|
| |
Some Maple worksheets
from my 3rd year
Number Theory and Cryptography course
(related Word 7 notes may be accessed
here)
A note of late September 2026.
After many years of not looking at what I originally had on this page I am attempting a huge edit. Original Maple worksheets no longer function properly because they were created using Maple V. I am attempting to convert all original active .mws worksheets to html format. This will take some considerable time.
A good summary of the
essential content of my third year Number Theory and Cryptography course may be found in
my paper Number Theory and Cryptography (using Maple), which was published in:
David Joyner (Editor, and Organiser
of the United States Naval Academy Conference on Coding Theory, Cryptography, and Number
Theory, 1998), Coding Theory and Cryptography: From Enigma and
Geheimschreiber to Quantum Theory, Springer-Verlag, 2000, 124-143. (Added Nov.
2003: All papers from that conference are now available here)
Actually, the best introduction
to what I do in my third year course (it's how I now start my course every year)
is my public lecture Bill Clinton, Bertie Ahern, and digital signatures,
in the Public and Other Lectures corner of my web site.
 | I have prepared no less
than 13 worksheets on various aspects of the Lehmer-Selfridge, and Pocklington primality
tests. Here I make available just one of
those (24 KB).
|
| Modular exponentiation (8 KB) computations -
using the elementary, but powerful square-and-multiply method - are essential in
cryptographic applications, and I present one explanatory worksheet.
|
| A proper understanding of
the modular exponentiation method requires an understanding of binary expansions (17 KB).
|
|
When I first prepared the work in the previous worksheet, I set about attempting to find interesting examples of primes, whose primality could be established using Henry Cabourn Pocklington's work. A beautiful, quirky example was one which I dubbed a 'millennium' prime, and that worksheet - mill_pri.mws - is available
here.
The html text version of that may be downloaded here.
The mathematics columnist, Ivars Peterson, wrote about this prime in Science News in his column of January 16th 1999. You may read IP's archived article
here.
|
| I teach my students two
really serious factorization methods: John Pollard's two remarkable and beautiful ones
dating from 1974 and 1975. The first of these is the one known as his 'p-1' method (73 KB).
|
| The second of Pollard's
methods that I teach is his 'Monte-Carlo' (or 'rho') method (the one
that derives from Pollard's imaginative use of the 'birthday paradox'). I am not
entirely happy with the worksheet I have prepared on this method, but here it is (39 KB) nevertheless. It is presented using
the 'Floyd cycle-finding' improvement to Pollard's original work.
|
|
|
|
