Efficient Montgomery-like formulas for general Huff's and Huff's elliptic curves and their applications to the isogeny-based cryptography
In this paper for elliptic curves provided by Huff’s equation Ha,b : ax(y 2 − 1) = by(x − 1) and general Huff’s equation Ga,b : x(ay − 1) = y(bx − 1) and degree 2 compression function f(x, y) = xy on these curves, herein we provide formulas for doubling and differential addition after compression, which for Huff’s curves are as efficient as Montgomery’s formulas for Montgomery’s curves By = x + Ax + x. For these curves we also provided point recovery formulas after compression, which for a point P on these curves allows to compute [n]f(P ) after compression using the Montgomery ladder algorithm, and then recover [n]P . Using formulas of Moody and Shumow for computing odd degree isogenies on general Huff’s curves, we have also provide formulas for computing odd degree isogenies after compression for these curves. Moreover, it is shown herein how to apply obtained formulas using compression to the ECM algorithm. In the appendix, we present examples of Huff’s curves convenient for the isogeny-based cryptography, where compression can be used.