MA3NTC: Number Theory and Cryptography

Module code: MA3NTC

Module provider: Mathematics and Statistics; School of Mathematical, Physical and Computational Sciences

Credits: 20

Level: Level 3 (Honours)

When you'll be taught: Semester 1

Module convenor: Dr Chris Daw, email: chris.daw@reading.ac.uk

Pre-requisite module(s): BEFORE TAKING THIS MODULE YOU MUST ( TAKE MA1FM AND TAKE MA1LA AND TAKE MA1RA1 ) OR ( TAKE MA0FMNU AND TAKE MA1LANU AND TAKE MA1RA1NU ) (Compulsory)

Co-requisite module(s):

Pre-requisite or Co-requisite module(s):

Module(s) excluded: IN TAKING THIS MODULE YOU CANNOT TAKE MA4NTC (Compulsory)

Placement information: NA

Academic year: 2024/5

Available to visiting students: Yes

Talis reading list: No

Last updated: 21 May 2024

Overview

Module aims and purpose

This module is an introduction to number theory and cryptography.

It aims to bring together a number of fascinating topics from both pure and applicable mathematics.

It will put on a rigorous footing many of the facts about integers that all students understand at an intuitive level. It will cover questions in number theory, which are easily appreciated, but whose solutions are far less easy. It will go on to cover two of the most vital tools of modern mathematics, namely, the RSA cryptosystem and error correcting codes.

Module learning outcomes

By the end of the module, it is expected that students will be able to:

1. Use the concepts of factorisation, prime numbers and congruence to solve problems in elementary number theory;
2. Manipulate arithmetic functions and Dirichlet series and make elementary estimates thereof;
3. Implement the RSA public key cryptosystem and use it to encode, decode and authenticate documents;
4. Construct error correcting codes capable of correcting a specific number of errors, and calculate probabilities of correct transmission.

Module content

The first half of the module studies in detail the basic theory of numbers. In particular, the following topics will be discussed:

• Divisibility, prime numbers, congruences (modular arithmetic), arithmetic functions, Dirichlet generating functions.

The second part of the module focuses on cryptography, and the related field of error correction. In particular, the following topics will be discussed:

• The RSA cryptosystem; error correcting codes.

Structure

Teaching and learning methods

Lectures supported by tutorials. Learning materials (lecture notes/reading lists, tutorial problem sheets, assessments) made available via Blackboard.

Study hours

At least 50 hours of scheduled teaching and learning activities will be delivered in person, with the remaining hours for scheduled and self-scheduled teaching and learning activities delivered either in person or online. You will receive further details about how these hours will be delivered before the start of the module.

Please note the independent study hours above are notional numbers of hours; each student will approach studying in different ways. We would advise you to reflect on your learning and the number of hours you are allocating to these tasks.

Semester 1 The hours in this column may include hours during the Christmas holiday period.

Semester 2 The hours in this column may include hours during the Easter holiday period.

Summer The hours in this column will take place during the summer holidays and may be at the start and/or end of the module.

Assessment

Requirements for a pass

Students need to achieve an overall module mark of 40% to pass this module.

Summative assessment

Penalties for late submission of summative assessment

The Support Centres will apply the following penalties for work submitted late:

Assessments with numerical marks

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of three working days;
• the mark awarded due to the imposition of the penalty shall not fall below the threshold pass mark, namely 40% in the case of modules at Levels 4-6 (i.e. undergraduate modules for Parts 1-3) and 50% in the case of Level 7 modules offered as part of an Integrated Masters or taught postgraduate degree programme;
• where the piece of work is awarded a mark below the threshold pass mark prior to any penalty being imposed, and is submitted up to three working days after the original deadline (or any formally agreed extension to the deadline), no penalty shall be imposed;
• where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

Assessments marked Pass/Fail

• where the piece of work is submitted within three working days of the deadline (or any formally agreed extension of the deadline): no penalty will be applied;
• where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension of the deadline): a grade of Fail will be awarded.

The University policy statement on penalties for late submission can be found at: /cqsd/-/media/project/functions/cqsd/documents/qap/penaltiesforlatesubmission.pdf

You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Formative assessment

Formative assessment is any task or activity which creates feedback (or feedforward) for you about your learning, but which does not contribute towards your overall module mark.

Problem sheets 

Reassessment

Additional costs