# Math 110

Applied Number Theory and Field Theory

### Course Information

**Lecture:**Tuesdays and Thursdays 12:50pm–2:05pm

Building 380 (Sloan Hall), Room 380-W (Basement Floor)

**Course Webpage:**http://web.stanford.edu/~jchw/2015Math110

Grades will be made available through Coursework: https://coursework.stanford.edu/portal/site/W15-MATH-110-01/

**Textbook:**Trappe and Washington, Introduction to Cryptography with Coding Theory

** Professor: ** Jenny Wilson

**Email:** jchw@stanford.edu

**Office Hours:** Tuesdays 9:00am–10:30am and Wednesday 6:30pm–8:00pm

Building 380 (Sloan Hall), Room 382-H (Second Floor)

**Course Assistant:** Jeremy Booher

** Email:** jbooher (at math.stanford.edu)

**Office Hours:** Tuesday 2:05pm–3:30pm and Wednesday 4pm–5:30pm

Building 380 (Sloan Hall), Room 381-L (First Floor)

**Course Description:** Number theory and its applications to modern cryptography.

Topics: congruences, finite fields, primality testing and factorization, public key cryptography, error correcting codes, and elliptic curves, emphasizing algorithms. WIM.

Grading Scheme: | |

Homework | 20% |

Quizzes | 10% |

WIM | 20% |

Midterm | 20% |

Final | 30% |

**Other suggested references:**

Stein,

*Elementary number theory: primes, congruences, and secrets*(freely available online)

Koblitz,

*A course in number theory and cryptography*(a more advanced book)

Hungerford,

*Abstract algebra: an introduction*(a rigorous introductory reference on algebra and number theory, in more depth than Trappe-Washington).

If you find other references you think should be added to this list, let me know!

**Students with documented disabilities:** Students who may need an academic accommodation based on the impact of a disability must initiate the request with the Office of Accessible Education (OAE). Professional staff will evaluate the request with required documentation, recommend reasonable accommodations, and prepare an Accommodation Letter for faculty dated in the current quarter in which the request is made. Students should contact the OAE as soon as possible since timely notice is needed to coordinate accommodations. The OAE is located at 563 Salvatierra Walk (723-1066, http://studentaffairs.stanford.edu/oae).

** Homework policy: ** Homework assignments will be posted to the course webpage. Homework is due on Thursdays, and collected at the beginning of lecture. Late homework is generally not accepted except under extenuating circumstances. Your homework solutions should be * neat, legible, and stapled *. You may work in groups and discuss homework problems with other students, but your solutions must be written up *independently and in your own words*. Mathematical proof and writing skills are a major focus of this course, and homework will be evaluated for effective communication and exposition.

Each student's lowest homework score will be dropped.

You are welcome to use other texts and online resources to review the mathematical theory and computational techniques we cover. You may not, however, seek out solutions to specific homework problems. Outside sources should be used to improve your understanding of the material, not as a shortcut to finish assignments with an incomplete understanding. Use your discretion.

You are encouraged to discuss the homework with classmates and work on difficult problems in groups.

** Quizzes: ** There will be short quizzes throughout the quarter, typically 10 minutes. I will give advance notice about each quiz and hints about what it will cover. The quizzes are intended to encourage the class to regularly review the material, to provide practice for the exams, and to give early feedback (both to you the student, and to me the teacher) about your progress.

Each student's two lowest quiz scores will be dropped.

### Homework

#### Recommended reading and exercises: Primer on Mathematical Proof-Writing

Homework 0 | Due: Thursday January 8 | |

MyFirstDocument.tex | ||

Template.tex | ||

Homework 1 | Due: Thursday January 15 | (Solutions) |

Homework 2 | Due: Thursday January 22 | (Solutions) |

Homework 3 | Due: Thursday January 29 | (Solutions) |

Homework 4 | Due: Thursday February 5 | (Solutions) |

Homework 5 | Due: Thursday February 12 | (Solutions) |

Homework 6 | Due: Thursday February 19 | (Solutions) |

Homework 7 | Due: Thursday February 26 | (Solutions) |

Homework 8 | Due: Thursday March 5 | (Solutions) |

Field Axioms Handout | ||

Homework 9 | Due: Thursday March 12 | (Solutions) |

### Quizzes

Quiz 1 | (Solutions) |

Quiz 2 | (Solutions) |

Quiz 3 | (Solutions) |

Quiz 4 | (Solutions) |

Quiz 5 | (Solutions) |

### Exams

The course will have one ** midterm exam ** on ** Tuesday 10 February (Week 6), in class. **

We will have a 2-hour midterm review session on Monday 9 February from 7pm–9pm in Building 380 (Sloan Hall), Room 380-F (Basement Floor). You're welcome to drop in for some or all of the review session. Come with questions!

Midterm | (Solutions and Grading Scheme) |

The

**final exam**will be

**Thursday 19 March at 7pm in 380-W**(our usual classroom).

The exams are closed book. Students may bring in a basic (non-programmable, non-scientific) calculator to the exams. Such calculators are available, for example, through the Stanford Bookstore.

Students may bring a single double-sided sheet of notes to the final exam (8.5 by 11 inches; standard letter size). The notes can be handwritten or typed, but you must prepare your notes yourself.

Review questions for the final exam are available here: Final Review Problems

We will have a 2-hour midterm review session on Tuesday 17 March from 4pm–6pm in Building 380 (Sloan Hall), Room 380-F (Basement Floor). Drop in for part or all of the session. Bring questions!

Final Exam (Solutions)

### WIM

The Writing in the Major (WIM) project is a short essay (about 4-7 pages) on a mathematical topic. The paper should be typeset in Latex.

A major focus of the project will be on *exposition*. The papers should be written at a level appropriate for a typical Math 110 student -- someone familiar with the material covered in our course, but not with the specific topic of the paper. WIM papers will be evaluated on the quality of the writing, including effective communication and writing style.

For advice on formal mathematical writing, see: Mathematical Writing, especially Section 1.

The Hume Center offers additional writing support.

**Detailed instructions:**WIM Guidelines and Grading Criteria

**Topics:**The following is a list of suggested topics. You may also write on a topic not listed here, but please have your topic and proposed project outline approved by Jenny and Jeremy.

The primitive element theorem

Continued fractions and applications to cryptography

The Miller-Rabin primality test

Zero knowledge proofs

**Topic-specific guidelines:**WIM Topics and Instructions

**Due Dates**

**Draft:**Tuesday 17 February (in class)

**Final copy:**Friday 6 March (e-mail to Jeremy by 5pm)

### Latex Resources

**Installing Latex:**

Installing Latex on Windows

Installing Latex on a Mac

For Linux you can install TeX Live and an editor such as Kile or TeXstudio

Alternative Tex distributions and editors, and installation instructions: Latex/Installation

A comparison of Tex editors

A list of online Tex editors

**Latex references**

Latex Wiki

Documents:

Latex cheat sheet (pdf)

Writing scientific documents using Latex (pdf)

The not so short introduction to Latex (pdf)

Tutorials:

Latex Tutorial: Getting to grips with Latex

Latex Tutorial: Everything you wanted to know about Tex, but were too afraid to ask

Video Tutorials:

Latex Tutorial Series (TeXmaker)

Latex Tutorial Series (TeXworks)

### Extraneous Reading

The following reading is strictly optional: it is not related to the course material and will not be discussed in the course. These are articles on math education and learning psychology which may be of interest to math students.