Lecture 37: A Look Back

William Stein

Date: Math 124 $\quad$ HARVARD UNIVERSITY $\quad$ Fall 2001

As we look back over our number theory course, several topics stand out: integers and congruences, factorization, public-key cryptography, continued fractions, binary quadratic forms, and elliptic curves. The integers and congruences are at the heart of almost everything we studied. We learned that integers factor as products of primes and got a taste of how to find such factorizations in some cases using Pollard's $(p-1)$ method and Lenstra's elliptic curve method. We learned the basics of the beautiful theory of binary quadratic forms, their composition law, and finiteness of the group of equivalence classes of binary quadratic forms of given discriminant. We also learned that every positive real number $\alpha$ has a continued fraction, and that it is eventually periodic if and only if $\alpha$ satisfies an irreducible quadratic polynomial. We learned about three public-key cryptosystems: the Diffie-Hellman key exchange, the RSA cryptosystem, which uses arithmetic in $(\mathbb{Z}/pq\mathbb{Z})^*$, and the ElGamal elliptic curve cryptosystem which is used by Microsoft in their digital rights management scheme. We spent the last month learning about the group law on an elliptic curve, torsion points and a big theorem of Mazur, about how modularity of elliptic curves is used in the proof of Fermat's Last Theorem, and about the Birch and Swinnerton-Dyer conjecture.