MA300-15 Mathematics of Cryptography
Introductory description
An introduction to mathematics of cryptography.
This will cover various cryptographic schemes, including public key cryptography, private key cryptography, hash functions, key exchange, digital signatures. Elliptic curve cryptography and its usage in many popular applications, such as bitcoin, will be described in detail. This module will build on concepts students have studied in second year algebra and number theory modules.
Module aims
Students will understand the mathematics of commonly used cryptographic methods. This module shows how pure mathematics is important in modern technology, and shows the importance of applied algebra and number theory.
Transferable skills include practice with python, group work, and presentation skills.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
- Private key cryptography
- Block ciphers
- Stream ciphers
- Hash functions
- RSA Public key cryptography
- Elliptic curve cryptography
- Key exchange
- Zero knowledge proof
Learning outcomes
By the end of the module, students should be able to:
- A theoretical and practical knowledge and understanding of the main methods of public key cryptography
- A theoretical understanding of some common methods of private key crypography
- An theoretical and computational understanding of elliptic curve cryptography
- Understanding of digital signature algorithms and their use
- Understanding of the mathematics of cryptocurrencies, block chains, and hash functions
- Competency with use of python applied for cryptographic purposes
- Ability to write a formal mathematical project
- Skills of working in a group
- Practice of presentation skills
Indicative reading list
- Knopse: A Course in Cryptography, AMS 2019
- Washington: elliptic curves: number theory and cryptography, Taylor and Francis, 2008
- Blake, Seroussi, Smart: Elliptic curves in cryptography, LMS 1999
- Gerhard: Modern Computer Algebra, Maplesoft, 2013
Research element
Potentially investigating real world applications of the cryptographic systems they will study in abstract in the classes.
Subject specific skills
Mathematical analysis of cryptographic systems
Coding in python and sage
Transferable skills
Group work skills
academic writing skills
presentation/speaking skills
Study time
| Type | Required |
|---|---|
| Lectures | 30 sessions of 1 hour (20%) |
| Private study | 45 hours (30%) |
| Assessment | 75 hours (50%) |
| Total | 150 hours |
Private study description
Around 3 hours per lecture private study. May be included in quiz and assignment work.
Regular group work for the project is expected.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group A
| Weighting | Study time | Eligible for self-certification | |
|---|---|---|---|
| Moodle Quiz | 10% | 5 hours | No |
|
short formative quizzes on moodle. 10 questions per quiz. Best 4 of 5 count. Due every other week. Multiple attempts allowed, but must be completed by fortnightly deadlines. |
|||
| Assignments | 40% | 20 hours | No |
|
5 written assignments, including sage/python components. Best 4/5 count |
|||
| Group Project | 40% | 40 hours | No |
|
Students will work in groups of about 5 students to produce a written project which will be around 25 pages long. This may include coding elements in sage/python. Length flexible, depending on coding content, diagrams, etc. |
|||
| Project presentation | 10% | 10 hours | No |
|
Group talk about the project |
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Assessment group R
| Weighting | Study time | Eligible for self-certification | |
|---|---|---|---|
| Project | 80% | No | |
|
Project selected from a list of topics in cryptography |
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| Project presentation | 20% | No | |
|
A talk about the project |
|||
Feedback on assessment
Written feedback
Pre-requisites
some familiarity with python programming or similar.
Courses
This module is Optional for:
- Year 1 of TMAA-G1P0 Postgraduate Taught Mathematics
-
TMAA-G1PC Postgraduate Taught Mathematics (Diploma plus MSc)
- Year 1 of G1PC Mathematics (Diploma plus MSc)
- Year 2 of G1PC Mathematics (Diploma plus MSc)
- Year 3 of UCSA-G4G1 Undergraduate Discrete Mathematics
-
UCSA-G4G3 Undergraduate Discrete Mathematics
- Year 3 of G4G1 Discrete Mathematics
- Year 3 of G4G3 Discrete Mathematics
- Year 4 of UCSA-G4G4 Undergraduate Discrete Mathematics (with Intercalated Year)
- Year 4 of UCSA-G4G2 Undergraduate Discrete Mathematics with Intercalated Year
This module is Core option list C for:
- Year 3 of UMAA-GV17 Undergraduate Mathematics and Philosophy
- Year 3 of UMAA-GV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Core option list F for:
- Year 4 of UMAA-GV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Option list A for:
-
UMAA-G105 Undergraduate Master of Mathematics (with Intercalated Year)
- Year 4 of G105 Mathematics (MMath) with Intercalated Year
- Year 5 of G105 Mathematics (MMath) with Intercalated Year
- Year 3 of UMAA-G100 Undergraduate Mathematics (BSc)
-
UMAA-G103 Undergraduate Mathematics (MMath)
- Year 3 of G100 Mathematics
- Year 3 of G103 Mathematics (MMath)
- Year 4 of G103 Mathematics (MMath)
- Year 4 of UMAA-G107 Undergraduate Mathematics (MMath) with Study Abroad
- Year 4 of USTA-G1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
- Year 4 of UMAA-G101 Undergraduate Mathematics with Intercalated Year
This module is Option list B for:
- Year 3 of USTA-GG14 Undergraduate Mathematics and Statistics (BSc)
This module is Option list C for:
- Year 3 of USTA-G1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
- Year 4 of USTA-G1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)