Programme/Approved Electives for 2022/23
Available as a Free Standing Elective
MAT-20025 Abstract Algebra.
Number Theory studies the properties of the natural numbers and the integers. It is one of the oldest and most beautiful areas of Pure Mathematics, and also one of the most active areas of modern research. Recently, ideas from Number Theory have been applied to problems in Cryptography, such as the design of ciphers and secret sharing schemes. This module will trace the development of the subject from ancient problems to these modern applications.
This module aims to introduce students to Number Theory, which is one of the oldest and most beautiful branches of Pure Mathematics, and also illustrate how some concepts from Number Theory have had unexpected applications to modern problems in cryptography.
Intended Learning Outcomes
evaluate different approaches to linear congruence to determine a full list of solutions, or provide a mathematical proof that none exist: 1,31,3state the definition of a Primitive Root modulo a natural number, and apply this definition to solve theoretical problems concerning the enumeration and properties of primitive roots: 3define Quadratic Residues and Legendre Symbols and, by stating, proving and applying suitable theorems, compute Legendre symbols in a variety of cases: 2,3explain and evaluate the construction and properties of a variety of ciphers, including symmetric and asymmetric (public key) systems: 2,3appraise the strengths and weaknesses of different cryptography systems, and make judgements on appropriate side-channel attacks to attack poorly implemented systems:
Lectures and Examples Classes: 36 HoursPreparation of Coursework: 24 Hours Independent Study: 88 HoursUnseen examination : 2 hours
1: Assignment weighted 15%
Description of Module Assessment
Number Theory AssignmentA take-home assignment comprising approximately 5 questions designed to assess students' progress towards the ILOs relating to number theory. Solutions will be uploaded to the KLE.
Students should expect to spend 10 hours on the assessment.2: Assignment weighted 15%
Cryptography AssignmentA take-home assignment comprising approximately 5 questions designed to assess students' progress towards the ILOs relating to cryptography. Solutions will be uploaded to the KLE.
3: Exam weighted 70%
2-hour unseen examinationThe examination paper will consist of no less than five and not more than eight questions all of which are compulsory. The examination will be closed book.