Elliptic Curves: Number Theory and Cryptography, Second Edition (Discrete Mathematics and Its Applications) 2nd Edition by Lawrence C. Washington (PDF)

Description

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 EditionChapters on isogenies and hyperelliptic curvesA discussion of alternative coordinate systems, such as projective, Jacobian, and Edwards coordinates, along with related computational issuesA more complete treatment of the Weil and Tate–Lichtenbaum pairingsDoud’s analytic method for computing torsion on elliptic curves over QAn explanation of how to perform calculations with elliptic curves in several popular computer algebra systemsTaking 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.

User’s Reviews

Editorial Reviews: Review … the book is well structured and does not waste the reader’s time in dividing cryptography from number theory-only information. This enables the reader just to pick the desired information. … a very comprehensive guide on the theory of elliptic curves. … I can recommend this book for both cryptographers and mathematicians doing either their Ph.D. or Master’s … I enjoyed reading and studying this book and will be glad to have it as a future reference.―IACR book reviews, April 2010Praise for the First Edition There are already a number of books about elliptic curves, but this new offering by Washington is definitely among the best of them. It gives a rigorous though relatively elementary development of the theory of elliptic curves, with emphasis on those aspects of the theory most relevant for an understanding of elliptic curve cryptography. … an excellent companion to the books of Silverman and Blake, Seroussi and Smart. It would be a fine asset to any library or collection. ―Mathematical Reviews, Issue 2004e Washington … has found just the right level of abstraction for a first book … . Notably, he offers the most lucid and concrete account ever of the perpetually mysterious Shafarevich–Tate group. A pleasure to read! Summing Up: Highly recommended. ―CHOICE, March 2004 … a nice, relatively complete, elementary account of elliptic curves. ―Bulletin of the AMS

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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

⭐Excellent modern exposition.

⭐I found there to be several antiquated texts on number theory but fortunately, this one provides readers with a descent exposure to elliptic curve cryptography. If you have familiarized yourself with the likes of Rohrlich’s work on Galois theory, elliptic curves, and root numbers, move along. Pure mathematicians also need not read.That being said, this book is written to teach rather than to impress. The first five or six chapters are not unique but otherwise necessary to establish the groundwork for this text (i.e. torsion points, elliptic curves over finite fields, the discrete logarithm problem etc.) Chapter 6 does provide a brief foray into elliptic curve cryptography with sections on the Diffie-Hellman key exchange, the ElGamal public key encryption, digital signatures, and ECIES. Similar to many undergraduate math texts, the material is presented in a typical lemma-proof, theorem-proof format but first establishes several definitions that are relatively easy to follow along. The chapter on the discrete logarithm problem also includes a section on attacks with pairings (i.e. the MOV attack and the Frey-Rück attack), which show that in some situations, the Tate-Lichtenbaum pairing can be used to solve discrete logarithm problems.While this book provides an excellent starting point, you will see the author introduce more sophisticated notions in the second half of the text that look at elliptic curves over Q, zeta functions, groups and fields. There is also a discussion on different coordinate systems (i.e. Jacobian coordinates, Edwards coordinates etc.), which you will see have their own unique computational advantages. Some prior exposure to concepts from abstract algebra or group theory would certainly make for an untroubled reading of these pages.The text also includes a chapter on hyperelliptic curves that can be applied to modern techniques in cryptography (i.e. data encryption) for cryptosystems based on the discrete logarithm problem. The author takes you through a very lengthy 2-page proof of Cantor’s algorithm where you will (hopefully) see how it can be used to compute in the Jacobian of a hyperelliptic curve.***Remark about chapter 13 (on hyperelliptic curves): Coming with a background in abstract algebra, I felt there was a lot more that could have be done here. It has been well documented that HECC is superior in providing better security, which naturally sparks an interest into cases where genus g=2,3,4. In order to study hyperelliptic curve cryptography, you must FIRST know how many hyperelliptic curves over finite fields can be applied to the cryptosystems. You can easily prove that the Jacobians of two isomorphic hyperelliptic curves are isomorphic as abelian groups. This result can then be applied to two isomorphic hyperelliptic curves to show that both are the same from the cryptographic point of view. Then we are left with the problem of counting the number of isomorphism classes of hyperelliptic curves, which I think could have been an *excellent* topic for this book to cover.Moving on, I will simply add that the author has made huge strides in providing dutiful readers with a comprehensive coverage of ECC. The inclusion of isogenies in this ‘modern’ rendition make for an interesting read and discuss the homomorphisms between elliptic curves, which you will see can be used to count points on elliptic curves over finite fields. This book allows readers to grasp a very thorough understanding of some pretty abstract concepts while providing meaningful insight into number theory and its modern applications to cryptography.

⭐Wenn man mehr über Elliptische Kurven wissen will, dann ist dieses Buch eine gute und umfangreiche Quelle. Nur der Preis ist unverschämt (insbesondere wenn man weiss, dass bei wissenschaftlichen Büchern für den Autor nur ein Trinkgeld raus springt).

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