# 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?

• On your remark about sage: an $\ell$-isogeny is an isogeny whose kernel is a cyclic subgroup of order $\ell$ of the elliptic curve $E(\overline{\mathbf F_p})$, over the algebraic closure of the finite field. Therefore there are $\ell$-isogenies even for $\ell$'s that do not divide the order of $E(\mathbf F_p)$, the group of points defined over the finite field. Jan 22, 2016 at 14:11
• If request for reference recommendations are off-topic, why reference request tag is exist? Jan 22, 2016 at 14:58
• @MeysamGhahramani See this meta question on the tag. Specifically DW's answer. I think your edit is good, so I reopened. Jan 22, 2016 at 16:16
• "This show that in future we can use such post-quantum cryptographic systems in practical world"; actually, this does not address the most fundamental question: is this system secure (against either conventional computers or quantum ones). IMHO, this system has not been vetted well enough for us to give a confident affirmative to this question. Jan 22, 2016 at 16:26
• @MeysamGhahramani: could you then point out the papers analyzing the security of EC Isogeny? That's not a criticism; I haven't seen anything, and (as you said) that system would be its advantages, should we have confidence in its security. Jan 22, 2016 at 19:11