AMI Springfestival 2013: Can Mathematics keep a secret?

### Interview with Gerhard Frey by Ingrid Vos and Claire Hallewas

**Mathematics and numbers**

There are two answers to the question why Gerhard Frey chose to study mathematics. One is he liked the subject at school. But then again, he also liked history. The other answer is that with mathematics you can discover things just by thinking and you donât have to learn things by heart. So with mathematics he could be the laziest man of the world.

We interview Gerhard Frey before the 3TU.AMI Springfestival held at the TU Delft, where he will be the key note speaker with a presentation with the intriguing question: Can Mathematics keep a secret? Gerhardâs specialty is number theory. âDuring the start of my study at the University of Tubingen I had a beginner course by the quite famous Peter Roquette. He sparked my interest in number theory. Number theory starts very simple. You use your fingers to count and you learn about numbers, calculation, prime numbers and polynomials in school. In number theory you want to find relations between numbers with properties. For example: Euclid stated that there are an infinite number of prime numbers and he gave a beautiful and simple proof. The other extreme is Fermatâs last theorem that is also very simple to state but it was very hard to prove it. In number theory you need a lot of theory to prove simple things. It is fascinating. Everyone is interested in numbers. People playing a Sudoku are playing with numbers. Number theory proved to have many big challenges for me, but if I maybe had another instructor at my beginnerâs courses I might have ended up in another field of mathematics.â

**Fermatâs last theorem**

Peter Roquette was Gerhardâs supervisor during his PhD work in Heidelberg. âAfter my Diploma (master) I had the choice to go into the industry, work at a school or continue at the university. Nowadays it is quite difficult for young people to get a position at the university, but at that time it easierâ, tells Gerhard. âFrom the 1960âs onwards until the end of the last century it was a golden age for number theory. We proved so many results . It was nice to be part of a revolution.â When we ask him why so many mathematicians were fascinated by Fermatâs last theorem, Gerhard says that he has to disappoint us. âIt is a very simple question. Everyone can understand it. Still once every week I receive a letter, by someone who claims to have found a proof that only takes two pages, but it is always nonsense. But for a long time you didnât have the power to attack it. The famous mathematician Gauss tried to solve it. He asked himself: Why should it be so important? I can think of similar questions once a week.â It took 358 years to publish the proof. Gerhard explains: âDuring the last 200 years people started to develop theories that would hopefully help to prove the theorem. The key word is algebraic number theory. Fermatâs last theorem was a great motivation to develop this theory. But only in the sixties and seventies of the last century theories were raised to give a method on how to solve it. You have to find the right method and structure for the problem to apply the theory . My fascination with Fermatâs last law was to find a structure that we could apply. Algebraic manipulations of formulas gave no way to solve the equation. We took a totally new approach and attached Galois representations to Fermat's Last Theorem. This turned out to be an important step in the final proof by Andrew Wiles.â

**The importance of conferences**

Gerhardâs career changed from the theoretical side to the more applied side, when he started to work for the Institute for Experimental Mathematics in 1990 in Essen. There were four chairs. Gerhard had the chair for number theory. The other chairs were on group theory, computer science and digital communication. âIt was a big luck to work with professor Han Vinck. He is an engineer with a great background in mathematics. We became interested in coding theory. In one conference we held in Essen somebody spoke about cryptography and a participant asked questions about public key cryptography. We saw immediately that we could apply the whole machinery we used for solving Fermatâs last theorem to this question. Again we found the structure that was behind and we understood that our expertise could help to provide useful theory . So the next 15 years I led a double life, working both in theoretical and in applied mathematics. We had the good luck to attract many very clever young people who wished to work in this new area of cryptography.â

Gerhard stresses the importance of conferences for the work of a mathematician. âIt is important to travel around the world and to exchange problems and questions. As a mathematician you donât have a lab or machines, so you are free to go to conferences that are important to improve your ideas.â

In 2005 Gerhard's work had consequences for the electronic passport in Germany. âOur institute was involved in developing the mathematical background. We helped to find criteria for the right elliptic curves, which are the backbone of the theory, which were put in use in the passport. This passport is a very complex thing, with a lot of protocol. When a passport and a machine communicate they ask each other in milliseconds to solve a mathematical problem. This can be done only if both passport and machine have certain knowledge and thus it helps to establish whether it is the right passport and the right machine. â

