There is a link between the Langlands program and elliptic curves as well as a link between elliptic curves and cryptography. I am wondering how a thing in the Langlands program can be translated to something interesting in cryptography?
For example, do the proof of Langlands functoriality conjecture leads to a breakthrough result in cryptography? Or do functoriality for endoscopy pairs have been already used in cryptography? If yes, what are the intermediary products in the theory of elliptic curves?
I am relatively comfortable with the Langlands program and theory of elliptic curves individually (but don't know much about how one field interacts with the other). Any reference/clue will be highly appreciated.
Arithmetic geometry in general has applications to areas such as cryptography and physics. There is much research about the interplay between arithmetic geometry and the Langlands correspondence for number fields.
However, I am not aware of an immediate "breakthrough result" for cryptography following from the Langlands program. I am sure that arithmetic geometry offers new techniques for cryptography, but I am no expert here. There was a thematic semester on The correspondences between Geometry, Arithmetic and Cryptography, which has discussed also "the link between automorphic forms and Galois theory in the Langlands programme, and bridges between the world of error-correcting codes and cryptography."