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Efficiently computable homomorphisms allow elliptic curve point multiplication to be accelerated using the Gallant-Lambert- Vanstone (GLV) method. We extend results of Iijima, Matsuo, Chao and Tsujii which give such homomorphisms for a large class of elliptic curves by working over ${\mathbb F}_{p^2}$ and demonstrate that these results 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 0.84 the time of the previous best methods for elliptic curve point multiplication on general curves.

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