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This paper presents method for obtaining high-degree compression functionsusing natural symmetries in a given model of an elliptic curve. Such symmetriesmay be found using symmetry of involution $[-1]$ and symmetry of translationmorphism $\tau_T=P+T$, where $T$ is the $n$-torsion point which naturallybelongs to the $E(\mathbb K)$ for a given elliptic curve model. We will studyalternative models of elliptic curves with points of order $2$ and $4$, andspecifically Huff's curves and the Hessian family of elliptic curves (likeHessian, twisted Hessian and generalized Hessian curves) with a point of order$3$. We bring up some known compression functions on those models and presentnew ones as well. For (almost) every presented compression function,differential addition and point doubling formulas are shown. As in the case ofhigh-degree compression functions manual investigation of differential additionand doubling formulas is very difficult, we came up with a Magma program whichrelies on the Gr\"obner basis. We prove that if for a model $E$ of an ellipticcurve exists an isomorphism $\phi:E \to E_M$, where $E_M$ is the Montgomerycurve 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 asMontgomery's. However, it seems that for the family of elliptic curves having anatural point of order $3$, compression functions of the same efficiency do notexist.

High-degree compression functions on alternative models of elliptic curves and their applications