Hunting Giant Primes: The Mathematics Behind Modern Cryptography

Overview

Modern digital security relies on deep ideas from number theory. Prime numbers, modular arithmetic, and efficient mathematical algorithms underpin technologies used every day, from secure web browsing and online banking to cryptocurrencies and emerging post-quantum cryptographic systems.

This project introduces students to research at the intersection of computational number theory and cryptography. Depending on the interests and background of the student, the project may involve investigating large prime numbers, developing efficient algorithms for modular arithmetic, exploring residue number systems (RNS), benchmarking cryptographic computations, or conducting computational experiments related to contemporary cryptographic problems.

The topic is suitable for students interested in mathematics, computer science, cybersecurity, algorithms, or high-performance computing. Specific research questions will be tailored to the student's interests and experience.

Research engagement

The student will engage in:

• Literature review of computational number theory and cryptography research.
• Reading and analysing research papers and technical reports.
• Designing computational experiments.
• Implementing and evaluating mathematical algorithms.
• Participating in regular research meetings and discussions.

Research activities

Potential activities may include:

• Investigating algorithms for discovering and verifying very large prime numbers.
• Exploring Mersenne primes and distributed prime-search projects.
• Studying residue number systems and their applications to fast modular arithmetic.
• Implementing and benchmarking algorithms for primality testing or integer factorisation.
• Evaluating arithmetic techniques used in modern cryptographic systems.
• Developing software prototypes in Python, C/C++, or related languages

The student will work closely with the supervisory team and may also interact with postgraduate students and researchers working in cryptography and computational mathematics.

Research skills

Students will gain experience in:

• Research methodology and literature review.
• Mathematical problem solving.
• Algorithm design and analysis.
• Scientific programming and software development.
• Experimental evaluation and benchmarking.
• Technical writing and presentation of research outcomes.
• Communicating complex mathematical ideas to technical audiences.

Outcomes

The expected outcome is a research report and presentation describing the student's investigation and findings. Depending on the chosen project direction, outcomes may include:

• New computational results or benchmarks.
• Software implementations of mathematical algorithms.
• Comparative studies of arithmetic techniques.
• Exploratory research that may form the basis of an Honours or HDR project.

Skills and experience

Essential:

• Strong interest in mathematics, algorithms, computer science, cybersecurity, or cryptography.

Desirable:

• Completion of undergraduate units in mathematics, algorithms, data structures, computer security, or related areas.
• Experience with programming in Python, C, C++, Java, or similar languages.
• Curiosity about mathematical research and problem solving.

Start date

2 November, 2026

End date

19 February, 2027

Location

Gardens Point campus / online

Additional information

This project forms part of ongoing research activities in computational number theory, cryptography, and post-quantum cryptography within the School of Computer Science. Students will have access to research computing resources, relevant software tools, and regular supervision meetings throughout the project.

Keywords

• Cryptography
• Number Theory
• Post-Quantum Cryptography
• Algorithms

Contact

craig.costello@qut.edu.au

31382870

craig.costello@qut.edu.au