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

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Over the last two or three decades, elliptic curves have been playing an increasingly important role both in number theory and in related fields such as cryptography. For example, in the 1980s, elliptic curves started being used in cryptography and elliptic curve techniques were developed for factorization and primality testing. In the 1980s and 1990s, elliptic curves played an important role in the proof of Fermat’s Last Theorem. The goal of the present book is to develop the theory of elliptic curves assuming only modest backgrounds in elementary number theory and in groups and fields, approximately what would be covered in a strong undergraduate or beginning graduate abstract algebra course. In particular, we do not assume the reader has seen any algebraic geometry. Except for a few isolated sections, which can be omitted if desired, we do not assume the reader knows Galois theory. We implicitly use Galois theory for finite fields, but in this case everything can be done explicitly in terms of the Frobenius map so the general theory is not needed. The relevant facts are explained in an appendix.

The book provides an introduction to both the cryptographic side and the number theoretic side of elliptic curves. For this reason, we treat elliptic curves over finite fields early in the book, namely in Chapter 4. This immediately leads into the discrete logarithm problem and cryptography in Chapters 5, 6, and 7. The reader only interested in cryptography can subsequently skip to Chapters 11 and 13, where the Weil and Tate-Lichtenbaum pairings and hyperelliptic curves are discussed. But surely anyone who becomes an expert in cryptographic applications will have a little curiosity as to how elliptic curves are used in number theory. Similarly, a non-applications oriented reader could skip Chapters 5, 6, and 7 and jump straight into the number theory in Chapters 8 and beyond. But the cryptographic applications are interesting and provide examples for how the theory can be used.
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