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S2011: MAT4930 7554 Number Theory (Special Topics) MWF6 LIT219 (SW)

# Number Theory & Cryptography

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].

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.

### General Info

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).
 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.

## Resources on The Web

2. Historical Cryptography (Trinity College)>.
3. MathPages. (I haven't reviewed this.)
4. Sample chapters from the Handbook of Applied Cryptography. (I have not reviewed this book.)

#### 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 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.

#### End-of-semester NT Individual project

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 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.