Explore the foundations of elliptic curve isogenies in public key cryptography with David Jao from the University of Waterloo in this 43-minute lecture. Delve into the history and core concepts of elliptic curves, finite subgroups, and their applications in cryptographic systems. Examine the Supersingular Isogeny Diffie-Hellman (SIDH) and Isogeny-based Cryptographic Random Self-reducible Assumption (ICRSA) protocols. Investigate quantum cryptanalysis techniques, including the Quantum Fourier Transform and lattice approaches, as they relate to post-quantum cryptography. Gain insights into the random case scenario and control mechanisms within this advanced cryptographic framework.