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

# Algebra/Cryptology Seminar

Fall 2001

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

Tuesday, December 4, 11:00 a.m., in S&E 215

### "A probabilistic comparison of various voting schemes"

Prof. Julian Edward
(Florida International University)

Abstract: A proposal satisfies the Pareto criterion if its passage will hurt no one and make at least one person better off.
In this talk, we establish a probabilistic model that enables us to compare various voting schemes using norms based on the Pareto criterion. A utilitarian norm is also considered.

Tuesday, November 27, 3:30 p.m., in S&E 215

### Zeta Functions

Stephanie Fitchett

Abstract.  Zeta functions provide an exciting link between Algebraic Geometry and Public Key Cryptography as they help describe the points in an elliptic curve over a finite field.  I will share whatever I can learn about zeta functions between now and Tuesday.

Tuesday, November 13, 3:30 p.m., in S&E 215

### Elliptic Curve Cryptography

Daniel Soucek

Abstract: Elliptic curves form a group under certain addition.  This group has a richer structure than the commonly used multiplicative group of integers mod p.  We  discuss EC analogs to existing public key cryptosystems and the benefits of using such.  Examples will be provided as time permits.

Tuesday, November 6, 3:30 p.m., in S&E 215

### 2 - Dimensional  R -Algebras with Identity

Jack Freeman

Abstract.  We observe that these algebras will have the form  R[x]/(p(x))  where  deg(p(x)) = 2 .  Each such is isomorphic to just one of

R[x]/(x^2+1) ,  R[x]/(x^2-1) ,  or  R[x]/(x^2),

according to the discriminant of  p(x) being negative, positive, or zero.  These are respectively: The fields of complex numbers  C , the direct product of two copies of the field of real numbers  R x R , and the dual numbers over the reals  R .

We derive general coordinate-free Cauchy-Riemann equations satisfied by polynomials over these algebras.  Also,  2nd order PDE's for these polynomials are given, being of elliptic, hyperbolic, or parabolic type, depending on the isomorphism class of the  2 -dimensional algebra.

In passing, the question raised earlier, ``Why Do We Multiply As We Do ?'', will be given its abiding and definitive answer !

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

### Electronic Elections

Markus Schmidmeier

Abstract.  We present an application of modern public key cryptology to the election problem.

(A)     Is it possible that every citizen has (at most) one vote, and yet, voters remain anonymous ?

(B)     Is there someone who can verify the correctness of the counting process ?

We discuss a setup suggested in the recent book on Modern Crypology by Beutelspacher et al.  Their answer to (A) and (B) is yes, even better:  (A) Even non-voters can remain anonymous, and (B) any person with a computer and internet acces could verify the correctness of the counting !

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

### Ring Units in Public Key Cryptology

Markus Schmidmeier

Tuesday,  October 16, 15:30 p.m., in S&E 215

### Non-Noetherian Dedekind-like Rings

Jim W. Brewer

Abstract.  We present naturally occuring examples of non-noetherian Pruefer domains, which are Dedekind-like in the sense that all finitely generated ideals are invertible.  In particular, we will focus on the ring of all algebraic integers.

Tuesday, October 9, 3:30 p.m., S&E 215

### On Decomposing Ideals into Products of Comaximal Ideals

Jim W. Brewer

Abstract.  Recall that two ideals  Q , Q'  in a domain  D  are comaximal provided  Q + Q' = D .  We consider integral domains  for which each nonzero ideal  A  (or each nonzero principal ideal aD ) can be written as a product  A = Q1 Q2 ... Qn, where the  Qm are pairwise comaximal and have some additional property.  We determine the structure of those integral domains having this property in the following cases:

• Each  Q has prime radical
• Each  Qm  is primary
• Each  Qm  is the power of a prime ideal.

Tuesday, October 2, 3:30 p.m., S&E 215

### A New Public Key Cryptosystem

Spyros Magliveras

Tuesday, September 25, 3:30 p.m., S&E 215

### Cyptology Today

Spyros Magliveras
Abstract.  After a short review of classical cryptosystems we present a first introduction to modern public cryptology.

Tuesday, September 18, 3:30 p.m., S&E 215

### A Sort of Division Algorithm

Fred Richman

Abstract.  Heyting showed that for a polynomial  over what is now called a "Heyting field" you can figure out whether divides another  polynomial  by doing some calculations with the coefficients of  and  f  and seeing if the results are zero. If  is monic, you could just use the division algorithm and look at the coefficients of the remainder. Heyting's contribution is to allow any coefficient of  p to be a unit, not just the leading coefficient.

In retrospect, Heyting proved this theorem for any local ring without nilpotent elements. The natural generalization to arbitrary rings (possibly without nilpotent elements) is to require that the coefficients of generate the ring as an ideal. The question is essentially one of whether the  R -module map   R[X] -> R[X]   given by multiplication by  has a left inverse, at least sort of.

Here is the Current Seminar Program.