Elliptic Curves and Their Applications to Cryptography : An Introduction

Rating:
Summary: A balanced, nice introduction to elliptic curve cryptography
Review: 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.