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

Algebra-Cryptology Seminar

Spring 2002

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



Tuesday, April 30, 2002, 3:30 p.m. in S&E 215

Mysterious Shadows:
Gessel's Application of Rota's Revival of Bissard's Calculus

Heinrich Niederhausen

Abstract.  "As a matter of fact, the feeling of witchcraft that has hovered over umbral calculus is probably what kept it from dying altogether. ... At long last, it was realized that umbral calculus could be made entirely rigorous by using the language of Hopf algebras, and this was done in a lengthy treatment. However, although the notation of Hopf algebra satisfied the most ardent advocate of spic-and-span rigor, the translation of 'classical' umbral calculus into the newly found rigorous language made the method altogether unwieldy and unmanageable. Not only was the eerie feeling of witchcraft lost in the translation, but, after such a translation, the use of calculus to simplify computation and sharpen our intuition was lost by the wayside." (Gian-Carlo Rota, 1994).

"When I first encountered umbral notation it seemed to me that this was all there was to it; it was simply a convenient notation for dealing with exponential generating functions, or to put it bluntly, it was a method for avoiding the use of exponential generating functions when they really ought to be used. The point I want to make in this paper is that my first impression was wrong: none of the results in this paper (with the exception of ...) can be easily proved by straightforward manipulation of exponential generating functions." (Ira Gessel, 2002).



Tuesday, April 23, 2002, 3:30 p.m. in S&E 215

A Method for Computing Multiples of Points
in Finite Elliptic Curve Groups

Ron Mullin


Tuesday, April 16, 2002, 3:30 p.m. in S&E 215

Fast Public Key Cryptography on the Internet

Rita Agrelo

Abstract.  The public key cryptographic system NTRU has attracted a lot of interest recently as it is much faster than traditional public key cryposystems and as it appears to be secure since its security is based on, and perhaps equivalent to, the lattice basis reduction algorithm.

In this talk we discuss the NTRU algorithm and present our internet implementation MARS.



 

Tuesday, April 9, 2002, 2 p.m. in S&E 215

Torsion, Splitting and Idempotency
(II)

Dr. Jorge E. Viola-Prioli
(Simon Bolivar University, Caracas, and FAU)

Rings for which every filter is idempotent have been discussed in our previous talk and characterized as a finite product of fields, when commutativity is assumed. That result will be taken as our reference when dealing with the same question in a general context, when commutativity is abandoned.
Our goal is to obtain a decomposition of the given ring in terms of more manageable ones. Are the rings Noetherian? Are they von Neumann? How many torsion classes are we to expect? Results by Teply, Fenrick, van den Berg  and mine will be cited. Overall, the different tools used will be mentioned.



Tuesday, April 2, 2002, 3:30 p.m. in S&E 215

Torsion, Splitting, and Idempotency

Dr. Jorge E. Viola-Prioli

Abstract.  In this talk, the first of a two-talk series, we will present the main aspects of torsion in its more general context.  Illustrations will include the most heavily studied torsion theories.  The existence of a ring of quotients is tightly connected with "nice torsion functors": here is where idempotency comes into the picture.  How is a ring such that all its torsion theories are "nice" (in a sense to be specified in the talk)?  We will address this question and characterize completely the commutative rings with this property.  Finally, a rather surprising consequence will be proved.


Tuesday, March 26, 2002, 4:00 p.m. in S&E 215

Operator Identities

Jack Freeman

Abstract.  We give a unified approach to the classification and analysis of special operators on algebras, including endomorphisms, derivations, and the Baxter, Reynolds, and averaging operators.  The key is a reformulation of operator identities as algebraic closure properties of operator graphs.  The Baxter-Hausdorff operator will then be introduced and shown to to play a natural, unifying role in this context. 


Tuesday, March 12, 2002, 3:30 p.m. in S&E 215

Constructions for efficient IPP Codes

Dr. Tran van Trung
(University of Essen, Germany)

Abstract.  Identifiable parent property (IPP) codes are introduced to provide protection against illegal producing of copyrighted digital material.  In this talk we present the results of a joint work with Sosina Martirosyan in which we construct IPP codes explicitely by means of recursion techniques.  The first method directly constructs IPP codes, wheras the second method yields perfect hash families that are then used to derive IPP codes.  In fact, we have constructed an infinite class of IPP codes having the best known asymptotic behavior and more importantly allowing a traitor tracing algorithm which has a runtime of  O(M)  in general, where  is the number of codewords.  Even more faster tracing algorighms for this class of codes can be realized.  We thus have solved an open fundamental problem about the existence of IPP codes enabling efficient tracing algorithms.


