Elliptic Curves

We introduce elliptic curves and describe how to put a group structure on the set of points on an elliptic curve. We then apply elliptic curves to two cryptographic problems--factoring integers and constructing public-key cryptosystems. Elliptic curves are believed to provide good security with smaller key sizes, something that is very useful in many applications, e.g., if we are going to print an encryption key on a postage stamp, it is helpful if the key is short! Finally, we consider elliptic curves over the rational numbers, and briefly survey some of the key ways in which they arise in number theory.

Figure 6.1: The Elliptic Curve $y^2 = x^3 + x$ over $\mathbb {Z}/7\mathbb {Z}{}$