Joint Statistics and Applied Mathematics Colloquium: Christelle Vincent (UVM)

Abstract: One of the most exciting developments in computer engineering right now is the rapid improvement of quantum computers. However, their potential capabilities are deeply concerning for cryptographers. Indeed, once they are built at scale, quantum computers will be able to solve problems that are "hard" on classical computers, and on whose hardness current public-key cryptographic schemes rely for their security. For this reason new hard problems must be proposed and studied to secure the internet in the near post-quantum era.

After a brief overview of the field of post-quantum cryptography, we introduce the problem of "navigating the supersingular l-isogeny graph" -- which, roughly speaking, asks to give an isogeny between two given supersingular elliptic curves. This problem is believed to be both classically hard and quantum-safe, and the primitives based on variations of it have very attractive performance parameters. The mathematics of this problem is deep and rich, however, and over the years it has become clear how careful one must be when using this problem to create cryptographic primitives to avoid the possibility of attacks.