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REU 2013 Research Projects

 

 


Number Theory, Elliptic Curves and Cryptography
 

Mentored by Dr. Liljana Babinkostova
Cryptography is naturally a 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. This is a continuation of previous work with several REU students.
References:
  • L. Babinkostova, K.W. Bombardier, M.C. Cole, T.A. Morrell and C.B. Scott, Algebraic Properties of Generalized Rijndael-like Ciphers, arXiv:1210.7942
  • L. Babinkostova, K.W. Bombardier, M.C. Cole, T.A. Morrell and C.B. Scott, Elliptic Reciprocity, arXiv:1212.1983 

Geometry, Topology and Complexity
 

Mentored by Dr. Jens Harlander
The projects explore questions from the areas geometric and computational group theory, and low dimensional topology. The symmetries of a mathematical object form a group. In topology and geometry one encounters groups presented by generators and relations. When a group is given by a presentation, it can be difficult to determine even basic properties of the group; for example, whether the group is finite or infinite. One can use combinatorial, geometric, and computational techniques to answer such question. This typically involves the study of graphs and tessellations of the sphere, the Euclidian and the hyperbolic plane. Of particular interest are knots, braids, and 3-manifolds.


Game Theory and Algebraic Structures


Mentored by Dr. Marion Scheepers
The mathematical theory of games has had several successes in clarifying otherwise complex problems. The game-theoretic framework is especially useful for analyzing problems calling for a solution that is optimal in one way or another. There are several such problems in the theory of finite algebraic structures, many of which are known to be inherently difficult. In the projects offered this summer we will use a game theoretic approach to analyze a new class of problems of this kind in finite groups. The problems are inspired by events taking place during the developmental program of certain single cell organisms. Game theory provides several natural ways in which to introduce mathematical techniques that have been developed for other purposes to the arena of this class of problems. Students will be immersed in the fundamentals of game theory while doing research on this class of problems about finite groups.

  • Richard K. Guy and Richard J. Nowakowski, Unsolved Problems in Combinatorial Game Theory, MSRI Publications, vol. 62 (2002), 457-473 [pdf]
  • K.L.M. Adamyk, E. Holmes, G. Mayfield, D.J. Moritz, M. Scheepers, B.E. Tenner and H.C. Wauck, Sorting permutations: Games, Genomes and Graphs, arXiv1410.2353