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Or: JK Homepage.

Modified:
Sunday, 01Mar2020
Printed:
Thursday, 03Dec2020

Page:
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Spr2011: MAT4930 7554 Number Theory (Special Topics) MWF6 LIT219 (SW)

This was a 1-semester course (Spring of 2011) for students who have had the rudiments of Number Theory, and would like to learn more, and to study applications of NT to Mathematical Cryptography, and to Coding in general [i.e, data-compression codes, isomorphism codes].

- Available are all the quizzes so far, and some potential future quiz/text questions (pdf). [Monday, 21Mar2011]
- Alice and Bob made an appearance on Class-Y (pdf), to the delight of Cryptographers –and to the dismay of Eve.
- Persuasive Paula persisted, pushing the permitting of practicality
[i.e, calculators], for
Class-X (pdf).
Of course, they were permitted on
Home-X (pdf)
which explored the theme of
*Collision Algorithms*. [It was due at BoC on Monday, 21Mar2011.] - The decipherment of a substitution cipher appears in The Gold Bug, by Edgar Allan Poe.
- If she saw this, (TAotDM) , WWKKnow?
- Some
codes, mostly prefix (txt).
Here are some
*very-preliminary,-still-being-proofread*notes on codes (pdf).We'll take a look at Caesar ciphers and affine ciphers. A changing Caesar cipher is called a Vigenere cipher. (Shall we listen to (Blaise de) Vigenere?). One can generalize to Hill-type codes.

- The little bitsy Class-W (pdf) addressed the triumvirate of Huffman, Elias and Ziv. It complemented the squarish Home-W (pdf). (Please let me know immediately of errors on the exam, or team-dynamic problems. Ta.)
- For the curious, Huffman's original 1952 paper (pdf) is online.
- Here is the semester schedule (txt), This calendar may have to change somewhat during the semester, so please check on it once-a-week or thereabouts.
- I janned up some emacs-lisp code for the Miller-Rabin probabilistic primality test (textfile).

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).

The following, from
Wikipedia (Enigma_machine),
is *edited* and *abbreviated*.

In December 1932, the Polish Cipher Bureau first broke Germany's Enigma ciphers. Five weeks before the outbreak of World War II, on 25 July 1939, in Warsaw, the Polish Cipher Bureau gave Enigma-decryption techniques and equipment to French and British military intelligence. …[A]llied codebreakers were able to decrypt a vast number of messages that had been enciphered using the Enigma. The intelligence gleaned from this source was codenamed “Ultra” by the British.

[After World War II] Winston Churchill told Britain's King George VI:

It was thanks to Ultra that we won the war.Though the Enigma cipher had cryptographic weaknesses, in practice it was only in combination with other factors (procedural flaws, operator mistakes, occasional captured hardware and key tables, etc.) that those weaknesses allowed Allied cryptographers to be so successful.

Our Cryp class has a LISTSERV archive. I will email to each student how to post-to and read-from the Archive. (The archive is at a private URL, only for the use of the folks in our class.)

Our
Cryp syllabus
has important URLs and eddresses, and a partial topic-list
It also mentions the **Class photo Day**,
and *letters-of-recommendation*.

Various math czars who help out.

Time | Projector | Blackboard | Chalk | Humor | E-Probs |
---|---|---|---|---|---|

Kaitlin | Julius & Jay | Kyle | Trevor | Kyle | Jay & ? |

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 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.
- The Legendre and Jacobi symbols.
*Possibly*: Meshalkin Isomorphism code.*Maybe*: Lenstra's*Elliptic Curve*factorization algorithm.

The EoS
Project-Z (pdf)
was due, slid
u
n
d
e
r
my office door (Little Hall 402, Northeast corner)
,
no later than **noon, Friday, 22Apr2011**.

*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.