School of Computing and Mathematics

Faculty of Natural Sciences

For academic year: 2020/21 Last Updated: 30 May 2020

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,

state and prove the Chinese Remainder Theorem, and apply it to solve problems involving systems of simultaneous linear congruences: 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: 1,2,

explain and evaluate the construction and properties of a variety of ciphers, including symmetric and asymmetric (public key) systems: 1,2,

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 and prove the Chinese Remainder Theorem, and apply it to solve problems involving systems of simultaneous linear congruences: 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: 1,2,

explain and evaluate the construction and properties of a variety of ciphers, including symmetric and asymmetric (public key) systems: 1,2,

appraise the strengths and weaknesses of different cryptography systems, and make judgements on appropriate side-channel attacks to attack poorly implemented systems: 1,2,

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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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.

Continual Assessment

Continuous assessment will consist of written coursework, problem sheets, class tests, or any combination thereof.