  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.
