Third Year
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Some Maple worksheets from my 3rd year
Number Theory and Cryptography course
(related Word 7 notes may be accessed here)

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.

 

An introduction to the Pohlig-Hellman crytpographic method (14 KB). 

 

An introduction to the Fermat factorization method (21 KB), also. 

 

A worksheet to illustrate the Speed of the Euclidean Algorithm (14 KB). 

 

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). 

 

An important number theoretical concept is the order of a modulo n, and here I present a worksheet An introduction to the order of a modulo n (38 KB). 

 

Primality testing is a wonderful topic for students. Here I present a working through of H.C. Pocklington's remarkable 1914-16 paper (29 KB). The html text version of that may be downloaded here.

 

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 (31 KB). The html text version of that may be downloaded here.

The excellent mathematics columnist, Ivars Peterson, wrote about this prime in Science News in his column of January 16th 1999. You may read IP's article here.

 

Euler pseudoprimes (14KB)

 

A very basic Proth worksheet (16KB)

 

D.H. Lehmer's primality test illustrated in Lehmer's examples (18KB).

 

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.

 

For use in Maple lab, the cryptalphabet worksheet.

 

I will have many more worksheets to add here when time allows.

 

 

Contact details 

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

This page was last updated 21 November 2005 12:16:29 -0000