ABSTRACT
Although the history of elliptic curves is well over a century old and was initially developed in the context of classical analysis, these essentially algebraic constructs have found their way into other areas of mathematics in the modern day. Elliptic curves have had impact, at a deep level, on both applied mathematics, for instance in the area of cryptology, as well as in pure mathematics, such as in the proof of FLT. Indeed, a key ingredient in the resolution of Fermat’s equation, (1.44) on page 41, involved certain elliptic curves, which we will explore in §10.3. Moreover, as we shall see later in this chapter, elliptic curves are used in factoring algorithms, primality testing, as well as the discrete log problem, upon which certain elliptic curve ciphers base their security. In fact, elliptic curve methods are widely considered to be some of the most powerful and elegant tools available to the cryptographic community. To see the beauty, complexity, and power of this topic, we must begin with foundational material. Some of what follows is adapted from [64].
