*13/12/2023*

## What is the subject of your project funded by the ERC ?

We aim at mixing mathematics with computer science, particularly Numbers Theory and algebraic geometry. Pure mathematicians who want to solve a problem often need computations. For example they can calculate many particular cases to identify common patterns and extract ideas on theorems to be demonstrated.

But most of the time, they just use low-efficiency methods that only work on small examples. I believe that one needs to use the theoretical progress realized in computer science to provide powerful tools to help mathematicians. This approach is essential to solve abstract mathematical problems as well as to realize efficient software.

In my project we are interested among others in mathematical objects that require Numbers Theory and geometry. And that will probably be part of the design of 3^{rd} millennium cryptosystems.

## What is the originality of your approach?

My idea is to combine theoretical computer science results, among others Complexity Theory and certificates to prove the correctness of calculus, to serve mathematics and symbolical arithmetic.

In the meantime, the results will be validated by freely available implementations. This requires two different areas of expertise that are hard to find together. My colleagues in Lfant project-team are perfect examples for these skills. We are part of a computer science and automatic research center and at the same time, we are members of the Mathematics Institute of Bordeaux and its team on Numbers Theory known throughout the world.

Mathematicians need computer-implemented algorithms and I think it is time to add more computer science to the story.

## How did you get into computer science and become a head of an Inria research-team?

I have always been very motivated by applicative aspects since I enjoy the idea that high range mathematics can provide something tangible that others will be able to use whether in security of information or throughout software distribution.

I have specialized in computer science during my studies in mathematics at Augsburg University and I wrote my thesis about security of hyperelliptic cryptosystems.

When a research team from Ecole Polytechnique that focused especially on cryptology offered me a post-doctoral position, I was very tempted, all the more so that I am a huge Francophile. When in France, the possibility of getting a permanent job in an Inria research team was very attractive to me, with the prospect of focusing completely on my research.

## How do you plan on using your ERC grant?

My purpose is to strengthen the team and make it last. The grant will allow me to hire 3 post-doctoral students, possibly recruiting them abroad, which is important because the double skill (both mathematics and computer science) is very rare everywhere. The fact of not being pressured by organisms’ schedules for applications will help me attract them… Perhaps will they be willing to stay in the team then! I have also planned to hire a research engineer for 5 years to help us with the development of the software PARI/GP made by the team and that will be used by mathematicians throughout the world. In addition, I will put up the money for a thesis and organize a couple of symposiums in Bordeaux.

### Cryptology : The third generation will be at crossroads between mathematics and computer science.

The cryptographic system that currently provides safety for credit cards, online purchases or any other https-protected website dates back from the seventies. It is based on the existence of two keys, one of them public, meant to cipher the message and the other one private, only known by the receiver and the only one capable of deciphering it.

To design such a system, one needs to identify procedures that are easy in a way, but difficult in the other. For example, it is easy (for a computer!) to find the result of the multiplication of two numbers of over 300 digits but it is very difficult to find back those numbers from the result of 600 digits.

It appears that pure mathematics offer solutions to this type of problems. The following generation of cryptosystems that provide electronic passport or identity cards security has switched multiplications for operations on elliptic curves, sophisticated objects from algebraic geometry.

Andreas Enge goes further: 3^{rd} generation systems that would use even more complex curves. The whole point is to get to know what level of security those curves can provide and how to find concrete curves keeping those promises of security with most efficiency. Mathematicians’ work that cannot be made without computers and calculus will probably require new algorithms and huge computation capacity.