MA6054 - Cryptography and Number Theory (2021/22)
|Module specification||Module approved to run in 2021/22|
|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||144|
|Running in 2021/22||
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
• 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.
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.
There will be a series of mini tests every 2-3 weeks lasting c20minutes, allowing students an early opportunity to demonstrate their understanding of basic topics and techniques and to receive feedback on them [LO1, LO2, LO3]. The marks for these are aggregated to form a mark for component sequence 001. Students who fail this component of assessment on aggregate will be reassessed by a consolidated test covering all topics.
Additionally, a 1.5-hour test will take place at the end of the module where students will be tested across the whole of the material studied with longer questions covering both theory and practice in number theory and cryptography [LO1, LO2, LO3].
Students are permitted to bring two sides of handwritten notes to each assignment to aid the consolidation of learning.
Core Texts: On-line lecture notes
W. Stallings (2016); Cryptography and Network Security: Principles and Practice (7th Edition); Prentice Hall.
J. Holt, J. Jones (2002); Discovering Number Theory; W.H. Freeman and Company.
Singh, S (2002) The Code Book: The Secret History of Codes and Code-breaking; Fourth Estate
Katz, J (2014) Introduction to Modern Cryptography; Chapman and Hall
Aumasson, J-P (2018) Serious Cryptography; No Starch Press