We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 05 August 2013
The problem of determining the order of the group of rational points on an elliptic curve over a finite field – the point counting problem – is of critical importance in applications such as primality proving and cryptography. As seen in the summary section of Chapter V, for cryptographic applications, we require the curve to be non-supersingular, and the group order to be divisible by a large prime factor, which in practice may be several hundred bits long (160 bits is sometimes considered a minimal requirement). Therefore, the problem is difficult, and it requires innovative solutions that are both mathematically challenging and computationally effective.
The point counting problem is introduced in this chapter, where general methods for finite groups, as well as some Easier’ cases of elliptic curve groups, are discussed. More advanced methods applicable to broader classes of curves are discussed in Chapters VII and VIII.
Main Approaches
Three main techniques are presently used to determine elliptic curves suitable for cryptography:
• Generate random curves and compute their group orders, until an appropriate one is found.
• Generate curves with given group order using the theory of complex multiplication (CM). […]
To save this book to your Kindle, first ensure [email protected] 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.
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.
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.
