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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 note of late March 2017.
You may encounter a problem opening my Maple worksheets (as indeed I do myself) here at my web site - it would appear to depend on the internet
browser being used.
Thus, if I attempt to open one of my worksheets using Interner Explorer there is never a problem, whereas if I use Firefox then all that one sees
- this is just an example - is something like this:
{VERSION 3 0 "IBM INTEL NT" "3.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 0 128 0 128 1 0 1 0
0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0
0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "2D Output" 2 20 "" 0 0 0 128 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "
" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0
0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE ""
and so on (almost at infinitum)...
I asked Maplesoft for advice on this and they recommended doing the following (and I found it worked):
- Don't attempt directly to open the worksheet (by clicking on my link), instead right click and save the worksheet to (say) the download folder.
- Now that the worksheet is in the download folder (cut it out if you wish and put into whatever folder you wish) you may open the worksheet
in the usual way (I should add that while I see this 'works', I have absolutely no idea as to why it does... ) [END OF NOTE].
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).
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| 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.
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| A proper understanding of
the modular exponentiation method requires an understanding of binary expansions (17 KB).
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| 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.
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| 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).
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| 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.
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