Tuesday, February 26, 2002, 3:30 p.m. in S&E 215

Polynomial Rings over a von Neumann regular Ring

Jim Brewer

Abstract.   If  R  is a commutative von Neumann regular ring, the polynomial ring  R[X]  has many interesting properties. In this talk, we will review the basic results on von Neumann regular rings and discuss the polynomial ring R[X] , contrasting it to the ring  F[X]  for  F  a field.


Tuesday, February 19, 2002, 3:30 p.m., in S&E 215

On Power Series over Non-Commutative Rings

Dr. Jorge E. Viola-Prioli
(Universidad Simón Bolívar (Caracas, Venezuela) and FAU)

Abstract.  This talk will be centered on the development of strongly prime rings, from its origins in the early seventies to these days.  A brief overview will allow presenting many of the properties of these rings, which are now spead throughout the literature.  After that, one particular question that remained open for over twenty years, and was finally solved recently by F. Cedo, will be discussed.  Much of this seminar will be devoted to a proof of that result, in a manner that differs from Cedo's.  Overall, this talk emphasizes the sharp contrast between commutative and non-commutative algebra.



Thursday,  February 14, 2002, 3:30 p.m. S&E 215

Problems in Numerical Semigroups

Dr. Kurt Herzinger (US Air Force Academy, Colorado Springs, CO)

Abstract.  A numerical semigroup  S  is a submonoid of the non-negative integers with the property that there exists a largest integer not in  S .  We will discuss the properties of this structure and related structures.  Further, we will examine various open problems related to numerical semigroups and commutative ring theory.



Tuesday,  February 5, 2002, 3:30 p.m. S&E 215

R - Automorphisms of  R[X]

Jim Brewer

Abstract.  If R is a commutative ring and X is an indeterminate, then a (ring) R - endomorphism of R [X] is completely determined by its action on X. In this talk, we reprise some old results of Gilmer characterizing when such an endomorphism is an automorphism.



 

Tuesday, January 29, 2002, 3:30 p.m. S&E 215

Subgroups and Lattices over Tiled Orders

Markus Schmidmeier

Abstract.  In  his habilitation dissertation written in 1989, Wolfgang Rump has determined the indecomposable lattices over the tiled orders given by the exponent matrices

     (  0  1  1  )     (  0  2  2  )     (  0  2  2  )
     (  1  0  1  )     (  1  0  2  )     (  1  0  2  )
     (  n  n  0  )     (  2  2  0  )     (  3  3  0  )  .

It is the aim of this talk to understand how his result is related to the classification of subgroups of finite abelian groups.


Tuesday, January 22, 2002, 3:30 p.m., S&E 215

A Century of Subgroup Problems

Markus Schmidmeier

Abstract.  The study of subgroups of finite abelian groups has been a topic in the mathematical literature since almost one hundred years.  In this talk we aim at a description of some of the main developments.

After some early results about subgroups, or characteristic subgroups, of finite abelian groups  [Miller 1904], [Hilton 1907], the problem of classifying the possible subgroups  A' of a finite abelian group  A , up to automorphisms of the big groups  A , seems to have been posed by Birkhoff.   In his manuscript from 1934, ``Subgroups of Abelian Groups'', he shows that the subgroup problem can be posed as a problem of classifying matrices up to certain row and column operations.   - A recurring theme in the research of Baer is the development of a unified theory for subgroups of abelian groups, and projective spaces over finite fields. -

Subgroups of finite abelian groups are often studied within the context of various branches of representation theory, and in my talk I would like to mention some recent results obtained by different approaches:  The combinatorial approach (subgroups as trees) by Hunter, Richman, Walker;  the representation theory of posets (Arnold, Simson),  and the representation theory of lattices over tiled orders (Rump).


Tuesday, January 15, 2002, 3:30 p.m., S&E 215
 

Enigma

Fred Richman

Abstract.  "Enigma" is the name of the most famous cipher device of World War II. Developed and used by the Germans, over 30,000 were made. The original commercial device, which came out in 1918, was later modified for military use.

I'll describe the device, how it was used, and how Polish mathematicians were able to break it in the 1930's. There will even be a theorem about permutations, although I'm not sure that it is "the theorem that won World War II". After Poland was overrun, the British codebreakers at Bletchley Park inherited the information and techniques that the Poles had developed. They faced a more formidable Enigma with the same basic design.



There will be a Cryptology Section at the 33rd International Conference on Combinatorics, Graph Theory and Computing.

Some links to Further EventsCollege Calendar, Cryptology meetings(UCL),
Algebra Conferences (FDLIST), Algebra Conference Venice 2002, Weekend Algebra Conference New Orleans 2002, Gainesville Algebra Year 2002-3 ,


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 1999Spring 2000, Fall 2000, Spring 2001, Fall 2001


Last modified:  , by Markus Schmidmeier