reference resource for researchers in academia, government and industry who are concerned with issues of data security. Because of the comprehensive treatment, the book is also suitable for use as a text for advanced courses on the subject.

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Understanding DSA.......1999-09-12

I saw a copy once and read the intro which alluded to removing the mystery from elliptic curve cryptography, disdaining the popular 'myth' that it's "very complicated."

Advances in Cryptology ASIACRYPT 2005: 11th International Conference on the Theory and Application of Cryptology and Information Security, Chennai, India, ... (Lecture Notes in Computer Science)

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Advances in Cryptology ASIACRYPT 2005: 11th International Conference on the Theory and Application of Cryptology and Information Security, Chennai, India, ... (Lecture Notes in Computer Science)

Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

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ASIN: 3540306846

Book Description

This book constitutes the refereed proceedings of the 11th International Conference on the Theory and Application of Cryptology and Information Security, ASIACRYPT 2005, held in Chennai, India in December 2005.

The 37 revised full papers presented were carefully reviewed and selected from 237 submissions. The papers are organized in topical sections on algebra and number theory, multiparty computation, zero knowledge and secret sharing, information and quantum theory, privacy and anonymity, cryptanalytic techniques, stream cipher cryptanalysis, block ciphers and hash functions, bilinear maps, key agreement, provable security, and digital signatures.

Advances in Cryptology - EUROCRYPT 2004: International Conference on the Theory and Applications of Cryptographic Techniques, Interlaken, Switzerland, ... (Lecture Notes in Computer Science)

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Advances in Cryptology - EUROCRYPT 2004: International Conference on the Theory and Applications of Cryptographic Techniques, Interlaken, Switzerland, ... (Lecture Notes in Computer Science)

Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

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ASIN: 3540219358

Book Description

This book constitutes the refereed proceedings of the International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2004, held in Interlaken, Switzerland in May 2004.

The 36 revised full papers presented were carefully reviewed and selected from 206 submissions. The papers are organized in topical sections on private computation, signatures, inconditional security, distributed cryptography, foundations, identity based encryption, elliptic curves, public-key cryptography, multiparty computation, cryptanalysis, new applications, algorithms and implementation, and anonymity.

Advances in Cryptology -- ASIACRYPT 2006: 12th International Conference on the Theory and Application of Cryptology and Information Security, Shanghai, ... (Lecture Notes in Computer Science)

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Advances in Cryptology -- ASIACRYPT 2006: 12th International Conference on the Theory and Application of Cryptology and Information Security, Shanghai, ... (Lecture Notes in Computer Science)

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ProductGroup: Book
Binding: Paperback

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ASIN: 3540494758

Book Description

This book constitutes the refereed proceedings of the 12th International Conference on the Theory and Application of Cryptology and Information Security, ASIACRYPT 2005, held in Shanghai, China in December 2006.

The 30 revised full papers presented were carefully reviewed and selected from 314 submissions. The papers are organized in topical sections on attacks on hash functions, stream ciphers and boolean functions, biometrics and ECC computation, id-based schemes, public-key schemes, RSA and factorization, construction of hash function, protocols, block ciphers, and signatures.

Advances in Cryptology -- EUROCRYPT 2003: International Conference on the Theory and Applications of Cryptographic Techniques, Warsaw, Poland, May 4-8, ... (Lecture Notes in Computer Science)

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Advances in Cryptology -- EUROCRYPT 2003: International Conference on the Theory and Applications of Cryptographic Techniques, Warsaw, Poland, May 4-8, ... (Lecture Notes in Computer Science)

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ProductGroup: Book
Binding: Paperback

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ASIN: 3540140395

Book Description

This book constitutes the refereed proceedings of the International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2003, held in Warsaw, Poland in May 2003.

The 37 revised full papers presented together with two invited papers were carefully reviewed and selected from 156 submissions. The papers are organized in topical sections on cryptanalysis, secure multi-party communication, zero-knowledge protocols, foundations and complexity-theoretic security, public key encryption, new primitives, elliptic curve cryptography, digital signatures, information-theoretic cryptography, and group signatures.

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