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How to decompose a public key into subgroups EC?

br flag

Is it possible to decompose the public key into its own subgroups? Suppose we know the order P with which the public key was generated (Qx, Qy)

How can the public key (Qx, Qy) be decomposed into subgroups of small orders?

I saw in SageMath it is possible to work with Elliptic Curves

M = EllipticCurve (GF (p), [0.7])

I am just getting familiar with SageMath and am having a hard time working on creating a generator on order. Are there examples of these works?

Manish Adhikari avatar
us flag
A public key is a GROUP ELEMENT, which is a point with coordinates taken from a finite field in elliptical curves, AND NOT a group itself. I don't know what you mean by "decompose public key into its own subgroups". The EC group used is (generally) a large prime order (sub)group in the curve. Which means it has no non-trivial subgroup and every valid public key point generates the entire (sub)group.
kelalaka avatar
in flag
Do you mean determine which subgroups it belongs? Well, factor the order of the group and use the Lagrange theorem of each of the factors; if $5$ is a factor and $[5]P = \mathcal{O}$ then the order of $P$ is 5...
Dew Debra avatar
br flag
@kelalaka Yes, it is kelalaka that I thank for the quick reply. How do I do this in SageMath? I add `E = EllipticCurve(GF(p), [0,7]) Grp = E.abelian_group() g = Grp.gens()[0] numElements = g.order()` ,but unfortunately I get an error
kelalaka avatar
in flag
[cardinality and factor?](https://crypto.stackexchange.com/a/95667/18298)
kelalaka avatar
in flag
Note, for direct SageMath errors, there is https://ask.sagemath.org/
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