Counting superspecial Richelot isogenies by reduced automorphism groups (Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties)

The recent cryptanalysis by Costello and Smith [10] employed the subgraphs whose vertices consist of decomposed principally polarized abelian varieties, hence it is important to study the subgraphs in isogeny-based cryptography. Katsura and Takashima [22] initiated the investigation of the decomposed abelian surface subgraphs in the genus-2 case. This paper surveys the work, aiming to provide a kind of handbook for applying our results to cryptography

Ribet bimodules and the specialization of Heegner points

We describe the specialization of Heegner points on Shimura curves at primes of bad reduction. Moreover, we give some reciprocity laws relating the Galois action on these points to natural actions on the set of singular points and the set of connected components of the fiber

Isogeny-based post-quantum key exchange protocols

The goal of this project is to understand and analyze the supersingular isogeny Diffie Hellman (SIDH), a post-quantum key exchange protocol which security lies on the isogeny-finding problem between supersingular elliptic curves. In order to do so, we first introduce the reader to cryptography focusing on key agreement protocols and motivate the rise of post-quantum cryptography as a necessity with the existence of the model of quantum computation. We review some of the known attacks on the SIDH and finally study some algorithmic aspects to understand how the protocol can be implemented