Conference for Parents at Collège Stanislas

On Tuesday, March 23, Emmanuel Royer gave a conference aimed at the parents of students at Collège Stanislas.

The oldest traces of mathematics that we know of date back three thousand years before our era. At that time, it was about measuring and sharing.

Little by little, mathematics became an abstract science, whose goal is to define and demonstrate. A pioneer of this approach was Euclid, with his Elements, three hundred years before our era.

A game of the mind, developed for the honor of the human spirit, should we therefore mock the uselessness of mathematics? Should we always seek to make them concrete?

In 1640, Pierre de Fermat wrote a letter that has remained famous. A magistrate, commissioner of requests at the parliament of Toulouse, he spends his free time doing mathematics. He studies the properties of numbers and states, in this letter from 1640, what high school students today know as Fermat’s Little Theorem.

The professional mathematicians of the time — teachers, bridge builders, road constructors… — did not take these works very seriously, which fall under what is now called number theory, and questioned their usefulness.

At the same time, the need to transmit messages by coding them developed, so that only the people to whom they are intended could understand them. Coding is ancient: Caesar already had a method, which is, however, easy to decode. Vigenère, from Saint-Pourçain-sur-Sioule, popularized in the 16th century a method known as the Vigenère cipher. Although more robust, it was discovered, three centuries later, how to decode messages thus encrypted.

These methods, aside from their lack of robustness, present an essential weakness: it is only possible to send a coded message to a recipient that one has met beforehand.

To allow the sending of coded messages to third parties that one has not met before, Rivest, Shamir, and Adleman proposed a method in 1977. This method is based on Fermat’s Little Theorem, once deemed useless. Abstraction, via the algebraic notion of a group — also invented without an immediate application goal — has allowed generalizations of this method. These generalizations are what we use daily, in the 21st century, to conduct financial transactions or exchange messages via our smartphones.

Their security relies on a simple principle: if an integer of several hundred digits is the product of only two integers worth at least 2 (for example, 6 is the product of 2 and 3, and nothing else, but 6 does not have several hundred digits…), it is very — very — long to find these two numbers.

However, one cannot guarantee that a method will not be invented one day to identify them. Worse, a new type of computer, called a quantum computer, would allow this calculation to be performed if it were sufficiently developed. Certainly, quantum computers are still far from being operational, but is it serious to ignore the risk they pose to current coding, solely on the grounds that their development remains uncertain (especially since these computers represent tremendous opportunities in other fields, such as health)?

It is therefore essential to invent, right now, coding methods that would remain robust, regardless of the computer used to attempt decoding. This is the task that mathematicians are already engaged in, relying on the knowledge accumulated over the previous centuries: knowledge sufficiently abstract to become concrete in a wide range of situations. And because no one knows what tomorrow will bring, it is necessary not to consider as useless what seems too abstract today.

Find the entire series of popularization lectures in mathematics at Collège Stanislas (French institution of the AEFE network) for the 2025-2026 academic year.

Emmanuel Royer gave this conference on the occasion of Mathematics Week twice in Auvergne (France), in Saint-Sandoux and Clermont-Ferrand.

Professor at Université Clermont-Auvergne, Emmanuel Royer is welcomed in delegation for institutional functions by CNRS, to lead the CRM-CNRS.