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The endomorphism ring problem and supersingular isogeny graphs

Thesis type
(Thesis) M.Sc.
Date created
2021-08-06
Authors/Contributors
Author: Cha, Do Eon
Abstract
Supersingular isogeny graphs, which encode supersingular elliptic curves and their isogenies, have recently formed the basis for a number of post-quantum cryptographic protocols. The study of supersingular elliptic curves and their endomorphism rings has a long history and is intimately related to the study of quaternion algebras and their maximal orders. In this thesis, we give a treatment of the theory of quaternion algebras and elliptic curves over finite fields as these relate to supersingular isogeny graphs and computational problems on such graphs, in particular, consolidating and surveying results in the research literature. We also perform some numerical experiments on supersingular isogeny graphs and establish a number of refined upper bounds on supersingular elliptic curves with small non-integer endomorphisms.
Document
Identifier
etd21560
Copyright statement
Copyright is held by the author(s).
Permissions
This thesis may be printed or downloaded for non-commercial research and scholarly purposes.
Supervisor or Senior Supervisor
Thesis advisor: Chen, Imin
Language
English
Member of collection
Download file Size
input_data\22238\etd21560.pdf 1.14 MB

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