Syllabus for Number Theory, Dr. Ellen Gethner

Number Theory/Applied Number Theory: CSC 4800/5800 and Math 4110/5110 (Spring 2016)

Professor Ellen Gethner

Syllabus

automatically updated on 8 January 2016

[ Instructor and Office Hours | Class Time and Room | Textbook | Prerequisites | Objectives | Grades and Policies | Schedule| Academic Deadlines | Student Page (not available yet) ]

Instructor

Dr. Ellen Gethner
Email: ellen dot gethner at ucdenver dot edu
Office: Lawrence Street Center LW 817
Phone: 303 315 1405
Office hours:

Tuesday and Thursday 11:45-1:45pm
Tuesday and Thursday 4:45-5:15pm

Class Time and Room

Tuesdays and Thursdays 3:30-4:45pm in Lawrence Street Center 844

Textbook

Elementary Number Theory, 6th edition by Kenneth Rosen, published by Pearson. Available at the Auraria book store and many other places.

Other Resources (send interesting links and I'll add them)

introductory prime number theory resources
Riemann Hypothesis
A page with free number theory textbook downloads
RSA Challenge (archived)
The Number Theory Web
Number Theory Visualizatons
Gaussian Primes What does the graphic at the top of this page mean?

Prerequisites

Either CSC 2411 (Discrete Structures) **or** Math 3000 (Introduction to Abstract Math).

Course Objectives

We will begin by acquiring knowledge of the basic tools and concepts in Number Theory such as Integers, Primes, Divisibility, GCD, Congruences, Wilson's and Fermat's Theorems, Pseudoprimes, and Multiplicative Functions (such as the Euler phi funcion). Then we will move on to more advanced topics such as Cryptology and Computer Security, Quadratic Residues, Continued Fractions, Diophantine Equations, and the Gaussian Primes. We will study computational and algorithmic aspects of each topic as appropriate.

Grades and Policies

Assignments and Quizzes: Homework assignments (6) will count toward 40% percent of your grade. You will not hand in homework assignments: each assignment will have an associated in-class quiz (30 minutes), which will be given only once. Homework 6 material will be tested on the final exam (along with other material). The lowest score will be dropped.
- Each assigment will have an assortment of straightforward problems as well as challenging problems.
- To prepare for the quizzes, write up your ideas on all of the problems that you have given some thought to, even if you do not believe you have a complete solution.
- Be sure to be neat, write in complete sentences, and do not leave out any steps: clarity of exposition may contribute toward a positive score. Please talk to one another about the assignments, but the write-ups should be done individually so that you are sure to be prepared for the quiz.
Make-up quizzes: None. I will drop your lowest score. There are no exceptions.
Midterm Exam (30% for undergrads, 20% for grads) Given once only during class on Thursday 10 March 2016.
Final Exam (30% for undergrads, 20% for grads) Given once only during class on 5 May 2016.
CS Graduate Students will give a research paper presentation and software demonstration (20% of final grade) . 35 minute research presentation to be given to your colleagues in class. Early in the semester I will hand out a research guidelines document. You are expected to find, read and thoroughly understand the (peer-reviewed legitimate and substantive) paper and the contents of the bibliography. In addition you will implement the ideas about which you have learned.You will prepare and present a conference-style presentation on your paper, complete with citations and a software demonstration at the end of the semester. A guide will be posted with the list of possible papers with more explanation.
Math graduate students will conduct one class period on the topic of their choice (20% of final grade). We will meet individually to discuss options. This assignment is very flexible: be creative and take advantage of the opportunity to teach us something about which you are passionate (related to Number Theory).

Schedule

Lecture	Date	Topic(s)	Comments	Reading	Assignment, Quiz, and Exam Schedule
One	19 and 21 January 2016	Introduction to Number Theory	Vocabulary and Definitions; Mathematica demo on the 21st	Chapter 1.1-1.5, 2.1-2.3 and class notes	Homework problems are in this week's mathematica notebook
Two	26 and 28 January 2016	Vocabulary and Definitions, continued	Primes and greatest common divisor	Chapter 3.1 and 3.2. Class notes.
Three	2 and 4 February 2016	GCD, Euclidean Algorithm, Fundamental Theorem of Arithmetic		Chapter 3.3, 3.4, and 3.5 and class notes	Quiz 1 on Thursday; Homework problems are contained in the mathematica notebook
Four	9 and 11 February 2016	Fermat Numbers, Congruences, Chinese Remainder Theorem	Homework problems are contained in the mathematica notebook	3.6, 4.1, 4.2, 4.3 and class notes
Five	16 and 18 Feb 2016	Wilson's Theorem, Fermat's Little Theorem, Pollard Factoring Method		6.1 and class notes	Quiz 2 on Thursday; Homework problems are in this week's mathematica notebook
Six	23 and 25 Feb 2016	Pseudoprimes, Euler's Theorem (generalizes Fermat's Little Theorem), Euler's phi function		6.2, 6.3, 7.1 through example 7.4	Quiz 3 is on Thursday and covers material through 6.1; The next batch of homework problems are in this week's mathematica notebook
Seven	1 and 3 March 2016	Perfect Numbers, Mersenne Primes, Cryptography	Main topic on Thursday is public key cryptography, particularly RSA	8.1, 8.3 (not covered in class, but useful to know), 8.4 (RSA) and class notes
Eight	8 and 10 March 2016	Midterm (Thursday) plus the Diffie-Hellman Key Exchange, Playing Poker over the phone, Secret Sharing Schemes (Tuesday)	read 8.6 for Tuesday and exam one will be on Thursday 10th March		The midterm will cover material up through 7.3 in the Mathematica notebook "NumberTheoryWeek7_Part1"
Nine	15 and 17 March 2016	Another Threshold Scheme, Primitive Roots, Discrete Log Problem, and ElGammal Encryption	Alternative approaches to these topics are given in this week's Mathematica notebook	9.1-9.4 and class notes
null	21-27 March 2016	Spring Break: no classes
Ten	29 and 31 March 2016	Topic	Backdoors and how to build them	Guest speaker: Derek Kern	Quiz 4 on Thursday;
Eleven	5 and 7 April 2016	Elliptic Curve Cryptography		Class notes
Twelve	12 and 14 April 2016	Quadratic reciprocity		9.4 (again), 11.2, 11.2, and class notes	Quiz 5 on Thursday;
Thirteen	19 and 21 April 2016	1.Promoting Number Theory in High Schools or Birthday Problem and Number Theory, and 2. On the Applications of Mobius Inversion in Combinatorial Analysis, and 3. Hyper and Elliptic Curve Cryptography	Grad student presentations	Lecture slides
Fourteen	26 and 28 April 2016	1. The security, or lack thereof, of wireless technology, 2. The State of Elliptic Curve Cryptography, 3. Advanced Encryption Standard (AES), and 4. Speeding the Pollard and elliptic curve methods of factorization	Grad student presentations	Lecture slides
Fifteen	3 and 5 May 2016	Tuesday:1. Periodic Gaussian Moats and 2. An Algorithm for the Machine Calculation of Complex Fourier Series; Thursday EXAM	Exam plus grad student presentations	Lecture Slides	Thursday 5th May, Exam 2. Exam 2 is comprehensive and will include material from the week 12 Mathematica notebook on quadratic reciprocity.
Sixteen	Week of 9 May 2016	Monday office hours to be determined.