Arithmetic Using Compression on Elliptic Curves in Huff’s Form and Its Applications

In this paper for elliptic curves provided by Huff's equation $H_{a,b}: ax(y^2-1) = by(x^2-1)$ and general $G_{\overline{a},\overline{b}} : {\overline{x}}(\overline{a}{\overline{y}}^2-1)={\overline{y}}(\overline{b}{\overline{x}}^2-1)$ degree 2 compression function $f(x,y) xy$ on these curves, herein we provide formulas doubling differential addition after compression, which are as efficient Montgomery's $By^2 x^3 + Ax^2 x$. For also point recovery a $P$ allows to compute $[n]f(P)$ using the Montgomery ladder algorithm, then recover $[n]P$. Using of Moody Shumow computing odd isogenies have curves. Moreover, it is shown how apply obtained ECM algorithm. appendix, present examples convenient isogeny-based cryptography, where can be used.