Aspects of Elliptic and Hyperelliptic Curve Isogeny-based Cryptography

With the advent of quantum computers in mind, a lot of attention in the field of cryptography has shifted towards new and different mathematical hard problems. One of these shifts is the change from the discrete logarithm problem on a single elliptic curve to isogenies between distinct elliptic curves. The latter is the underlying hardness problem in one of the submissions to NIST's competition for a call for post-quantum standardization. The goal of my PhD thesis is to help with research in the field of post-quantum cryptography, with a focus on isogeny-based protocols. More specifically the objective is to help discover new primitives, speed up already existing schemes, find and/or prevent possible attacks, etc. A good starting point for my research would be a generalization of certain isogeny-based primitives to higher genus curves, something that I already started working on in my master's thesis. This generalization towards hyperelliptic curves can be formulated rather easily, but a lot of algebraically hard problems arise from the get-go that need to be investigated. Another path could be looking for new schemes based on the CSIDH primitive, which is an isogeny-based protocol only just formulated in early 2018. The field of cryptography advances very quickly, so keeping up to date with recent developments may move some of my attention towards other schemes based on for example lattices or multivariate polynomials.