Number Theory and Cryptography

Spring 2019

Instructor: Matt Kerr
Office: Cupples I, Room 114
e-mail: matkerr [at]
Office Hours: 2-3 Monday, 12-1 Wednesday, and 10:30-11 Friday


Math 233, 309, and 310; if you are missing one of those but want to take the course, come and talk to me.

Class Schedule:

Lectures are on Monday, Wednesday, and Friday from 11-12 in Cupples I Room 113, beginning Monday Jan. 14 and ending with a final exam review class on Friday April 26. Spring break is the week of March 11th, and Monday Jan. 21 is a holiday.

Midterm Exam 1: Friday, February 22 (in class) [solutions]
Midterm Exam 2: Monday, April 1 (in class)
Final Exam: Tuesday, May 7, 10:30AM - 12:30PM, in the same classroom.

Regarding missed exams, see the Grading Policy section below. Calculators aren't allowed, but the exams will not be computationally heavy.


The course text will be these notes. While I'll sometimes stray from this in lecture, the exams and homeworks are based on the text (and nothing else). If you want to buy a supplementary textbook,

Hoffstein, Pipher, and Silverman, "An Introduction to Mathematical Cryptography", Springer, 2008.

is good and covers some of the same material.

Course Syllabus:

We will cover many of the basics of elementary number theory, giving you a base of concrete knowledge from which you will be able to approach modern algebra, algebraic number theory and analytic number theory. The course will also serve as an introduction to one of the most important real-world applications of mathematics, namely the use of number theory and algebraic geometry in public key cryptography.

Brief Outline:

I. Primes and Divisibility
II. Congruences
III. Introduction to Cryptography
IV. Diophantine equations
V. Elliptic cryptosystems
VI. Algebraic Numbers


There will be a weekly homework due on Wednesday at 11:59PM, starting the second week. (You'll turn it in via Crowdmark.) I am available for help in office hours. (Regarding late homework, cf. the Grading Policy below. The grader will be Caleb Ji.) The assignments (and later, their solutions) will be distributed through Crowdmark.

Some Interesting Links:

Advanced Encryption Scheme: Wikipedia, NIST
Public Key Cryptohistory: Ellis, HPS
Computational complexity: P vs. NP, Minesweeper
PARI: PARI/GP home, reference card, tutorial, Conrad, Stein

Grading Policy:

Homework is worth 40% of your final grade; Midterm Exam 1 and Midterm Exam 2 are worth 15% each; and the Final Exam is worth 30%. I will drop your lowest 2 homework scores. Grades will be kept track of on blackboard.

If you have to miss an hour exam for a legitimate reason, you will be given a makeup exam. Of course verified illness and serious family emergency are legitimate reasons. (For the final exam, those are the only acceptable reasons.) Regarding other conflicts, e-mail me as soon as you know about them.

In general, credit will be given for late homework only in the event of illness or emergency. You may discuss homework with other students (calculators/computers are of course also allowed), but you should not have duplicate solutions. This link takes you to the standard university policies on academic integrity.