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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)
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), available
here. My paper (requested by
David Joyner) was published in Coding Theory and Cryptography: From Enigma and Geheimschreiber to Quantum Theory, Springer-Verlag, 2000, 124-143. It was one of a number of papers published in the proceedings of the U.S. Naval Academy (Annapolis) Conference on Coding Theory, Cryptography, and Number Theory (1998), a conference organised by David Joyner. David sought my permission to use my course materials
here.
All papers from that conference (including ones by two historic figures,
Peter Hilton and
William Tutte)
are available here.
A personal note. In those early still experimental days of my using Maple in my teaching I used to benefit greatly from the soon-to-vanish Maple Users Group. It functioned as follows: someone would post a question along the lines of "how can you use Maple to do such-and-such?", and everyone in the Group would see this, and everyone would see the (generally) several helpful replies.
On one occasion, however, David Joyner posted a question asking to hear from people around the world who were using Maple to teach NT+Crypto... I sent David a few tentative things I was doing... , and he asked for more, then more. It was quite a confidence booster for me, since I - as a beginner - imagined I was far from best practice. David then asked me to read a paper at his conference, and I attempted to obtain travel funding here in Ireland... I tried our government Department of Education (at a time when the Minister had made it known he was about to travel to the US with a view to examining best practice with regard to the use of computers in education...), I tried our Department of Foreign Affairs... No luck with any of them.
(Fr.) Sam Clyne (President of my College; he and I had many an unfortunate 'run-in') offered to cover my travel expenses, but I had my reasons for declining his generous offer. Poor Sam, how I regret that we just didn't get on, but I have to record to Sam's credit that when my father died in March 1995, he travelled all the way from Dublin to Bailieboro to assist at my father's funeral service (where the principal celebrant was Fr. Noel McIntyre, a former pupil of my father's). [end of personal note.]
Aims/Objectives.
This NT+Crypto 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'.)
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.
Encryption, decryption and digital signatures. Download here.
Primality testing: Lucas, Proth, Pocklington, Lehmer, Selfridge.
The Lucas-Proth-(Pocklington)-(Lehmer-Selfridge) theorems. Download here.
The (Lucas)-(Kraitchik)-Lehmer-Selfridge theorem. Download here.
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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