School of Computing and Mathematics

Faculty of Natural Sciences

For academic year: 2020/21 Last Updated: 25 February 2021

MAT-30038 - Number Theory and Cryptography

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MAT-20025 Abstract Algebra.

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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.

evaluate different approaches to linear congruence to determine a full list of solutions, or provide a mathematical proof that none exist: 1,2

1,2

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: 1,2

define Quadratic Residues and Legendre Symbols and, by stating, proving and applying suitable theorems, compute Legendre symbols in a variety of cases: 2

explain and evaluate the construction and properties of a variety of ciphers, including symmetric and asymmetric (public key) systems: 2

appraise the strengths and weaknesses of different cryptography systems, and make judgements on appropriate side-channel attacks to attack poorly implemented systems:

1,2

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: 1,2

define Quadratic Residues and Legendre Symbols and, by stating, proving and applying suitable theorems, compute Legendre symbols in a variety of cases: 2

explain and evaluate the construction and properties of a variety of ciphers, including symmetric and asymmetric (public key) systems: 2

appraise 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 Hours

Preparation of Coursework: 24 Hours

Independent Study: 88 Hours

Unseen examination : 2 hours

Preparation of Coursework: 24 Hours

Independent Study: 88 Hours

Unseen examination : 2 hours

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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-hour unseen online 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 online and open book.