MA436      Half Unit
Mathematics of Cryptocurrencies and the Blockchain

This information is for the 2025/26 session.

Course Convenor

Availability

Requisites

Pre-requisites:

Students must have completed MA407 before taking this course.

Additional requisites:

Students should be comfortable with proofs and proof techniques used in pure mathematics. Students must have completed Algorithms and Computation (MA407) or have taken an equivalent course to provide a basic knowledge in analysis of algorithms

The course will cover the core aspects of distributed computing that are relevant to blockchain technology, from the pioneering work of computer scientists such as Lamport, Dolev, Reischuk and others in the 1980s, right up to some of the state-of-the-art protocols that are in use today. In particular, the course will cover topics amongst: 

•  The definition of various `consensus' problems, such as Byzantine Agreement, Byzantine Broadcast, Atomic Broadcast, and State Machine Replication.  
•  Reductions between consensus problems. 
•  Timing models: the synchronous, partially synchronous, and asynchronous settings. 
•  Fault models: crash failures, omission faults and Byzantine faults. 
•  Protocols for consensus in synchrony: The Dolev-Strong and Phase-King protocols. 
•  Impossibility results for consensus protocols in the synchronous setting: the lower bounds of Dolev and Reishchuk and  Fischer, Lynch and Merritt. 
•  Protocols for State Machine Replication in partial synchrony: Tendermint, PBFT and Hotstuff. 
•  Complexity measures: communication complexity and latency. 
•  The FLP impossibility theorem for the asynchronous setting: deterministic consensus is not possible.
•  Reliable Broadcast in asynchrony. 
•  Payment systems. 
•  Randomised protocols for State Machine Replication in asynchrony. 
• "Layer 2" solutions: rollups and the Lightning Network. 

20 hours of lectures and 10 hours of seminars in the Winter Term.
2 hours of lectures in the Spring Term.

Students will be set weekly home-works, which they are invited to hand in on Moodle. These will home-works will be marked, with feedback given to the student each week.  

Some relevant references are: 

• Consensus in 50 pages: https://lewispye.wordpress.com/wp-content/uploads/2023/01/consensus-in-50-pages7-1.pdf
• Foundations of distributed consensus and blockchain, Elaine Shi: http://elaineshi.com/docs/blockchain-book.pdf
• Distributed algorithms, Nancy Lynch. 
• Foundations of blockchains, Tim Roughgarden: https://timroughgarden.github.io/fob21/ 
• Introduction to Reliable and Secure Distributed Programming, Cachin, Guerraoui, and Rodrigues. 
• Distributed Thoughts: https://decentralizedthoughts.github.io/

Exam (65%), duration: 120 Minutes in the Spring exam period

Continuous assessment (10%) weekly

Video (25%) in Spring Term Week 1

This component of assessment includes an element of group work.

For the group-work summative assessment, students will be given a list of papers that develop and extend material presented in lectures. Students will be randomly assigned to groups of three or four, and each group will then have to choose a paper from the list. Students will then meet (outside class) to read and discuss the paper and related papers. They will be asked to prepare (between them) a 20 minute video presentation summarising their findings. Students will also submit an individual contribution and reflection form, to help identify how they feel they collaborated as a group, and whether any issues (such as individual students contributing significantly less than others) arose during the process.