Horizontal isogenies and endomorphism rings of supersingular elliptic curves
Open Access
- Author:
- Scullard, Gabrielle
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- June 20, 2024
- Committee Members:
- Pierre-Emmanuel Jabin, Program Head/Chair
Antonio Blanca, Outside Unit & Field Member
Mihran Papikian, Major Field Member
Kirsten Eisentraeger, Chair & Dissertation Advisor
Wen-Ching Li, Major Field Member - Keywords:
- elliptic curves
isogenies
arithmetic geometry
abelian varieties
number theory - Abstract:
- This dissertation considers two problems regarding the structure of isogenies between supersingular elliptic curves which are motivated by isogeny-based cryptography. The first problem is inspired by the setup of OSIDH, a recent proposal for an isogeny-based protocol whose security relies on the hardness of finding a ``horizontal'' isogeny between two ``oriented'' supersingular elliptic curves. A basic question is to ask if it is harder to find horizontal isogenies between oriented curves than it is to find isogenies between oriented curves. Under certain conditions, the answer is no. We give conditions for which all or most isogenies of fixed degree between oriented supersingular elliptic curves are horizontal, and we classify the exceptions. Our work can be applied to extend an attack on OSIDH by Dartois and de Feo. The second problem is to compute the endomorphism ring of a supersingular elliptic curve. Most problems in isogeny-based cryptography have polynomial-time reductions to computing the endomorphism ring of a supersingular elliptic curve, so an efficient algorithm for computing endomorphism rings of supersingular elliptic curves would have consequences for isogeny-based protocols. We give a deterministic polynomial time algorithm which computes the endomorphism ring of a supersingular elliptic curve from the input of two noncommuting isogenies. Our algorithm uses techniques of higher-dimensional isogenies to navigate towards the local endomorphism ring in the Bruhat-Tits tree.