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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. |