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Spr2013: MAT4930 7554 Number Theory (Special Topics) MWF6 LIT221 (NW)
Number Theory & Cryptography
(A previous version of
this course
has two nice photos of the class.
)
This is a 1-semester course interested in Mathematical Cryptography
(major topic) and other aspects of Coding:
Data Compression and
Error-correcting codes.
We also did one example of an Isomorphism code.
It was accessible to anyone with an introductory course or
who has read the first chapter or two of a beginning text.
(Mainly, I'd like you to know how to
“add two numbers mod N”,
and what the Euler phi function is.)
Good choices for self-study are:
- Elementary Number Theory by James Strayer;
or
- A Friendly Introduction Number Theory by Joseph Silverman;
or
- Elementary Number Theory by David Burton;
or
- the text by Vanden Eynden.
Here are the
Courageous Ones, who registered:
(Click for a larger version. Which student has spinach in his teeth?
[Perhaps it is visible in the photo at the page-bottom?])
Various math czars who help out.
| Chalk |
Phone |
Time |
Computer/Proj |
Blackboard |
Humor |
E-Problems |
| Dillon |
Baker |
Hannah! |
Thom |
everyone |
me |
Jackson & James G. |
End-of-semester NT Individual project
The Project-Y
is due, slid
u
n
d
e
r
my office door (Little Hall 402, Northeast corner)
,
no later than noon, Friday, 26Apr2013.
The project must be carefully typed,
but diagrams may be hand-drawn.
At all times have a paper copy you can hand-in; I do
NOT accept
electronic versions.
Print out a copy each day, so that you always have the latest version to
hand-in; this, in case your printer or computer fails.
(You are too old for My dog ate my homework.
)
Please follow the guidelines on the
Checklist 
(pdf, 3pages) to earn full credit.
ENCODING:
a b c d e f g h i j k l m n o p q r s t u v w x y z ' . ? ! ,
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Three Examples of simple ciphers:
CAESAR: Shift of 9:
a b c d e f g h i j k l m n o p q r s t u v w x y z ' . ? ! ,
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 0 1 2 3 4 5 6 7 8
MULTIPLICATIVE-CIPHER: Mult by 5:
a b c d e f g h i j k l m n o p q r s t u v w x y z ' . ? ! ,
0 5 10 15 20 25 30 3 8 13 18 23 28 1 6 11 16 21 26 31 4 9 14 19 24 29 2 7 12 17 22 27
AFFINE-CIPHER: Mult by 5, then add 9. x |-> [5*x]+9 (mod 32)
a b c d e f g h i j k l m n o p q r s t u v w x y z ' . ? ! ,
9 14 19 24 29 2 7 12 17 22 27 0 5 10 15 20 25 30 3 8 13 18 23 28 1 6 11 16 21 26 31 4
Our Teaching Page
has useful information for students in all of my classes.
It has my schedule,
LOR guidelines,
and Usually Useful Pamphlets.
One of them is the
Checklist (pdf)
which gives pointers on what I consider to be good mathematical
writing.
Further information is at our
class-archive URL
(I email this private URL directly to students).
Approx. Syllabus
- A review of modular arithmetic.
- Versions of The Euclidean Algorithm (the
Lightning Bolt
alg).
- Possibly: LBolt over the Gaussian Integers.
Proving unique factorization in the Gaussian Integers. Using LBolt to write
certain primes as sums-of-two-squares.
- Euler phi function, Fermat's Little Thm.
Euler-Fermat Thm (EFT). The Legendre and Jacobi symbols.
- The RSA Cryptosystem.
- The Chinese Remainder Thm (CRT) and a brief introduction to Rings and Ring-isomorphism.
- Huffman codes. Huffman's theorem on
minimum expected coding-length codes.
Uniquely-decodable codes and the Kraft-McMillan theorem.
- Elias delta code and the Ziv-Lempel adaptive code.
- Diffie-Hellman Cryptosystem.
Shank's Baby-step Giant-step method for trying to break the Diffie-Hellman protocol.
- Pollard-ρ factorization algorithm.
Descent-from-the-Top algorithm for computing the mod-M mult. order of an element.
- Possibly: Pollard's p-1
factorization algorithm.
- Miller-Rabin algorithm. Possibly: Polytime testing whether N is a prime-power.
- Multiplicative functions. Dirichlet convolution.
- Possibly: Meshalkin Isomorphism code.
Looking into the Future, here is part of
the Spring 2014, NT & Cryptography, page:
Our
syllabus mentioned some
algorithms we will learn.
There were some
nice microquizzes during the semester.
The exams were:
JK Home page
![[Image: Updated]](Images/updated.gif)
[Wednesday, 27Mar2013]
Our textbook is
An Introduction to Mathematical Cryptography
(Undergraduate Texts in Mathematics).
Resources on