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Aut2007: MAT4930 5662 Number Theory (Special Topics) MWF2 LIT203

Number Theory and
Elliptic curve Cryptography
We may use more advanced computing devices...

Cryptographically curious? —Then C'mon in! 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 ChecklistThe 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).

Our major topic is Number Theoretic codes, in particular Elliptic curves and Cryptography.

The various Number Theory czars who help out:
Chalk Blackboard Time Phone-list
Jackie Zach Cameron Catherine

End of semester project

Here is the final project (pdf), to be done individually. It was due,
slid u n d e r room 402 Little Hall my office door (Little Hall 402, Northeast corner) , 2PM, Friday 7Dec2007 (Pearl Harbor Day). The final project must be carefully typed.

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.

Our textbook is Cryptanalysis of Number Theoretic Ciphers.
Author: Samuel Wagstaff ISBN: 978-1584881537
Year: 2003 Publisher: Chapman & Hall
Photo of text cover

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

(Note: There is a 2002 edition and a 2003 edition. Here is a list of errata for the two editions. The 2003 edition is preferable, since it corrects typographical errors of the 2002 edition.)

Reference books

Among many fine NT books, here are a few:

The prerequite material can be learned in a few weeks of reading during the summer. 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 immediately above), together with these Wikipedia pages:

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

Here are syllabi and exams and notes from my five earlier Number Theory courses:
Number Theory 1 and 2 , from Spring 2000 and Spring 2001.
NT1 and NT2 & Cryptography , Spring 2006 and Autumn 2006.

NT1, Spring 2007.

Potential topics

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 (see links below). 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?

The answer: 84 The answer is determined from two methods: 1. Finding the common multiple of 12, 6, and 7 (which is 84). 2. Taking 14 (the age up to which would be considered a boy; one-sixth of his life) multiplied by 6, which equals 84.

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