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

► Two distinct efficient endomorphisms are discovered to coexist on some GLS elliptic curves. ► 3-dimensional GLV method for fast point multiplication is generalized to these curves. ► Implementation of 3-dimensional GLV method on these curves is 10 percent more efficient than that of 2-dimensinal GLV method. 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.