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Instructor: Matt Kerr Office: Cupples I, Room 114 e-mail: matkerr [at] math.wustl.edu Office Hours: Tuesday 4-5 and Friday 2-3 Class Schedule: Lectures are on Tuesday and Thursday from 2:30-3:50 in Seigle L006, beginning Tuesday Jan. 13 and ending on Thursday Apr. 23. Spring break is the week of March 9th. Midterm Exam 1: Thursday, Feb. 26 (in class) Midterm Exam 2: Tuesday, Apr. 7 (in class) Final Exam: 3:30-5:30, May 6, 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. Textbook: 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. (There is a second edition published in 2014 as well. It doesn't matter which one you have.) Course Goals: 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. Divisibility (Euclidean algorithm, Fundamental theorem of arithmetic, distribution of primes)) II. Congruences (modular arirthmetic, Chinese remainder theorem, primality testing, factorization) III. Introduction to Cryptography (RSA encryption, Diffie-Hellman key exchange) IV. Diophantine equations (Pell's equation, elliptic curves) V. Elliptic cryptosystems (El Gamal, Lenstra's algorithm) VI. Algebraic Numbers (rings of integers, ideals, Fermat's equation) Assignments: There will be a weekly homework due on Wednesday at 11:59PM (starting the second week), to be turned in via Gradescope. I am available for help in office hours. (Regarding late homework, see the Grading Policy below.) The assignments will be distributed through Gradescope, and solutions posted on Canvas. 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 30% of your final grade; Midterm Exam 1 and Midterm Exam 2 are worth 15% each; and the Final Exam is worth 40%. I will drop your lowest 2 homework scores. Grades will be kept track of on Canvas. 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. |