Finding the subgroup in isogeny-based cryptography Isogeny-based cryptography is one of the newest post-quantum cryptography. Hardness of this system is based on finding isogeny between two elliptic curves. Also this is theorem:


  Elliptic curves are isogenous over $F_p$ if and only if they have equal
  number of points.


Recently, isogeny based public key methods are widely used in articles.  This methods are implemented in computer systems in milliseconds and in android systems in seconds. This show that in future we can use such post-quantum cryptographic systems in practical world. 

I studied many articles about this method, and I saw only mathematical methods, not practical methods or source code. In program such as the MAGMA we can find all isogenous curves with given curves defined over rational field, but I looking for finite field. Also in sage we can find $l$-isogeny with subgroup of order $l$, but in several time our defined curve have not any subgroup with order $l$.

This is one of interesting article. In this article we have some computational example (page $15$). Can you help me fo understand this example. When I send mail to its authors I face with mail error.

How can we find cyclic subgroup of order $l$ of the elliptic curve $E(\overline{\mathbf F_p})$, over the algebraic closure of the finite field?
While there is a sub-exponential attack to compute isogenies on ORDINARY elliptic curves (the basis for the Rostovev and Stulbunov paper that you reference) there is not (yet at least) a sub-exponential attack to compute isogenies on SUPERSINGULAR elliptic curves.  The cryptosystem proposed by DeFeo, Jao, and Plut back in 2011 is based on Supersingular rather than ordinary elliptic curves.  Microsoft Research recently published a paper on this cryptosystem.  The paper contains a good overview of research on this problem to date. An preprint of Microsoft's paper is found here.