Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-24hb2 Total loading time: 0 Render date: 2024-03-18T23:35:10.928Z Has data issue: false hasContentIssue false

11 - Cryptography based on hyperelliptic curves

Published online by Cambridge University Press:  05 April 2014

Richard E. Blahut
Affiliation:
University of Illinois, Urbana-Champaign
Get access

Summary

The widespread success of cryptography based on elliptic curves motivates the investigation of other curves for possible cryptographic uses. However, elliptic curves are the only plane curves that admit a definition of point addition in such a way that the points of the curve form a group. This does not mean that other curves cannot be used. It only means that the points of those curves must be organized to form a group in some other way. It is more complicated to find group structures based on other curves. In general, the curve X must be embedded into a larger algebraic structure on which a suitable group operation can be defined. Hyperelliptic curves are a class of curves that lead to such a group structure. A hyperelliptic curve is associated in a natural way with an abelian group called the jacobian of the hyperelliptic curve. In contrast to the curve itself, the jacobian of a hyperelliptic curve does admit a suitable group structure. Based on the group structure of its jacobian, a hyperelliptic curve can be used to construct a cryptographic system. Most of the chapter is devoted to the task of defining the jacobian of a hyperelliptic curve and its relevant computational algorithms. The usual methods of cryptography constructed on a large finite group are then immediately applicable.

Because an elliptic curve is a special case of a hyperelliptic curve, this chapter also serves to extend our understanding of elliptic curves.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×