[en] The aim of this project is to explain, as detailed as possible, how the isogeny-based cryptosystem Commutative Supersingular Isogeny Diffie-Hellman (CSIDH) works from a mathematical point of view. In order to do so, we need to introduce the concept of elliptic curve and give some of their properties, which allow us to establish the basis of this cryptosystem. Furthermore, we also present some notions of Algebraic Number Theory, particularly those that are related to elliptic curves. Once this theoretical basis is built, we proceed to describe the algorithm, paying special attention to the mathematical components. Finally, using SageMath– a mathematical software which includes elliptic curves implementations– we program the algorithm and check its efficiency by doing several tests.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: Xavier Guitart Morales