Unit rationale, description and aim
With the development of modern electronic cryptographic methods, Number Theory has evolved from a purely abstract subject into one of crucial practical importance. Essentially all secure electronic communication now relies upon elementary number theory results. The unit also considers the ethics surrounding the use of strong cryptography and the tension between the needs of vulnerable and sometimes oppressed peoples and the need to restrict criminal behaviour.
In this unit, the study of cryptographic algorithms such as RSA is supported by consideration of topics from number theory including congruences, primality and factorisation. The ethics surrounding the use of cryptographic systems will be considered, as well as the counter-intuitive idea that a study of cryptanalysis strengthens cryptography.
The aim of this unit is to provide students with an understanding of the central role number theory has in modern cryptography and knowledge of a wide variety of number theoretic ideas and techniques. Students will also have an appreciation of the risks and benefits of some of the requirements needed to support secure cryptography. The applications of cryptography will support a consideration of how the careful use of these tools can promote social good, but also how careless use can result in significant harm.
Learning outcomes
To successfully complete this unit you will be able to demonstrate you have achieved the learning outcomes (LO) detailed in the below table.
Each outcome is informed by a number of graduate capabilities (GC) to ensure your work in this, and every unit, is part of a larger goal of graduating from 糖心原创 with the attributes of insight, empathy, imagination and impact.
Explore the graduate capabilities.
Describe key concepts in number theory, cryptograp...
Learning Outcome 01
Explain basic problems in cryptography and secure ...
Learning Outcome 02
Apply appropriate mathematical strategies—such as ...
Learning Outcome 03
Content
Topics will include:
- Basic number theory, divisibility results, primes.
- Greatest common divisor, Euclid’s algorithm
- Introduction to Congruences – definitions and arithmetic
- Euler’s -function, powers, logs and squares modulo m, Fermat’s little Theorem
- Diophantine equations.
- The Chinese Remainder Theorem
- Introduction to cryptography
- RSA
- Digital signatures and key exchange problems, including the Diffie-Hellman algorithm for publicly exchanging a private key
- Introduction to elliptic curve cryptography
Assessment strategy and rationale
To successfully complete an undergraduate Mathematics sequence, students need an understanding of a variety of basic Mathematical topics and an ability to apply that understanding to a variety of problems. To succeed at problem solving in Mathematics, students must have these skills at their fingertips and be able to recall them and choose an appropriate approach under some pressure. The assessment strategy chosen tests and supports student learning by providing opportunities to develop and test their problem-solving skills through the unit.
Any detailed discussion of cryptography is obliged to address the ethical and social aspects of using or not using those techniques. The assessment schedule includes a group project that will require students to develop arguments for or against the use of cryptography in a recent historical event. These reports will be submitted via Turnitin.
At specified times, through the semester, students will sit two supervised test components to ensure students have fully integrated the learning and can independently bring a variety of strategies to bear to solve a variety of problems.?
Typing of Mathematical notation either requires a significant investment of time, or knowledge of advanced Mathematical typesetting software. The lecturer may request that assignments and tests are handwritten.
Overview of assessments
Assessment Task 1: Group Project Addressing ethi...
Assessment Task 1: Group Project
Addressing ethical aspects of cryptography
20%
Assessment Task 2: Content and Application Test 1...
Assessment Task 2: Content and Application Test 1:
This test will assess your conceptual understanding and computational skills in relation to topics covered in the unit.
40%
Assessment Task 3: Content and Application Test 2...
Assessment Task 3: Content and Application Test 2
This test will assess your conceptual understanding and computational skills in relation to topics covered in the unit.
40%
Learning and teaching strategy and rationale
As is common in Mathematics a variety of Active Learning approaches are used to promote the best acquisition of skills and understanding. Lectures will typically provide explanations of the material to be covered, along with examples of the applications of that material, as well as formal and informal tasks for students that will reinforce that learning.
Splitting the taught content of the lectures into short topics will ensure increased student engagement with the material by allowing students to pause, reflect, and consolidate their understanding before progressing to new material.?Face-to-face/online tutorials then provide opportunities for students to develop and practice the skills introduced in lectures enabling them to build understanding, competence and confidence through?guided problem-solving, discussion, and peer interaction.?In all cases this should be supported using available online technology.