From Fermat's Little Theorem to the Notion of Group

On Wednesday, January 14, Emmanuel Royer gave a presentation to the final year students of Collège Stanislas in Montreal on Fermat’s little theorem and the notion of group, an introduction to a series that will take place on February 4, during which RSA cryptography and elliptic curve cryptography will be discussed.

Every prime number unfailingly measures one of the powers -1 of some progression, and the exponent of the said power is a sub-multiple of the given prime number -1 (…). I would send you the proof, if I were not apprehensive of being too lengthy.
Pierre de Fermat, October 18, 1640.

The aim is to show how, from rather applicationless questions of amateur mathematicians about prime numbers (Fermat was a lawyer, professional mathematicians built bridges), mathematics generalized the context (or should we say: reduces it to the essential?) before returning to applications, inventing public key cryptography.

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

A professor at the University of Clermont-Auvergne, Emmanuel Royer is welcomed in delegation for institutional functions by the CNRS, to lead the CRM-CNRS.