Efficient 3-dimensional GLV method for faster point multiplication on some GLS elliptic curves

Abstract

We discover that two distinct efficient endomorphisms can both exist on some Galbraith–Lin–Scott (GLS) elliptic curves Galbraith et al. (2009) [4]. By using them we generalize the Gallant–Lambert–Vanstone (GLV) method Gallant et al. (2001) [5] for faster point multiplication on these curves to dimension 3, and give some implementation result which shows that our 3-dimensional GLV (3GLV) method runs in 0.897 the time of 2-dimensional GLV (2GLV) method as Galbraith et al. did in Galbraith et al. (2009) [4] for the point multiplication on these curves.