# module specification

## MA6054 - Cryptography and Number Theory (2024/25)

Module specification Module approved to run in 2024/25
Module title Cryptography and Number Theory
Module level Honours (06)
Credit rating for module 15
School School of Computing and Digital Media
Total study hours 150

 105 hours Guided independent study 45 hours Scheduled learning & teaching activities
Assessment components
Type Weighting Qualifying mark Description
In-Course Test 40%   Series of mini tests
Unseen Examination 60%   Final Exam
Running in 2024/25

(Please note that module timeslots are subject to change)
Period Campus Day Time Module Leader
Spring semester North Friday Afternoon

## Module summary

The module is an introduction to modern ideas in cryptography. It proves the background to the essential techniques and algorithms of cryptography in widespread use today, as well as the essentials of number theory underlying them.

The module looks at symmetric ciphersystems and their use in classical cryptography as well as public key systems developed to support internet commerce and deliver data security for private individuals.

The module will enable students to understand the mathematics underpinning key algorithms, how they operate using small values and how computer packages such as MAPLE allow us to apply them at a more realistic scale.

## Prior learning requirements

Students are expected to be familiar with concepts in number theory (e.g. greatest common divisors and Euclid’s Algorithm) such as would be encountered in an introductory course in discrete mathematics.  The module is open to Study Abroad students.

## Syllabus

Number theory:
• Solution of linear congruence equations; Fermat’s Little Theorem and generalisations; Quadratic congruence equations; Primality testing and prime factorisation techniques; Primitive roots; Discrete logarithms.
Theory of fields:
• Euclidean algorithm and the calculation of inverses.
Cryptography and cryptanalysis:
• History and Terminology, Symmetric Encryption. Classical cryptography and techniques for cryptanalysis based on frequency analysis, Block Cipher Principles.
• Asymmetric and Public Key Encryption: RSA public key cryptosystem; El Gamel and Diffie Hellman protocols based on discrete logarithms.
• Modern developments in number theory and cryptography: fast factorisation methods, cryptography based on elliptic curves.

Learning Outcomes LO1 - LO3

## Balance of independent study and scheduled teaching activity

Students’ learning is directed via face-to-face learning activities centred on lectures and seminars. There is full provision of documents related to the module in electronic format that can be accessed by students all the time. The documents include lecture notes, slides, guidance to further reading and relevant mathematical packages, and exercises and tests. In particular, the MAPLE computer algebra system is used to explore number theoretic properties for the sort of large natural numbers used to generate keys for modern cryptosystems. This will develop transferable skills applicable across the mathematics subject area.

Students will also reflect on the issues raised by the module for personal data privacy and consider where they feel the boundaries between personal liberty and a secure society should be set.

## Learning outcomes

On successful completion of this module, students should be able to:
LO1 Appreciate proofs of theorems in Number Theory and their application to cryptography
LO2 Use appropriate encryption and decryption algorithms for a given ciphersystem; perform cryptanalysis of received cipher text;
LO3 Construct and perform arithmetic, including finding inverses, in finite and infinite fields.

## Bibliography

Core Texts:    On-line lecture notes