oa Designing A Cryptosystem Based On Invariants Of Supergroups

The work of our group falls within the area of Cyber Security, which is one of Qatar's Research Grand Challenges. We are working on designing a new public key cryptosystem that can improve the security of communication networks. The most widely used cryptosystem at present (like RSA) are based on the difficulty of factorization of numbers that are constructed as product of two large primes. The security of such systems was put in doubt since these systems can be attacked with a help of quantum computers. We are working on a new cryptosystem that is based on different (noncommutative) structures, like algebraic groups and supergroups. Our system is based on the difficulty of computing invariants of actions of such groups. We have designed a system that uses invariants of (super)tori of general linear (super)groups. Effectively, we are building a "trapdoor function" enabling us to find a suitable invariant of high degree and do the encoding of the message quickly and efficiently but which provides an attacker with computationally very expensive and difficult task to find an invariant of that high degree. As with every cryptosystem, the possibility of its break have to be scrutinized very carefully and the system has to be investigated independently by other researchers. We have established theoretical results about minimal degrees of invariants of a torus that are informing possible selection of parameters of our system. We continue getting more general theoretical results and we are working towards an implementation and testing of this new cryptosystem. A second part of our work is an extension from the classical case of algebraic groups to the case of algebraic supergroups. We are concentrating especially on general linear supergroups. We have described the center of the distribution superalgebras of general linear groups GL(m|n) using the concept of an integral in the sense of Haboush and computed explicitly all generators of invariants of the adjoint action of the group GL(1|1) on its distribution algebra. The center of the distribution algebra is related via the Harish-Chandra map to infinitesimal characters. Understanding of these characters and blocks would lead us to the description of the linkage principle, that is of composition factors of induced modules. Finding and proving linkage principle for supergroups over the field of positive characteristics is one of our main interests. This extends classical results from the representation theory that are giving scientists, mathematicians and physicists, a tool to find a theoretical model where the fundamental rules of symmetries of the space-continuum are realized. Better theoretical background could lead to better understanding of the experimental data and predictions confirming or contradicting our current understanding of the universe. As happened many times in the past, finding the right point of view and developing new language can often lead to different level of understanding. Therefore we value the theoretical part of our work the same way as the practical work related to the cryptosystems.