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 publickey cryptosystems it is wellknown 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 nround 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 Rijndaellike
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
3manifolds.
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 gametheoretic 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), 457473 [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
