Cryptography and Group Theory

Cuny City College, New York NY

Investigators

Abstract

The possibility that quantum computers might eventually make today's cryptographic protocols obsolete and thereby make hitherto private information public, has recently led to an ongoing and very active effort to develop cryptoysystems based on group theory. One such system, using the so--called braid group, was developed in part by Michael Anshel (a member of the City College of New York Computer Science Department and CAISS, the Center for Algorithms and Scientific Software, a research center at CCNY), in 1999, and can be viewed as a group--theoretic analogue of the famous key-exchange idea of Diffie--Hellman. Today, as already indicated, a great deal of research is being carried out in this general area of group theory and cryptography. The Graduate Center of the City University of New York hosts the famous New York Group Theory Seminar, where current research in group theory is discussed on a weekly basis. There is also now a seminar devoted to cryptography at the Graduate Center and the Stephens Institute in Hoboken has a team of researchers working in part on cryptography. So a conference at this time in this general area seems particularly appropriate and important. One tends to marvel at the intricate patterns that one sees in the tiled floors of St. Marks cathedral in Venice and the remarkable symmetry of everyday crystals. Such symmetry also arises in the study of elementary particles and also in knot theory, which surprisingly arises in physics. They also play a part in efforts to understand the intricacies of the three--dimensional world that we live in. There is an algebraic structure known as a group which captures the notion of symmetry in a highly sophisticated way, allowing for the classification of different types of crystals and provides a means for studying and describing the minute particles which make up the material of everyday life. These groups can encode in a remarkably simple way, a huge amount, indeed an infinite amount, of information. Such encodings are often hard to decode and so are extremely promising tools in obtaining secure new ways for hiding and transmitting information. Thus they offer novel means for safe electronic communication. With the birth of the internet and the explosion of commerce over the net, secure transmission of information is now a critical aspect of today's world. The proposed conference will be devoted to bringing some of the ideas involved to the wider mathematical community in the hope that it will lead to broader understanding and application of them.