Third year
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Number Theory and Cryptography
(related Maple worksheets may be accessed here, and 
written and Maple examination papers may be accessed here)

Aims/Objectives. This course (which builds on my second year introductory Number Theory course) aims to introduce the student to some fundamental applications of Number Theory to the age-old subject of cryptography, and to impart an understanding of the fundamental difference between 'private-key' and 'public-key' cryptography; that understanding is further developed by practical Maple-lab work. The course also introduces the student to some further related topics in Number Theory, especially primality testing and factorisation methods (the difficulty of the latter being the critical underpinning of the current security of the revolutionary Rivest-Shamir-Adelman public-key cryptographic method, which is studied in this course).

Students taking this course will have already acquired a respectable facility in the use of Maple (the Computer Algebra System) in their first and second year of studies, and will further add to their expertise in this course. 

Learning outcomes. At the end of this course, a student should have a sound knowledge of some fundamental ideas in this important and fast changing intellectual discipline. (Incidentally, three of our recent graduates - two BEd and one BA, all young women - have developed their interest in 'Information Security' by proceeding to postgraduate studies at Royal Holloway College's Masters Course in Information Security, and the first of them - who was awarded a PhD scholarship - has just completed her PhD thesis there. I shall make an announcement when she successfully 'defends her thesis'.)

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)

Aviel D.Rubin (formerly) of AT&T Labs-Research and New York University Computer Science Department maintains a List of crytpo and security courses from the USA, through Ireland, and ending in the UK.

My 3rd year Number Theory and Cryptography course consists of a study of:

A (very brief) history of cryptography, aimed at creating a familiarity with the terms plain and ciphertext, encryption, decryption, and digital signature, and an understanding of the crucial difference between private-key and public-key cryptography. 

All of the technical requirements for the private Pohlig-Hellman, and public Rivest-Shamir-Adleman (RSA) methods: the two prime version of the Euler-Fermat theorem, modular exponentiation, and computing the decryption power from the encryption power. 

Here are some Word 7 notes relating to these topics:

The mathematical basis of the Pohlig-Hellman (PH) and Rivest-Shamir-Adleman (RSA) cryptographic methods. Download here (335 KB), or in zip format (52 KB).

Encryption, decryption and digital signatures.
Download here (271 KB), or in zip format (47 KB).

Primality testing: Lucas, Proth, Pocklington, Lehmer, Selfridge.

The Lucas-Proth-(Pocklington)-(Lehmer-Selfridge) theorems.
Download here (410 KB), or in zip format (63 KB).

The (Lucas)-(Kraitchik)-Lehmer-Selfridge theorem.
Download here (307 KB), or in zip format (44 KB). 

Factorization methods: the elementary, but important Fermat method, plus the two 1970's Pollard methods ('p-1' and 'rho')

Some Maple worksheets for this course may be obtained from the corresponding  Maple section of my site, and also in my Public lectures section.

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Contact details 

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

This page was last updated 01 December 2006 20:10:13 -0000