Constructions of elliptic curves endomorphisms

Let $\mathbb K$ be an imaginary quadratic field. Consider an elliptic curve $E(\mathbb F_p)$ defined over prime field $\mathbb F_p$ with given ring of endomorphisms $o_\mathbb K$, where $o_\mathbb K$ is an order in a ring of integers $\mathbb Z_\mathbb K$.
An algorithm permitting to construct endomorphism of the curve $E(\mathbb F_p)$ corresponding to the complex number $\tau\in o_\mathbb K$ is presented. The endomorphism is represented as a pair of rational functions with coefficients in $\mathbb F_p$. To construct these functions we use continued fraction expansion for values of Weierstrass function. After that we reduce the rational functions modulo prime ideal in finite extension of $\mathbb K$. One can use such endomorphism for elliptic curve point exponentiation.