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There are a variety of elliptic curves which are often used in cryptography for things like key exchanges. It seems it would be problematic if points on the elliptic curve can be periodic. What I mean is given starting point on the curve $P = (x,y)$. Can it be that there is some m such that $mP = P$? If there are, what are the requirements to find such a curve? I'm curious as to the behavior of points on a variety of elliptic curve and if you can find periodic points (and if there is a formula for finding such a thing).

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  • $\begingroup$ The security of the elliptic curve cryptography relies on the hardness of the discrete log on the Elliptic curve group. For secure groups the cost $\mathcal{O}(\sqrt{n})$ where $n$ is the order of the base point. Just use Curves over 200-bit to be secure. Like Curve 22519. The order of an element is inevitable since we use finite fields in Cryptographic Curves. $\endgroup$ – kelalaka Mar 7 '20 at 21:55
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If $mP=P$, then $(m-1)P = \mathcal{O}$, so $P$ is a torsion point of order dividing $m-1$. Depending on the field of definition of your elliptic curve, you can find torsion points of various orders. For example, over $\mathbb{Q}$, you can find torsion points of order $m-1=1,2,3,4,5,6,7,8,9,10$, and $12$.

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  • $\begingroup$ Are there fields where we do not have this property? As in, depending on my choice of elliptic curve I am safe on picking any point and not ending up with a period point? $\endgroup$ – user757611 Mar 7 '20 at 21:38
  • $\begingroup$ If the field of definition is finite, then all points are torsion points. If the field of definition is the rationals or a number field, then yes, you can pick elliptic curves that do not have any torsion points. $\endgroup$ – Álvaro Lozano-Robledo Mar 9 '20 at 0:47

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