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Usually Spring: MAT4930 Number Theory & Cryptography

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

Nostalgic? See 2019g, 2016g, 2014g, 2013g (with photos), 2011g (photos), 2007t.
There is a closely allied Number Theory course.

Prerequisites: UF's Sets & Logic course (or equivalent)  OR  permission of the professor.

(I would like you to know: What a prime number is, and What mathematical induction is and What an equivalence relation is. Helpful, but not required, is knowing how to “add/multiply two numbers mod-N,  and the definition of the Euler phi function.)

See Useful NT Texts, below, for self-study suggestions.



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 by 9:   x ↦ 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 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:   x ↦ [5*x] (mod 32).   [Uses 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 Amazon.com.

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.



NT&Cryptography topic list




Various math czars who help out.

Time Computer/Proj Blackboard Humor E-Problems Phone
? ? ? ? ?



Web Cryptographic resources

  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. Historical Cryptography (Trinity College).
  4. Sample chapters from the Handbook of Applied Cryptography. (I have not reviewed this book.)
  5. Crypto-Gram Newsletter. Has an applied approach to cryptography.


End-of-semester NT&Crypto   Individual Optional Project (IOP)


Prof. ...will be due, typed, stapled, slid completely under my office door (402 Little Hall, northeast corner, top floor) no later than [Time & Date].

The Project must be carefully typed, but diagrams may be hand-drawn and scanned into the PDF.

The first page is to be the Problem Sheet, with Honor Code signed and blanks filled in.

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.


The following is abridged from Wikipedia, the free encyclopedia

Diophantus of Alexandria - (Greek: Διόφαντος ὁ Ἀλεξανδρεύς , circa 200/214 – circa 284/298)  was a Greek mathematician of the Hellenistic era. Little is known of his life except that he lived in Alexandria, Egypt ...

He was known for his study of equations with variables which take on rational values and these Diophantine equations are named after him. Diophantus is sometimes known as the father of Algebra. He wrote a total of thirteen books on these equations. Diophantus also wrote a treatise on polygonal numbers.

In 1637, while reviewing his translated copy of Diophantus' Arithmetica (pub. ca.250) Pierre de Fermat wrote his famous Last Theorem in the page's margins. His copy with his margin-notes survives to this day.

Although little is known about his life, some biographical information can be computed from his epitaph. He lived in Alexandria and he died when he was 84 years old. Diophantus was probably a Hellenized Babylonian.

A 5th and 6th century math puzzle involving Diophantus' age: He was a boy for one-sixth of his life. After one-twelfth more, he acquired a beard. After another one-seventh, he married. In the fifth year after his marriage his son was born. The son lived half as many as his father. Diophantus died 4 years after his son. How old was Diophantus when he died?


What is the answer, with reasoning?   (It is in the source-file.)






Spring 2019, NT&Crypto:


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



Spring 2016, NT&Crypto:


Various math czars who help out.

Time Computer/Proj Humor E-Problems Phone
Tad James S. Chris L. Autumn Austin J.



Spring 2014, NT&Crypto:




Spring 2013, NT&Crypto:

This was 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.


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 page-bottom?])
2013Sprint NT/Crypto class


Various math czars who help out.

Chalk Phone Time Computer/Proj Blackboard Humor E-Problems
Dillon Baker Hannah! Thom everyone me Jackson & James G.

2013Sprint NT/Crypto class



Spring 2011, NT&Crypto:


Welcome sign

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].


A good-looking 2011Spring NT&Cryptographers

Various math czars who helped the course run smoothly.

Time Projector Blackboard Chalk Humor E-Probs
Kaitlin Julius & Jay Kyle Trevor Kyle Jay & ?

The smiling 2011Spring NT&Cryptography class



Autumn 2007, NT & Elliptic curve Cryptography:

The prerequite material can be learned in a few weeks. It is knowledge of:


Modular arithmetic. Euclidean algorithm for the greatest common divisor of two integers. Euler phi-function and Fermat's Little Theorem. The Legendre and Jacobi Symbols. The statement of the Quadratic Reciprocity Theorem.

All of this is in any standard beginning Number Theory book, e.g, Strayer or Silverman (A Friendly Intro to NT) or Burton. It is also in chapters 1, 2, and 4.1-4.3 in Shoup's free online text (linked further below), together with the following Wikipedia pages:

  1. Legendre symbol.
  2. Jacobi symbol.
  3. Quadratic reciprocity.

Our textbook is Cryptanalysis of Number Theoretic Ciphers.
Author: Samuel Wagstaff ISBN: 978-1584881537
Year: 2003 Publisher: Chapman & Hall
Photo of text cover
(Note: There is a 2002 edition and a 2003 edition. Here is a list of errata for the two editions.)

Reference books

Among many fine NT books, here are a few:
I thank the Number Theory czars who helped:
Chalk Blackboard Time Phone-list
Jackie Zach Cameron Catherine



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____End: Number Theory