Number Theory and Cryptology

To introduce number theory as one of the main enablers of modern cryptography and as a topic of its own interest.
This course is only for students of de education variant.

Number theory is a classical discipline in mathematics and has been studied already in ancient times. It is the study of relations among the integers. Cryptography is the art of secretly transmitting information and is as such as old as people trying to hide their secrets. In recent years cryptography has changed a lot -- away from a
science that was mostly related to military and secret service to an omnipresent enabler of online banking, eCommerce, and secure email to mention just a few.

Cryptography is an exciting and motivating topic with a touch of a spy novel and thus a great background for math projects. A solid background in number theory is essential to understand the cryptography deployed e.g. in Internet browsers. Even though your future pupils will not be expected to build their own crypto algorithm
they should be able to understand the framework in which they are operating, not the least to make valid decisions which services to trust. While this course cannot cover all topics of security it will give a solid background of the mathematics involved and show several
examples, some of which have been tried in classes at school.

We will loosely follow Koblitz' "A Course in Elementary Number Theory and Cryptography". It is though not necessary to purchase the book to
follow the course; relevant material will be presented at the
blackboard.

Here is the rough schedule of the course:
We will review fundamental results such as the Euclidean
Algorithm and the Chinese Remainder theorem and study algorithmicversions thereof together with an analysis of the runtime.

From modular arithmetic we can understand the RSA cryptosystem and the original version of Diffie-Hellman key exchange. The integers modulo a prime p form the simplest case of a finite field. Finite fields are an important building block of cryptography, in particular of public key
cryptography. We consider general finite fields and study their use in elliptic curve cryptography.

We end the semester with a study of factorization algorithms and - if time permits - an overview of less standard public key cryptography.

Literature: Neal Koblitz "A Course in Elementary Number Theory and Cryptography", Springer. It is though not necessary to purchase the book to
follow the course;

http://www.hyperelliptic.org/tanja/teaching/NTCrypto08/