Elliptic curves have played a large role in modern cryptography. Most notably,
the Elliptic Curve Digital Signature Algorithm (ECDSA) and the Elliptic Curve
Di e-Hellman (ECDH) key exchange algorithm are widely used in practice today for
their e ciency and small key sizes. More recently, the Supersingular Isogeny-based
Di e-Hellman (SIDH) algorithm provides a method of exchanging keys which is conjectured
to be secure in the post-quantum setting. For ECDSA and ECDH, e cient
and secure algorithms for scalar multiplication of points are necessary for modern use
of these protocols. Likewise, in SIDH it is necessary to be able to compute an isogeny
from a given nite subgroup of an elliptic curve in a fast and secure fashion.
We therefore nd strong motivation to study and improve the algorithms used
in elliptic curve cryptography, and to develop new algorithms to be deployed within
these protocols. In this thesis we design and develop d-MUL, a multidimensional
scalar multiplication algorithm which is uniform in its operations and generalizes the
well known 1-dimensional Montgomery ladder addition chain and the 2-dimensional
addition chain due to Dan J. Bernstein. We analyze the construction and derive many
optimizations, implement the algorithm in software, and prove many theoretical and practical results. In the nal chapter of the thesis we analyze the operations carried
out in the construction of an isogeny from a given subgroup, as performed in SIDH.
We detail how to e ciently make use of parallel processing when constructing this
isogeny. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2018. / FAU Electronic Theses and Dissertations Collection
| Identifer | oai:union.ndltd.org:fau.edu/oai:fau.digital.flvc.org:fau_40929 |
| Contributors | Hutchinson, Aaron (author), Karabina, Koray (Thesis advisor), Florida Atlantic University (Degree grantor), Charles E. Schmidt College of Science, Department of Mathematical Sciences |
| Publisher | Florida Atlantic University |
| Source Sets | Florida Atlantic University |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation, Text |
| Format | 148 p., application/pdf |
| Rights | Copyright © is held by the author with permission granted to Florida Atlantic University to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder., http://rightsstatements.org/vocab/InC/1.0/ |
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