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Explore the structural properties of supersingular elliptic curve isogeny graphs in this 56-minute mathematical lecture. Examine how these graphs, which have isomorphism classes of supersingular elliptic curves over finite fields as vertices and isogenies of fixed degree as edges, function as optimal expander graphs with apparent "random" characteristics. Discover their applications in post-quantum cryptographic schemes designed to resist quantum computer attacks, while investigating potential hidden structures that could impact system security. Analyze various graph-theoretic structural properties of supersingular isogeny graphs over finite fields $\mathbb{F}_{p^2}$ and their subgraphs induced by vertices defined over $\mathbb{F}_p$. Learn about collaborative research findings from work conducted with Sarah Arpin from Virginia Tech and undergraduate student Taha Hedayat from the University of Calgary. Gain insights into the intersection of algebraic geometry, graph theory, and cryptography through this detailed mathematical exposition recorded during the thematic meeting on "Arithmetic, Geometry, Cryptography and Coding Theory" at the Centre International de Rencontres Mathématiques in Marseille, France.
