Analyzing the Galbraith-Lin-Scott Point Multiplication Method for Elliptic Curves over Binary Fields

Galbraith, Lin, and Scott recently constructed efficiently computable endomorphisms for a large family of elliptic curves defined over IF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> 2 and showed, in the case where q is a prime, that the Gallant-Lambert-Vanstone point multiplication method for these curves is significantly faster than point multiplication for general elliptic curves over prime fields. In this paper, we investigate the potential benefits of using Galbraith-Lin-Scott elliptic curves in the case where q is a power of 2. The analysis differs from the q prime case because of several factors, including the availability of the point halving strategy for elliptic curves over binary fields. Our analysis and implementations show that Galbraith-Lin-Scott point multiplication method offers significant acceleration for curves over binary fields, in both doubling- and halving-based approaches. Experimentally, the acceleration surpasses that reported for prime fields (for the platform in common), a somewhat counterintuitive result given the relative costs of point addition and doubling in each case.