Goto: Prof. King's page at Univ. of Florida.
Or: JK Homepage.

Modified:
Friday, 12Nov2021
Printed:
Sunday, 22May2022

Page:
http://squash.1gainesville.com/Include/thispage.shtml

Spr2013: MAT4930 7554 Number Theory (Special Topics) MWF6 LIT221 (NW)

(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. |

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

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).

- Here is our Class-X.
It was preceded by
Home-X, which
was due at
**BOC**on**Wednesday, 03Apr2013**. - Here are some
*preliminary*Notes on Codes (pdf). - A
topics guide (txt)
for Class-W
was posted on our Archive, and it provoked much posting in turn.
Our
Class-W
took place on
**Monday, 25Feb2013**. - For the curious, Huffman's original 1952 paper (pdf) is online.
- xkcd Cryptography
- Available are all the NT quizzes so far (pdf). [Wednesday, 27Mar2013]
- Empirical Entropy of English. (Thanks, Aidan.)
- The test of prerequisite knowledge was on Wednesday, 09Jan2013: A [harder than what I gave] sample-test (pdf) was available.

Our textbook is
An Introduction to Mathematical Cryptography
(Undergraduate Texts in Mathematics).

The homepage of ... Mathematical Cryptography, with a link to its Errata sheet.

Here are links to this book at The Publisher's site and at Amazon.com.

Authors: | Jeffrey Hoffstein, Jill Pipher, J.H. Silverman | ISBN: | 978-0-387-77993-5 |

Year: | 2008 | Publisher: | Springer |

Marston: | QA268 .H64 2008 | Electronic: | Chap. 1 and Chap. 2, Diffie-Hellman, etc. (Free for UF students) |

The homepage of ... Mathematical Cryptography, with a link to its Errata sheet.

Here are links to this book at The Publisher's site and at Amazon.com.

The Web

- Wikipedia. See also Primality testing and links to original AKS article and improvement.
- Historical Cryptography (Trinity College)>.
- MathPages. (I haven't reviewed this.)
- Sample chapters from the Handbook of Applied Cryptography. (I have not reviewed this book.)

- 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: