Mathematics & Cryptography

Complexity. Innovation. Seemingly unsolvable challenges.
We apply the latest technologies in a variety of ways.

Mathematics & Cryptography

Our mathematicians work in a wide range of fields, using a variety of techniques across many disciplines to solve complex real-world problems. Your career could involve working in many different areas and mathematicians here continually learn from one another, applying their skills collaboratively in multi-disciplinary teams. The strongest requirements for any mathematician at GCHQ are the ability to generate creative ideas and a willingness to learn new concepts.

Working at GCHQ, you will enjoy the opportunity to influence decisions made by government, the armed forces and law enforcement agencies. Mathematical expertise sits at the heart of our technical capability and you’ll be applying your analytical skills in areas such as cryptography, internet security, mobile communications, and understanding large data sets.

Underpinning much of our work is our high performance computing environment. We are involved in solving some of GCHQ’s hardest computational problems by writing highly-optimised implementations of algorithms on a variety of computers with specialised architectures. For many of our roles, an interest in computer security to go alongside mathematical ability is important.

We have produced some samples of the entrance test that will form part of the assessment for a role as a mathematician at GCHQ for those who are invited to an assessment centre, so that you will be familiar with the format when you come.

Sample GCHQ Mathematics Aptitude Test

mathematics & cryptography roles

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I understand how the work of GCHQ makes a difference to the UK


Mathematics & Applied Cryptography

We need to understand how cryptographic technologies are used in everyday life, and analyse weaknesses at a product, protocol, system or hardware level. We do this by combining ideas from across the whole mathematical spectrum with a wide range of computer security skills. We are particularly keen to increase the pool of applicants from joint maths and computer science backgrounds to support this work.

To study the algorithms that underpin these technologies, we need to apply a range of cryptanalytic techniques to understand potential weaknesses. This involves bringing in techniques from probability and statistics, group theory, combinatorics and complexity theory, alongside an enthusiasm for problem solving, and then applying our ideas as efficiently as possible, using some of Europe’s most powerful computers. Our strengths lie in the way in which lots of people can each contribute different parts of the expertise needed to solve our hardest problems, and we offer extensive training for all parts of our work.

Cryptographic Security

We devise the UK’s cryptography and provide consultation on its application in government systems. This may involve designing and implementing bespoke algorithms or protocols for use in constrained or difficult environments, or securing systems that will require protection for many years in the future. We need to understand how cryptographic technologies are used in everyday life, working closely with vendors and their products.

Public key cryptography is the cornerstone on which the modern digital world is built, while the next generation of cryptography will have at its core quantum-resistant algorithms, designed to resist attack from a hypothetical quantum computer. We lead UK research in these areas. To do this, we must understand a range of pure mathematical problems (such as integer factorisation and discrete logarithm computation), while, to evaluate new quantum protocols, we need to bring an understanding of quantum systems engineering principles to our cryptographic thinking.

Data Science and Information Processing

We design and implement applications of state-of-the-art machine learning and graph analysis techniques to discover patterns in large data stores or high-volume data sets and data streams, and build statistical models to enhance data processing and pattern recognition for the intelligence and cyber security missions.

We receive data in diverse formats, often unrecognised or corrupted, and need to be able to understand this data so we can extract the underlying information. A combination of mathematical modelling and a willingness to generate and test new ideas, building on limited knowledge, is required for this type of work.

Through partnerships across government and with industry and academia, including the recently formed Alan Turing Institute, GCHQ is building a vibrant data science community.

Maths and Cryptography Summer Programme

GCHQ is one of the few non-academic centres where you can use, stretch and develop your maths research skills in a real world setting. That’s why we look for committed mathematicians who are expecting a 1st class honours degree in maths, or a joint honours degree with maths as the main component who want to continue advanced mathematical research into their career. Join us on our nine week summer programme and you could be working alongside our experts on problems that have genuine practical importance, that might include:

  • Investigating cryptographic algorithms;
  • Assessing the security of cryptographic applications and protocols;
  • Developing novel mathematical techniques and analytical tools;
  • Identifying and modelling challenging data sets;
  • Implementing and optimising complex mathematical algorithms on high performance computers;
  • Understanding digital signals received over noisy channels

To apply you’ll need to have completed a minimum of three years of your degree and be on track to achieve a 1st Class Honours qualification in mathematics (Bachelors, Masters or equivalent) or Joint Honours subject with mathematics as the main component.

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