Diophantine equations and local-global principles: into the wild

Studying integer (whole number) solutions to polynomial equations is the oldest field in mathematics, containing problems that have remained unsolved for millennia. Furthermore, its applications to cryptography and security make it one of the most high-impact areas of pure mathematics. Cryptosystems rely on the computational hardness of mathematical problems to protect our data.

The realm of integer solutions to polynomial equations is a natural source of hard problems to underpin modern cryptosystems. For example, it can claim credit for the development of elliptic curve cryptography (ECC). This is a public key cryptographic system that has been widely used for over a decade by big players such as the USA National Security Agency and Microsoft.

For instance, ECC is used to protect our credit card details when we make purchases over the internet. Cybersecurity is of crucial national importance in protecting data at the individual, corporate and state level and its role in daily life is increasing as more of our economic, administrative and social interactions take place online.

The deep knowledge of elliptic curves needed for the development of ECC was gained by pursuing blue sky research in mathematics, of which the most famous recent example is Andrew Wiles' 1995 proof of Fermat's Last Theorem. This concerns one particular family of polynomial equations, namely x^n+y^n = z^n. When n=2, this is Pythagoras' equation relating the side lengths of a right-angled triangle.

There are infinitely many integer solutions to this equation (e.g. x = 3, y = 4, z = 5) and we even have a formula for them. However, when n is greater than 2, the behaviour is very different. Fermat conjectured in 1637 that there were no positive integer solutions to the equation x^n+y^n = z^n for n greater than 2.

The proof of this fact took more than 350-years and required the development of very advanced mathematical techniques.

In September 2019, Google announced that they had achieved 'quantum supremacy', having developed a quantum computer that performed a task in 200 seconds where a top-range supercomputer would take 10,000-years. This stunning achievement presents a looming crisis for the cryptosystems protecting our data. A quantum computer that can solve the mathematical problems underlying current cryptosystems in seconds rather than millennia would be able to decrypt encrypted data and compromise its security.

Security agencies and technology companies are urgently seeking new, and harder, mathematical problems to underlie post-quantum cryptographic systems and they are keen to collaborate with mathematicians to achieve this.

My proposal is to study integer solutions to a much larger and more complex class of polynomial equations than elliptic curves, using a wide variety of techniques from number theory, algebra, geometry and analysis. The modern approach looks first for so-called local solutions and then investigates whether a collection of them can be patched together to form a global (meaning integer) solution.

However, this local-global method is not always successful. I will study the reasons for its failure and conduct a statistical analysis of the frequency of these failures within families of equations. I will break new ground by tackling cases that have so far been untouched due to their complexity: the 'wild' in my title is an adjective used by mathematicians to describe mathematical objects whose behaviour is particularly difficult to handle.

Recent breakthroughs in number theory mean the time is ripe to grapple with these wild problems. I will collaborate with leading cryptographers to explore possibilities arising from my research for new hard mathematical problems that can be used to underpin cryptosystems that can resist attacks by quantum computers.