**Answers in your sleep**

Gerhard emphasizes the role of intuition in arriving to solutions for difficult problems. âIf you speak to a mathematician, you will often hear: I have the feeling that this canât be true. Experience is of course very important to get this intuition. I often got an answer to my work problems during sleep, but, alas, only to find out in most cases in the morning that the answer was completely wrong! I had a thirty minute bike ride to and from work and that was a good way to think ideas through or get new ones. Input that comes from discussion with others is also very important. So during the day you work on calculation and read papers, but the best ideas come when youâre not working!â

When asked what the biggest challenges in number theory are, Gerhard quotes the famous mathematician David Hilbert âWe must know and we will know.â He thinks that there are many unknown areas and old and new challenges to be solved. âFor example complexity theory. It tells you how difficult an algorithm has to be to solve a problem, and whether a âmachineâ can find a solution or not to a particular problem. Itâs sometimes a bit like the book Hitchhikerâs guide to the Galaxy by Isaac Asimov, that you have to build a gigantic machine to get an answer to a problem and the output is trivial like 42. One of the big theoretical classical challenges is of course the Riemann hypothesis. That is also one of the 10 Clay Mathematics Institute Millennium Prize Problems, who let you win a million dollar if you solve one of the problems. But in number theory there are also many seemingly small problems, who need a lot of structure to solve. Many of them could be done if we would understand the Galois group of rational numbers. â

**Young talent**

In Gerhardâs opinion there are many talented people in high schools, and universities offer good personal contact with the students to develop these talents. âThe hard thing is the PhD thesis, which will determine whether you are more suited for a career in industry or in research.â Gerhard sees a big pool of talent both male and female, but thinks we should do more to give them more security. âIt is a rat race. People have to communicate about their ideas in papers and at conferences, but at the same time there is a lot of competition. They often have a contract for 2 years and already have to apply for a new position after one year.â He thinks itâs a shame that till now there are very few women with a chair in mathematics, whereas in Germany the student numbers are 50 percent male and 50 percent female. This postdoc period is difficult. The woman often follows the man and it is also the time to start a family. âWe should give them more support.â

**No stress**

Gerhard now enjoys the fact that he no longer has the stress of having a chair and the responsibility for the scientific career of young people. âI still write papers, go to conferences and have discussions with friends, but it is more relaxed now.â He looks back on a great career. âMy biggest success was the attack on the Fermat theorem, that Wiles found proof for. Later we developed a method for elliptic curves that are still used in public key cryptography. A lot of the guidelines of national security agencies go back to the work we did in this area. I started out in the highest theoretical field, with the lowest possibilities for applications. Through contacts and discussions with electrical engineers, who were designing chips, we came up with new mathematical methods that have great applications in cryptography.â

Quantum computing is a new area that will propose big challenges. Gerhard says: âWe know from physics that quantum computing will work in principle and that it will break public key systems used today in milliseconds if a big enough computer will be built. Young colleagues will have to find other mathematical problems that will be resistant to quantum computing. During the next 20 years quantum computers may be there, but that is a story that we have been telling already for a long time. The difficulty is how to build and engineer such a computer. It is like hydrogen fusion. We know from the sun that it can be a source of energy, and from the 1940âs onward people have been trying to build such a system. But up till now, not a microVolt has come out. Maybe quantum computing will follow a similar path, because it is difficult to build. But as well it may be otherwise...â

**Cautious**

Coming back to the question âCan Mathematics keep a secret?â, Gerhard proposes the situation that the two interviewers want to exchange a secret, with him present in the room and him hearing the exchange and knowing the method of how we exchange the secret. Public key crypto give means to build up such a secret connection. âThis is an important method that is useful in everyday applications like smart cards, but also when your prime minister wants to have a telephone conversation with the president of the USA via a public net.â But there can always be leaks in the security if one is not careful. Gerhard remembers that thieves followed drivers of luxurious cars and recorded the beep that was given to open and close the door. And played that beep later to open the car door: not enough mathematics was used. So: Yes, mathematics can keep a secret, if you are cautious!

*Date: 13 June 2013*