Department of Mathematical Sciences
Florida Atlantic University
Boca Raton, Florida 33431-0991

Algebra-Cryptology Seminar

Spring 2003

All lectures take place in Science and Engineering, room 215.  Everybody is invited to participate.

On Tuesday, November 25, 2003, 2:00 p.m. at the Honors College (Jupiter)

Markus Schmidmeier

Subgroups of Finite Abelian Groups

Abstract.  In this talk I will to present to senior undergraduate and graduate students recent work on the classification problem for subgroups of abelian groups.
                   On Tuesday, October 28, 2003, at 2:00 p.m. in SE 215 we are going to have a

    Special Session

Get Ready for the Algebra Weekend !

Three speakers from our department have agreed to give a first presentation of their WAM-lecture in our seminar:

Carla Petroro

Decomposition Numbers for Subgroups of Finite Abelian Groups

Abstract: F. Richman and E.A. Walker have determined all possible indecomposable embeddings of a subgroup A in a finite p^n-bounded abelian group B, for each n<6.  In case n=2 for example, there are 5 such embeddings.  By the Krull-Remak-Schmidt theorem, an arbitrary embedding (A \subset B) is a direct sum of indecomposable embeddings, and the multiplicities of the indecomposables are determined uniquely.  In this talk we present formulas for these multiplicities in case n=2.

Tanya Seidel

On the Collineation Group of Hall-25

Abstract: We present the structure of the collineation group and how it acts on the points of the Non-Desarguesian Hall projective plane of order 25. By using elements of prime order in the collineation group, we introduce a method for constructing arcs in the projective plane.


Michal Sramka

 RSA (re)construction

Abstract: The security of the RSA public-key cryptosystem is based on the fact that given an integer n=pq, it is hard to find the prime factors p and q. By definition of the RSA encryption there always exists messages that are never encrypted. Some methods for constructing a cryptosystem with minimal number of such messages will be presented. Assuming that some or all of such messages are known, it is sometimes possible to factor n. A few approaches will be discussed.

                   On Tuesday, October 21, 2003, at 2:00 p.m. in SE 215

Joseph Israel, Ph.D.

will speak about

A Friendly Introduction to

Irreducible Modules over SU(n)

On Tuesday, October 14, 2003, at 2:00 p.m. in SE 215
Joseph Israel, Ph.D.

Amalgamation and Unimodality

Abstract.  A compact group H is called  a base of amalgamation of the category of compact groups if for all  embeddings f and g of  H  into compact groups F and G respectively, there  are  embeddings into a compact group H', f ' : F --> H',  g' : G  --> H'   such that
 
                          f ' o f  = g ' o g           (1)  

Such a group must be a Lie group, and must have all  its normal Lie subgroups either open or discrete. To date, the only  known  bases of amalgamation are the finite groups. The case of the circle group is not known.  In the above described set up, we can relax the condition that f ' and g' be embeddings, and merely require that they be nontrivial. An embedding f : H --> G is called universally completable, if for any embedding g : H --> F, nontrivial f ' and g' into a third compact group can be found satisfying the "commutativity" condition (1). The circle group is a base of amalgamation if all its faithful finite dimensional representations are universally completable. The universal completability of some maps can be shown by proving that  some discrete distributions are "unimodal". I am hoping to expand on all these notions.
                   On Tuesday, October 7, 2003, at 2:00 p.m. in SE 215
Ryan Karr, Ph. D. (Honors College)

will speak about

 
Finite Representation Type and Direct-Sum Cancellation II

Abstract.  In last weeks lecture we have seen that even in rings which are ``close'' to being Dedekind domains, Direct-Sum Cancellation may fail for finitely generated torsion-free modules.

The purpose of this lecture is to illustrate this phenomenon by presenting examples, as elementary as possible.
                   

                   On Tuesday, September 30, 2003, at 2:00 p.m. in SE 215
Ryan Karr, Ph. D. (Honors College)
will speak about
 
Finite Representation Type and Direct-Sum Cancellation

Abstract.  Consider the notion of finite representation type (FRT for short): An integral domain R has FRT if there are only finitely many isomorphism classes of indecomposable finitely generated torsion-free R-modules. Now specialize: Let R be of the form D+cO where D is a principal ideal domain whose residue fields are finite, c is a nonzero nonunit in D, and O is the ring of integers of some finite separable field extension of the quotient field of D. If the D-rank of R is at least four then R does not have FRT. In this case we show that cancellation of finitely generated torsion-free R-modules is valid if and only if every unit of O/cO is liftable to a unit of O. We also give a complete analysis of cancellation for some rings of the form D+cO having FRT.  We include some examples which illustrate the difficult cubic case.
On Tuesday, September 23, 2003, at 2:00 p.m. in SE 215

Ayan Mahanolobis

will speak about
Lucas Pseudoprimes
Abstract. The Lucas pseudoprime test is based on a divisibility condition of the Lucas sequence, which is a second order recursive sequence.  We are interested in this pseudoprime test for the following reason:

The usual method of primality testing of an integer  n  is as follows.  First, we test  n  for small factors, if none are found then we use the classical pseudoprime test (which is based on Fermat's Little Theorem) for various bases.

The problem with this method is that classical pseudoprime tests are ``dependent'' in the sense that if  n  is pseudoprime with respect to one basis  a  then it is ``likely'' that  n  will be pseudoprime with respect to another basis  b also.  According to a  conjecture, using the Lucas pseudoprime test together with the classical pseudoprime test breaks this dependence and hence produces a very effective primality test.  

On Tuesday, September 16, 2003, in SE 215 at 3:30 p.m.
Dr. Gretchen Matthews (Clemson University, SC)
will speak about
The Suzuki Curve and some Best Known Codes
Abstract.  In this talk, we will discuss properties of the Suzuki curve that play a role in constructing algebraic geometry codes.  We will show how the Suzuki curve may be used to construct codes over F_8 with better parameters than any known code.
On Tuesday, September 9, 2003, at 2:00 p.m. in SE 215
Michal Sramka
will speak about
Cryptanalysis of Video Encryption Algorithms
Abstract.  Cryptanalysis of two recently proposed MPEG video encryption algorithms will be presented - one is based on permuting Huffman codeword list, the other turns out to be a modification of a classical cipher. Some additional weaknesses of these encryption algorithms will be pointed out.
This is a report on joint work with Tanya E. Seidel and Daniel Socek.
On Tuesday, September 2, 2003, at 2:00 p.m. in SE 215
Dr. Wandi Wei
will speak about
 Some Applications of Geometry of Numbers and Diophantine Approximation to Cryptology
Abstract.  In the past several years, there was some progress in the applications of geometry of numbers and Diophantine approximation to cryptology. Some of this will be introduced here.

Comments and Suggestions are welcome !  Please contact Spyros Magliveras (spyros@fau.edu), Fred Richman (richman@fau.edu), Lee Klingler(klingler@fau.edu), or Markus Schmidmeier  (mschmidm@fau.edu).

For nostalgic reasons you can consult the Previous Programs of this seminar:  Fall 1999, Spring 2000,Fall 2000,Spring 2001,Fall 2001,Spring 2002,Fall 2002,Spring 2003,
Last modified:  , by Markus Schmidmeier