Quantum computers threaten to break forms of public-key cryptography which are currently in widespread use, creating the need for post-quantum cryptography. While some general purpose options exist, they cannot be effectively applied to the full range of use cases, hence the demand for new schemes remains. The subfield of isogeny-based cryptography works with mathematical objects called isogenies; morphisms between elliptic curves over finite fields. It has some promising candidates, however further cryptanalytic work is required to build confidence in their security. This thesis contributes to the cryptanalysis of isogeny-based schemes, studying the hard problems underlying the schemes. This includes theorising new algorithms to solve such problems, and determining when they are successful and unsuccessful. The approach taken is a mathematical one, combining areas of algebraic number theory including quaternion algebras and quadratic forms, together with algebraic geometry and graph theory. Some of the results include: new algorithmic reductions of isogeny problems; new formulae relating to endomorphism rings of supersingular elliptic curves; and new approaches to analyzing the structure of isogeny graphs. Throughout the thesis, many new questions are raised, prompting new avenues of research.
Cryptanalysis of Isogeny-based Cryptography
Clements, J. T. (Author). 30 Sept 2025
Student thesis: Doctoral Thesis › Doctor of Philosophy (PhD)