This paper presents method for obtaining high-degree compression functions using natural symmetries in a given model of an elliptic curve. Such symmetries may be found using symmetry of involution $[-1]$ and symmetry of translation morphism $\tau_T=P+T$, where $T$ is the $n$-torsion point which naturally belongs to the $E(\mathbb K)$ for a given elliptic curve model. We will study alternative models of elliptic curves with points of order $2$ and $4$, and specifically Huff's curves and the Hessian family of elliptic curves (like Hessian, twisted Hessian and generalized Hessian curves) with a point of order $3$. We bring up some known compression functions on those models and present new ones as well. For (almost) every presented compression function, differential addition and point doubling formulas are shown. As in the case of high-degree compression functions manual investigation of differential addition and doubling formulas is very difficult, we came up with a Magma program which relies on the Gr\"obner basis. We prove that if for a model $E$ of an elliptic curve exists an isomorphism $\phi:E \to E_M$, where $E_M$ is the Montgomery curve and for any $P \in E(\mathbb K)$ holds that $\phi(P)=(\phi_x(P), \phi_y(P))$, then for a model $E$ one may find compression function of degree $2$. Moreover, one may find, defined for this compression function, differential addition and doubling formulas of the same efficiency as Montgomery's. However, it seems that for the family of elliptic curves having a natural point of order $3$, compression functions of the same efficiency do not exist.

翻译：本文展示了在给定的椭圆曲线模型中使用自然对称来获取高度压缩函数的方法 。 这种对称可以使用进量 $ - 1 的对称和翻译形态的对称 $\ tau_ T= P+T$, 其中$T$是自然属于 $E (mathbb K) 的美元调解点。 我们将研究具有排序的椭圆曲线的替代模型( PP$ ) $ 和 $, 具体来说是 Huff 的曲线和 利滑曲线的赫萨家族( 如 Hesian $ - 1, 扭曲的Hesian 和通用的 Hesian 曲线的对称 3美元 。 我们把这些模型上的一些已知的压缩函数相同, 显示每个显示的压缩模型、 差异加值和点加值的双倍公式。 在高度压缩函数中, 变值是 $ 美元, 我们用磁带的货币 递增的货币值函数是 $ 。