Georgia Institute of Technology College of Sciences

This course will be an introduction to number theory and its applications to modern cryptography. Number theory, one of the oldest branches of mathematics, is about the endlessly fascinating properties of integers. The backbone of the course will be modular ("clock") arithmetic, which we will apply to calendar calculations (“What day of the week was March 17, 1903?”), music theory (the circle of fifths), security and randomness (how to flip a coin over the telephone), and the mathematics of card shuffling (magic tricks included!). We will learn how number theory is used in public key cryptography to securely transmit information over the internet: this leads naturally to discussions of factoring, primality testing, and the discrete logarithm problem.

The course will cover similar topics as Math 4150 but with more emphasis on examples and applications and less on abstract theory.   Specific topics to be covered include unique factorization (the Fundamental Theory of Arithmetic), divisibility criteria, the Euclidean algorithm, modular arithmetic, the Chinese Remainder Theorem, Fermat's Little Theorem and Euler's Theorem, primitive roots, public-key cryptography (including RSA, ElGamal, and digital signatures), primality testing, discrete logarithms, quadratic reciprocity, and the Prime Number Theorem. We will also provide an overview of the software package SAGE. Students are not expected to have significant computer programming experience but will be expected to write some simple code and do basic computations.

This course will be offered jointly a course for gifted Georgia high school students who have run out of traditional math courses to take in their schools. This is why it will be taught as an asynchronous online course with instruction done via web-based videos, handouts, interactive apps, and the course text.The official class meeting time will be used for exams and office hours. Exams will take place in the official course meeting location, office hours will be conducted online via Adobe Connect. Students will have weekly homework and will be required to submit the homework online in PDF format. Students will need access to a reliable Internet connection and a computer with sufficient capability to handle the processing requirements of live web conferencing.