Vladimir Shpilrain, The City College of New York.

Noncommutative Cryptography

This is an interdisciplinary mini-course focused on  cryptography, which is usually considered an area of computer science. However, there are areas of cryptography (most notably, public-key cryptography), where several different areas of mathematics find their important applications. Until recently, mathematics used in cryptography was "commutative", which means cryptographic primitives were based on commutative rings or commutative (finite) groups. This includes RSA, the most common public key cryptosystem in use today. It is employed, for instance, in the Netscape and Internet Explorer browsers.

Although the security of the internet does not appear to be threatened at this time because of the weaknesses of the existing protocols such as RSA, it seems prudent to explore possible enhancements and replacements of such protocols which depend on finite commutative groups. This is the basic objective of the present mini-course. Non-commutative groups were introduced into public-key cryptography by Wagner and Magyarik more than 20 years ago, but only relatively recently did this direction get serious attention of professional cryptographers worldwide, due to seminal work of Anshel and Goldfeld (1999). Since then, a very active research in non-commutative cryptography is going on, and we are going to describe these new promising research avenues, most of which employ classical as well as modern combinatorial group theory, with a focus on algorithmic problems and their complexity.