Abstract and time comment

Since 1995-96 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:


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), Springer-Verlag, 2000, pp 124-143.

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 e-commerce' - 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 public-key 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 Clinton-Ahern talk to college mathematics faculty: e.g. the inclusion of the Pohlig-Hellman 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 Pohlig-Hellman private -key cryptographic method
  • Demonstrate a simplified element of the renowned Rivest-Shamir-Adleman (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 (3-hour written, 2-hour 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 Two-Year Colleges Review, Vol 19, No 1, Fall 1997, 38-45.


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 hand-waving along the lines of:

  • that computation used a method known as modular exponentiation using the square-and-multiply 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 2-prime version of Fermat's little theorem (which is itself a special case of the so-called Fermat-Euler theorem)


Contact details 

After August 31st 2007 please use the following Gmail address: jbcosgrave at gmail.com

This page was last updated 18 February 2005 15:08:22 -0000