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Spr2019: MAT4930 2H22 17706, Number Theory & Cryptography MWF7 [13:55-14:45] LIT207 (CE)

Number Theory & Mathematical Cryptography We may use more advanced computing devices...

(Previous versions of [Spring 2016] and [Spring 2013] and [Spring 2011] NT&MC have nice photos of the class.)

The following is from Wikipedia, the free encyclopedia

Cryptography (or cryptology; derived from Greek κρυπτός kryptós "hidden," and the verb γράφω gráfo "write") is the study of message secrecy. In modern times, it has become a branch of information theory, as the mathematical study of information...

... Steganography (i.e., hiding even the existence of a message so as to keep it confidential) was also first developed in ancient times. An early example, from Herodotus, concealed a message - a tattoo on a slave's shaved head - by regrown hair. More modern examples of steganography include the use of invisible ink, microdots, and digital watermarks to conceal information .

Ciphertexts produced by classical ciphers always reveal statistical information about the plaintext, which can often be used to break them. After the Arab discovery of frequency analysis (around the year 1000), nearly all such ciphers became more or less readily breakable by an informed attacker. ... Essentially all ciphers remained vulnerable to cryptanalysis using this technique until the invention of the polyalphabetic cipher by Leon Battista Alberti around the year 1467. Alberti's innovation was to use different ciphers (ie, substitution alphabets) for various parts of a message (often each successive plaintext letter). He also invented what was probably the first automatic cipher device, a wheel which implemented a partial realization of his invention. In the polyalphabetic Vigenère cipher, encryption uses a key word, which controls letter substitution depending on which letter of the key word is used.

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 (using that 5 is coprime to 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 ' . ? ! , 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 important information for my students. (It has the Notes, Exams and Links from all of my previous courses.)
The Teaching Page has my schedule, LOR guidelines, and Usually Useful Pamphlets. One of them is the ChecklistThe checklist (pdf) which gives pointers on competant mathematical writing. Further information is at our class-archive URL (I email this private URL directly to students).
In all of my courses, attendance is absolutely required (excepting illness and religious holidays). In the unfortunate event that you miss a class, you are responsible to get all Notes / Announcements / TheWholeNineYards from a classmate, or several. All my classes have a substantial class-participation grade.

Photo of text cover 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)

Homepage of  Intro... Mathematical Cryptography, with a link to its Errata sheet.
Here are links to this book at The Publisher's site and at

Approx. Syllabus

Some Resources

Various math czars who help out.

Time Computer/Proj Blackboard Humor E-Problems Phone
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Web Resources on The Web

  1. If she saw this, AM HERE ABE SLANEY (TAotDM) , WWSKnow?
  2. The decipherment of a substitution cipher appears in The Gold Bug, by Edgar Allan Poe.
  3. Wikipedia. See also Primality testing and links to original AKS article and improvement.
  4. Historical Cryptography (Trinity College)>.
  5. MathPages. (I haven't reviewed this.)
  6. Sample chapters from the Handbook of Applied Cryptography. (I have not reviewed this book.)


End-of-semester NT Individual Project


The  IP (pdf),  is due, slid u n d e r room 402 Little Hall my office door (Little Hall 402, Northeast corner) , no later than   3PM, Thur., 25Apr2019.

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 Checklist (pdf, 3pages) to earn full credit.

Have a great summer, folks! Stop by in the Autumn 2019.


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____End: Number Theory