The Versatility of Elliptic Curves

Abstract

Elliptic curves come up everywhere in mathematics. We will now look at some of their uses in unexpected places. In this chapter, we will discuss some of these uses for elliptic curves. In particular, we will discuss a factorization algorithm based on elliptic curves, as well as some applications of elliptic curves over $\mathbb {Q}$ and $\mathbb {Z}$ to Diophantine equations. Before we can get to the elliptic curve factorization algorithm, we start with a non-elliptic curve factorization algorithm, that carries some of the same ideas as the elliptic curve method. This is the Pollard $p-1$ algorithm.