Elliptic Curves and Their Applications to Cryptography: An Introduction 1999th Edition by Andreas Enge (PDF)

Description

Since their invention in the late seventies, public key cryptosystems have become an indispensable asset in establishing private and secure electronic communication, and this need, given the tremendous growth of the Internet, is likely to continue growing. Elliptic curve cryptosystems represent the state of the art for such systems. Elliptic Curves and Their Applications to Cryptography: An Introduction provides a comprehensive and self-contained introduction to elliptic curves and how they are employed to secure public key cryptosystems. Even though the elegant mathematical theory underlying cryptosystems is considerably more involved than for other systems, this text requires the reader to have only an elementary knowledge of basic algebra. The text nevertheless leads to problems at the forefront of current research, featuring chapters on point counting algorithms and security issues. The Adopted unifying approach treats with equal care elliptic curves over fields of even characteristic, which are especially suited for hardware implementations, and curves over fields of odd characteristic, which have traditionally received more attention. Elliptic Curves and Their Applications: An Introduction has been used successfully for teaching advanced undergraduate courses. It will be of greatest interest to mathematicians, computer scientists, and engineers who are curious about elliptic curve cryptography in practice, without losing the beauty of the underlying mathematics.

User’s Reviews

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

⭐This book gives a straightforward introduction to the theory of elliptic curves as applied to cryptography and is written for those individuals without a strong background in abstract mathematics. The authors summarize the ideas behind public key cryptography in the first chapter and then move on to the group law on elliptic curves. The best part of this chapter is the explicit calculations the author gives for the group operations, espeically over fields of characteristic 2. The author chooses not to prove the associativity of the group operation geometrically bust uses the isomorphism between the curve and the degree zero part of its Picard group. This approach on the surface might seem abstract for those not having a background in algebraic geometry but the author does a good job of giving an intuition about the Picard group. A reader with a basic background in complex variable theory should find the discussion very understandable. Chapter 3 gives a thorough discussion of elliptic curves over finite fields, those being relevant to cryptographic applications. Explicit calculations are given for the Weil pairing and Hasse’s theorem is proved in detail. Dissapointingly, the author does not give explicit proofs for the cases of supersingular curves but instead refers the reader to the literature. More discussion on the supersingular case might be warranted here so as to give an idea on its limitations in elliptic curve cryptography. The chapter on the discrete logarithm problem is nicely written with discussions of the different attacks done in detail.Some pseudocode inserted in the text would have been nice. The last chapter concerns calculating the order of the group, and the author does a good job of discussing the Baby-Step Giant-Step algorithm and Schoof’s algorithm. The important idea of isogeny is discussed briefly in this chapter but proofs relating isogenies and modular polynomials are omitted entirely and the reader is referred to the literature. It would have been nice if the author could have taken this difficult theory and distilled it down to a form that is similar to the rest of the book. Such an intuitive discussion would have proven to be invaluable and would justify more the price of the book. A very expensive book but worth it for those very interested in elliptic curve cryptography.

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