Prof. Boaz Tsaban of the Department of Mathematics is best known for his contributions to the area of combinatorics. His research is focused on developing and applying an apparatus of combinatorial methods, solving fundamental problems, and improving our understanding of the connections between the continuous and the discrete. Prof. Tsaban is a recipient of the 2009 Krill Prize for Excellence in Scientific Research.
The security of information storage, processing and transmission are fundamental to the functioning of modern society, and the development of mechanisms to support these processes in public environments (public key cryptography) constitutes an important ongoing challenge.
The first, and perhaps greatest, breakthrough in the field was made in the 1970’s – the realization that two parties can agree on a secret key via communication over an insecure channel – a key exchange protocol (KEP). However, the classical KEPs have gradually become obsolete over the past decades, and few satisfactory substitutes have been proposed.
Tsaban and his team are exploring a novel approach to develop a new KEP (and related primitives) based on hard computational problems in infinite non-commutative groups. In a recent breakthrough, Tsaban has discovered a provable polynomial time cryptanalysis of the Commutator Key Exchange Protocol, the most important protocol in the field. Tsaban is a member of the Israeli team of researchers that cryptanalyzed the block cipher Zorro. He is an editor of the Walter de Gruyter journal Groups—Complexity—Cryptology.
Continuous objects such as continuous functions are usually analysed by means of “continuous” methods.
There are, however, fundamental problems concerning continuous objects, which do not lend themselves to classical methods. These problems can sometimes be approached by discretisation, which makes it possible to apply combinatorial methods.
This approach has been developed into a general theory, which Tsaban and his team are attempting to develop in an application-oriented manner. They are considering major and well known problems involving convergence of sequences of continuous functions, local-to-global correspondence between spaces of continuous functions and their domain space, hyperspaces and Ramsey theory, and Pontryagin-van Kampen duality for abelian topological groups.
Among his major achievements in this area, Tsaban and his research team proved that the quasi-normal convergence property for continuous real-valued functions on a domain implies the same property for Borel-measurable functions, revolutionizing how quasi-normal convergence is viewed and settling all previous problems about this property.
In addition, he has clarified the fine structure behind Menger’s and Hurewicz’s properties, and has developed general and uniform methods to produce examples possessing these properties.
Using Pontryagin van-Kampen duality and descriptive set theory, Tsaban and his collaborators discovered a powerful connection between topological groups and Shelah’spcf theory, yielding exact estimations for the character of free abelian topological groups over hemicompact k-spaces. This expanded results demonstrated previously by more basic methods.
Tsaban has also made fundamental contributions to Ramsey theory, which studies of patterns in partitions of combinatorially rich objects. He has recently established a qualitative extension of the Milliken-Taylor Theorem, one of the deepest theorems in the field, by developing and using methods from the algebra of maximal compactifications of semigroups and selection principles theory.