Number Theory and Cryptography

Credits	8 credit points
Instructors	Lange, T. (Technische Universiteit Eindhoven)
E-mail	tanja@hyperelliptic.org
Aim	To introduce number theory as one of the main enablers of modern cryptography and as a topic of its own interest.
Description	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 anomnipresent 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 algorithmthey 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 severalexamples, some of which have been tried in classes at school. Here is the rough schedule of the course:We will review fundamental results such as the EuclideanAlgorithm and the Chinese Remainder theorem and study algorithmic versions 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 keycryptography. 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.
Organization	Two hours of lectures followed by one hour of exercise classes.
Examination	active participation, final project
Literature	The following references can serve as additional reading Johannes Buchmann "Introduction to Cryptography", Springer Neal Koblitz "A Course in Elementary Number Theory and Cryptography", Springer. Kenneth Ireland and Michael Rosen "A Classical Introduction to Modern Number Theory", Springer
Prerequisites	Knowledge of linear algebra and of concepts such as groups, rings and fields is required.
Remarks	http://www.hyperelliptic.org/tanja/teaching/NTCrypto09/ The target audience for this course are students aiming for a teaching degree. The course is open to others but you need to check with your university whether you can get credit for it.

Last changed: 16-07-2010 15:08