For pairing-based cryptographic protocols to be both efficient and secure, the underlying genus 2 curves defined over finite fields used must satisfy pairing-friendly conditions, and have small rho-value, which are not likely to be satisfied with random curves.
In this thesis, we study two specific families of genus 2 curves defined over finite fields whose Jacobians do not split over the ground fields into a product of elliptic curves, but geometrically split over an extension of the ground field of prescribed degree n=3, 4, or 6. These curves were also studied extensively recently by Kawazoe and Takahashi in 2008, and by Freeman and Satoh in 2009 in their searches of pairing-friendly curves.
We present a new method for constructing and identifying suitable curves in these two families which satisfy the pairing-friendly conditions and have rho-values around 4. The computational results of the rho-values obtained in this thesis are consistent with those found by Freeman and Satoh in 2009. An extension of our new method has led to a cryptographic example of a pairing-friendly curve in one of the two families which has rho-value 2.969, and it is the lowest rho-value ever recorded for curves of this type. Our method is different from the method proposed by Freeman and Satoh, since we can prescribe the minimal degree n =3,4 or 6 extension of the ground fields which the Jacobians of the curves split over. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2011-11-08 18:57:59.988
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:OKQ.1974/6866 |
| Date | 09 November 2011 |
| Creators | CHOU, KUO MING JAMES |
| Contributors | Queen's University (Kingston, Ont.). Theses (Queen's University (Kingston, Ont.)) |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English, English |
| Detected Language | English |
| Type | Thesis |
| Rights | This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner. |
| Relation | Canadian theses |
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