Elliptic Curves for Prime Factorization of Integers

Elliptic curves are abelian groups known for their many applications ranging from the cryptographic standards underlying much of the modern internet, to number theory where they play a central role in Andrew Wiles' proof of Fermat's Last Theorem. In this talk, we introduce the basic theory of elliptic curves, beginning with the Weierstrass equation and a a brief overview of projective space, followed by a discussion of their group law. We then shift to the realm of factoring and describe how and why Lenstra's Algorithm works by leveraging elliptic curves to illuminate prime factors of composite integers. We conclude with some discussion of our own implementation of Lenstra's Algorithm.