MAT-30038 - Number Theory and Cryptography
Coordinator: Paul J Truman Room: MAC2.16 Tel: +44 1782 7 33246
Lecture Time: See Timetable...
Level: Level 6
Credits: 15
Study Hours: 150
School Office: 01782 733075

Programme/Approved Electives for 2022/23

None

Available as a Free Standing Elective

No

Co-requisites

None

Prerequisites

MAT-20025 Abstract Algebra.

Barred Combinations

None.

Description for 2022/23

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.

Aims
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: 1,3
1,3
state 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: 3
define Quadratic Residues and Legendre Symbols and, by stating, proving and applying suitable theorems, compute Legendre symbols in a variety of cases: 2,3
explain and evaluate the construction and properties of a variety of ciphers, including symmetric and asymmetric (public key) systems: 2,3
appraise the strengths and weaknesses of different cryptography systems, and make judgements on appropriate side-channel attacks to attack poorly implemented systems:

Study hours

Lectures and Examples Classes: 36 Hours
Preparation of Coursework: 24 Hours
Independent Study: 88 Hours
Unseen examination : 2 hours

School Rules

None.

Description of Module Assessment

1: Assignment weighted 15%
Number Theory Assignment
A 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 Assignment
A 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 examination
The 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.