, Volume 24, Issue 3, pp 446-469
Date: 04 May 2010

Endomorphisms for Faster Elliptic Curve Cryptography on a Large Class of Curves

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Abstract

Efficiently computable homomorphisms allow elliptic curve point multiplication to be accelerated using the Gallant–Lambert–Vanstone (GLV) method. Iijima, Matsuo, Chao and Tsujii gave such homomorphisms for a large class of elliptic curves by working over \({\mathbb{F}}_{p^{2}}\) . We extend their results and demonstrate that they can be applied to the GLV method.

In general we expect our method to require about 0.75 the time of previous best methods (except for subfield curves, for which Frobenius expansions can be used). We give detailed implementation results which show that the method runs in between 0.70 and 0.83 the time of the previous best methods for elliptic curve point multiplication on general curves.

Communicated by Nigel Smart
This is the full version of a paper published at Eurocrypt 2009.
The work of S.D. Galbraith supported by EPSRC grant EP/D069904/1.
The work of S.D. Galbraith described in this report has in part been supported by the Commission of the European Communities through the ICT program under contract ICT-2007-216676. The information in this document is provided as is, and no warranty is given or implied that the information is fit for any particular purpose. The user thereof uses the information at its sole risk and liability.
X. Lin thanks the Chinese Scholarship Council.
M. Scott acknowledges support from the Science Foundation Ireland under Grant No. 06/MI/006.