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#### Elementary Number Theory with Applications to Communication Systems

The course is divided into two parts. In part one, students are introduced to the fundamentals of number theory. This includes: Representation of integers to different basis, Fundamental theory of arithmetic, Linear and higher order congruences and their applications, Euler phi-function and its importance, Primitive Root definition, as well as Quadratic Residues and Legendre symbol.

In the second part of the course, applications of the above concepts are discussed. These applications include: wireless multiple access communications, frequency hopping radar and sonar, and cryptography. At the end of the course, students will have enough knowledge of number theory to find applications in their own field of interest and to understand the role of number theory in modern communication systems. In addition, they will understand aspects of wireless systems design.

Topics include:

Part One - Fundamentals

• Integers and Integer Representation
• Congruences, Linear congruences, Systems of Linear Congruences, Chinese Remainder Theorem
• Special Congruences (Fermat’s Little Theorem, Wilson’s Theorem), Eulers Theorem
• Primitive Roots (Primitive Roots for Primes, Existence of Primitive Roots, Index Arithmetic)
• Cryptology (Public key cryptography)
• Higher Order Congruences

Part Two - Multiuser Communications and Number Theory

• Frequency Hopping Sequences
• Application of FH sequences in Radar and Sonar
• Application of FH sequences in 4G LTE
• Application of FH sequences in a wireless propagation environment

Course typically offered: In-class in Winter and Online in Summer.

Prerequisites: A solid background in algebra and enough mathematical maturity and logical precision to be able to understand abstract arguments.

Next Steps: Upon completion of this class, consider enrolling in other Applied Mathematics coursework.