Cryptography has been in use for a few millennia. Some of the first cryptosystems relied on mathematics that was only developed recently. Many of these schemes have been broken.

The first public key cryptosystems developed (such as RSA and the Diffie-Hellman key exchange) rely heavily on number theory. Since their development we have seen that research in the field has exploded. Two major developments have taken place. First tools of theoretical computer science, in particular complexity theory and the analysis of algorithms, are used to analyze and ``prove'' the security of certain schemes and protocols. Second, besides number theory, we now see the use of combinatorics, elliptic curves, finite fields, graph theory, lattices, and ring theory, being used with unexpected applications, which will be surveyed.

This talk assumes that the audience is aware of the different disciplines in modern mathematics. No knowledge of cryptography is required.