Although crypto systems based on the hardness of solving discrete logarithms date back 40 years, they are more prevalent today than ever. The elliptic curve Diffie-Hellman protocol, for example, accounts for 90% of public-key agreements on the internet, meaning your device probably used discrete log based cryptography to fetch these very words. In this talk we’ll examine some of the more complex applications of discrete logarithm based cryptography, including homomorphic encryption, zero-knowledge proofs, distributed decryption, and secure multi-party protocols. Finally, we’ll tie these cryptographic primitives together with a case study of a cryptographically end-to-end verifiable internet voting system.

Consider a Grothendieck topos with a functorial cylinder with connections and a choice of generating cofibrations. Together, this induces a naive notion of fibration. We give sufficient coditions under which this gives rise to an (algebraic) model structure. The key point will be the glueing construction of Cohen et al., a recent combinatorial discovery that allows for constructively extending fibrations along trivial cofibrations without resorting to a theory of minimal fibrations.

I will define expander and Ramanujan graphs and give a variety of existence results. Most notably I will state a joint result of D. Puder, W. Sawin, and myself on so-called Ramanujan covers of graphs. Our result generalizes one of several recent celebrated joint results of A. Marcus, D. Spielman, and N. Srivastava.