Computability and Complexity in Elliptic Curves and Cryptography
Mentored by Dr. Liljana Babinkostova
Cryptography is naturally multidisciplinary field, drawing techniques from a
wide range of disciplines and connecting to many different subject areas. In
recent years, the connection between algebra and cryptography has tightened, and
established computational problems and techniques have been supplemented by
interesting new approaches and ideas. The projects from this research area have
their origins in Cryptography. For public-key cryptosystems it is well-known that the choice of
algebraic platform for the system influences the complexity of
implementing the cryptosystem and the level of security offered by the system.
Elliptic curve groups, which have applications in cryptography, give an example
of this. A major step in evaluating the cryptographic suitability of an elliptic
curve group is to find its order, and this can be a complex task. In the arena
of symmetric key cryptosystems, questions such as whether a block cipher
is a group and what is the group generated by n-round encryptions are important
as part of the evaluation of the security afforded by the cryptosystem. Some of the projects are concerned with the determination of the order of
elliptic curve groups that arise from the famous Bachet equation. In the arena
of symmetric key cryptosystems, the classical DES cryptosystem has been
replaced as standard by a new system called AES, the "advanced encryption
standard". Several questions that have been intensively studied for DES have not
yet received the same attention for AES. Some projects are concerned with
investigating algebraic aspects of AES-like crypto systems.
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Geometry, Topology and Complexity of Virtual Knots
Mentored by Dr. Jens Harlander
Knots, strings tangled in 3-space, are objects with which everyone is familiar. The
mathematical theory of knots is highly sophisticated, incorporating many classical areas
including topology, geometry, combinatorics and group theory. Currently, the study of
knots is finding application in fields as diverse as biology, physics and computing.
A knot, when drawn on a piece of paper, is a planar 4-regular graph. A virtual knot,
from a graph theoretic point of view, is an arbitrary (not necessarily planar) 4-regular
graph. Many questions which have been answered for classical knots are still unanswered
for virtual knots. Virtual knots play key roles in long standing conjectures, such as White-
head's asphericity conjecture. The projects are concerned with the topology,
geometry and complexity of virtual knots. We will investigate generic properties
of virtual knots with a large number of crossings and use computers for testing
some of our conjectures. More ...
Mathematics and genome remodeling in developmental time:
Algorithms for Ciliates
Mentored by Dr. Marion Scheepers
The theme of projects related to genome remodeling is the design and
analysis of algorithms that exploit genome remodeling processes to solve
mathematical problems. Genome remodeling in the single cell organisms known as
ciliates occurs as a programmed developmental process. Initial details about the
algorithmic nature of the ciliate genome remodeling program suggest mathematical
and technological consequences of being able to control individual program
steps.
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