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 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.
Two paws upsays renowned math-animal critic Archimedes Salamander. A Smash Hit!
Baby-Step Giant-Stepassignment, was due Monday, 05Nov2007. Here is a typeset version of problem 4.4 (pdf).
Here is the final project (pdf), to be done individually. It was due,
slid u n d e r my office door (Little Hall 402, Northeast corner) , 2PM, Friday 7Dec2007 (Pearl Harbor Day). The final project must be carefully typed.
|Year:||2003||Publisher:||Chapman & Hall|
(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.)
Elliptic Curves: Number Theory and Cryptography, by Lawrence C. Washington.
For students registered for the graduate version of this course, MAT6905 and MAT6910, this is a Required text. For students in MAT4930, this is a recommended text, and I have put a copy on reserve at the Marston Science Library.
Rational Points on Elliptic Curves (Undergraduate Texts in Mathematics), by Joseph H. Silverman and John Tate.
A Computational Introduction to Number Theory and Algebra, by Victor Shoup. The author has made freely available a PDF copy of his text for personal use.
Among other nice features, Shoup's text has a description+proof of the Polynomial-time Deterministic Algorithm (AKS primality test) for testing whether a positive integer is prime.
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,
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:
NT1, Spring 2007.
Pythagorean triples(a,b,c) of positive integers for which
Pythagorean Triples (pdf)at Usually Useful Pamphlets.
Lightning Boltalg), possibly over the Gaussian Integers. Using Lightning Bolt to write certain primes as sums-of-two-squares.
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 thefather 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 famousLast Theoremin 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.
appliedapproach to cryptography.
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