Cryptography and cryptographic protocols have become a key element of information systems, protecting data and communications to ensure confidentiality, integrity and authenticity of data. While most symmetric key systems (block ciphers such as DES and AES and stream ciphers) have relatively modest mathematical requirements, asymmetric or public key systems, as well as cryptographic protocols, have become increasingly mathematically sophisticated. Such systems rely for their security on the difficulty of specific mathematical problems such as integer factorization and the modular discrete logarithm problem.
It is important, however, to emphasize that no rigorous mathematical proof of security has ever been given for any of these systems. The difficulty of these problems is usually established anecdotally through frequent and unsuccessful attempts by specialists to provide computationally efficient solutions to them. Indeed, several problems thought to be very difficult, such as the integer factorization problem, have been shown to be somewhat less to considerably less intractable than previously believed. Furthermore, the possibility of quantum computing becoming practical would change this picture dramatically. If realized, most of the problems on which the security of public key cryptosystems rely drop from exponential complexity to polynomial, rendering currently deployed cryptographic systems useless. While the likelihood of this occurring in the short term is remote, this is an exciting area of research which may well lead to revolutionary advances in computation and secure information communication.
This program will engage the cryptographic and mathematical communities in Canada and abroad to increase awareness of recent developments in these fields and to initiate a greater degree of collaboration in attacking the important problems, particularly on the boundaries.
The specific areas of concentration will be:
- quantum computing and quantum cryptography
- algebraic curves and cryptography
- computational challenges arising in algorithmic number theory and cryptography
- unconditionally secure cryptography
- cryptographic protocols
- applied aspects of cryptography
The program will include series of one-week workshops, Graduate courses and distinguished lecturers. The scope of the program is ambitious in that it aims to bring together researchers from areas that seldom have the opportunity to interact in an atmosphere where problems at the intersections can be explored. Developments in certain areas of mathematics (for example, number theory, combinatorics, algebraic geometry, non-abelian groups and rings) and in cryptography are both numerous and rapid; however, it is often the case that lack of contacts and communication between cryptographers and mathematicians presents obstacles in achieving significant advances on both sides. The aim is to overcome these obstacles and foster new links between both areas.
This program is being coordinated with a related but distinct program at IPAM, on the topic of "Securing Cyberspace"
(see www.ipam.ucla.edu/programs/sc2006/ ).
Associated program activities include the Rocky Mountain Mathematics Consortium's Summer School on Computational Number Theory and Applications to Cryptography, to be held June 19 - July 7, 2006 at the University of Wyoming, in Laramie, Wyoming. In particular, the summer school courses will complement and prepare participants for the activities of the Fields cryptography program.
