Supersingular isogeny graphs in cryptography

We describe recent applications of expander graphs in cryptography, particularly supersingular isogeny graphs. The security of these cryptographic constructions relies on the assumption that computing paths in these graphs is practically infeasible, even with a quantum computer. One of these cryptographic constructions is currently considered for standardization by NIST. We also recall a related construction based on Lubotzky- Philips-Sarnak’s celebrated Ramanujan graphs. We describe an efficient path-finding algorithm for these graphs which was motivated by the cryptographic application, and we mention connections to the problem of optimal quantum circuit synthesis.