Abstract and time comment
Since 199596 I have taught a 3rd year undergraduate course
Number Theory and Cryptography
(
using Maple
) at St Patricks College. Before I had a web site (in June 1999), David Joyner at the United States Naval Academy (Annapolis) asked if I would allow my course Maple worksheets be displayed at their site; they are still there at:
http://web.usna.navy.mil/~wdj/crypto.htm
Subsequently I published a fairly full outline of my 3rd year course in:
Number Theory and Cryptography
(
using Maple
) in David Joyner USNA (Ed.), Coding Theory and Cryptography: From Enigma to Geheimschreiber to Quantum Theory (Unites States Naval Academy Conference), SpringerVerlag, 2000, pp 124143.
In September 1998 the U.S. President, Bill Clinton, and the Irish Prime Minister Bertie Ahern engaged in a 'historic, first digital signing of a treaty on ecommerce'  at Gateway 2000's Dublin plant (Gateway's then European headquarters), and using software produced by Baltimore, the Irish cyrptography company  between the U.S. and Irish governments, and in November 1998 I gave a
public
lecture (using Maple) 
Bill Clinton, Bertie Ahern, and digital signatures
 at my college, to explain the essential
revolutionary
ideas of
publickey cryptography
, that lay behind our governments' signing. The
Irish Times
 our leading daily newspaper  publicised my talk in advance, and invited me to contribute an explanatory article prior to my lecture (that article used to be freely available at the
Times
' site, but unfortunately they now charge for access). That talk  which took me about 70 mins to deliver  is available in active Maple mws format, or in html format in the Public and Other Lectures corner of my web site.
Since November 1998 I have used that public lecture to introduce my course to my students: my aim has been to show just how much one can cover with a general public audience. My third year course then entails putting
mathematical flesh
on that public lecture. My students have already studied Number Theory in their second year, that course consisting of three core areas: Congruences, the Euclidean Algorithm and its extension, and Fermat's 'little' theorem with applications, plus some other topics that I change from year to year.
Here in Chicago I offer a much revised version of my 1998 ClintonAhern talk to college mathematics faculty: e.g. the inclusion of the PohligHellman material is entirely new. No prior knowledge of Number Theory will be required to follow my talk, in which I will

Briefly
explain
the difference between
private
key and
public
key cryptography

Demonstrate
the PohligHellman
private
key cryptographic method

Demonstrate
a simplified element of the renowned RivestShamirAdleman
(RSA)
public
key
cryptographic
method

Illustrate
the relevance of the unsolved fast factorisation problem
to the security of the RSA method
(and, if I have time, also have a look at
phi of n
...)
The core content of my 3rd year Number Theory and Cryptography course, my preparatory 2nd year Number Theory course, and related examination papers (3hour written, 2hour Maple, and one set of student Maple solutions) are all available at my web site: visit my home page and follow the obvious links.
A related paper of mine is:
From divisibility by 6 to the Euclidean Algorithm and the RSA cryptographic method
, The American Mathematical Association of TwoYear Colleges Review, Vol 19, No 1, Fall 1997, 3845.
_______________
Time comment. Because of the amount of material I wish to cover, time constraints (45 minutes) dictate that I cannot devote as much time as I would wish even to sketchy mathematical details. The interested reader may, however, explore further, complete details of all the relevant mathematics at my web site.
In my talk, then, I will have to resort to a certain amount of handwaving along the lines of:

that computation used a method known as modular exponentiation using the squareandmultiply technique

that computation used the Euclidean algorithm

that computation used the
extended
Euclidean algorithm

what made that 'work' was Fermat's little theorem

what made that 'work' was the 2prime version of Fermat's little theorem (which is itself a special case of the
socalled FermatEuler theorem)
etc