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REU 2012 Research Projects


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.  More ...   

Research Papers: 

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 ...

Research Papers:

  • A. Earls, J. Harlander,  G. Islambouli, R. Keller and M. Yang, Labeled oriented trees that are not diagrammatically reducible,  (in preparation)

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.  More ...

Research Papers:

  • C. Anderson, M. Scheepers, M. C. Warner and H. C. Wauck, On permutations optimized for sorting by the ciliate decryptome (in preparation)