MAS2704 : Numbers and Ciphers

Semester 2 Credit Value: 10
ECTS Credits: 5.0


•       To develop concepts and techniques of mathematical proof, illustrated by results in number systems and cryptography.
•       To stimulate logical thinking and to develop students' skills at constructing mathematical arguments.
•       To provide an understanding of the application of ideas from Pure Mathematics to the real world.

Module summary

The secure transmission of information has been important for many centuries and mathematical systems have been one of the most important means of providing such security. Security is of even greater significance in the age of electronic communication. Cryptography is an important tool in achieving these aims. Many ciphers are based on mathematical constructions, many starting with properties of numbers. We shall begin by understanding the properties of numbers essential to constructing ciphers. We shall look at some of the fundamental symmetric ciphers, which are fast and efficient to implement, but which require both parties to possess a key. We shall study code-breaking techniques for such ciphers and see how ways of combining ciphers reduces vulnerability to such techniques. We shall then consider public-private key cryptography (asymmetric ciphers), in which everybody has access to a public key with which to encipher a message, but only the holder of the private key can decipher the resulting text. In particular we shall consider one of the most famous public-private key cryptosystems (RSA) that is based on the difficulty of factorising large numbers.

Outline Of Syllabus

-       The Division Algorithm, the Euclidean algorithm for highest common factors of integers, prime factorisation.
-       Zn as {0,1,..., n-1} with operations of addition and multiplication, inverses in Zn (when they exist), Fermat's Little Theorem.
-       Euler's totient function, Euler's Theorem.
-       Vigenere cipher, symmetric ciphers, affine ciphers, affine matrix ciphers, asymmetric ciphers, public key cryptography, the RSA cipher.

Teaching Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Guided Independent StudyAssessment preparation and completion16:006:00Revision for class test
Guided Independent StudyAssessment preparation and completion113:0013:00Revision for unseen exam
Guided Independent StudyAssessment preparation and completion12:002:00Unseen exam
Scheduled Learning And Teaching ActivitiesLecture11:001:00Class test
Scheduled Learning And Teaching ActivitiesLecture31:003:00Problem classes
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision lectures
Scheduled Learning And Teaching ActivitiesLecture261:0026:00Formal lectures
Scheduled Learning And Teaching ActivitiesDrop-in/surgery41:004:00Tutorials in the lecture room
Scheduled Learning And Teaching ActivitiesDrop-in/surgery120:102:00Office hours
Guided Independent StudyIndependent study120:0020:00Studying, practising and gaining understanding of course material
Guided Independent StudyIndependent study33:009:00Review of coursework assignments and class test
Guided Independent StudyIndependent study26:0012:00Preparation for coursework assignments
Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems. Tutorials are used to identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. In addition, office hours (two per week) will provide an opportunity for more direct contact between individual students and the lecturer: a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.

Assessment Methods

The format of resits will be determined by the Board of Examiners

Description Length Semester When Set Percentage Comment
Written Examination1202A85N/A
Written Examination402M10Class test
Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises2M5Coursework assignments
Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The coursework assignments will consist of two written assignment of approximately equal weight. The coursework assignments and the (in class, therefore 40 minute) coursework test allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

Reading Lists