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If Alice uses point doubling over a generator point P to create two new points and then hands them over to Bob, can Bob tell which of the two points is the greater point without having to bruteforce all possible integer multiples of P?

Assume Bob knows the elliptic curve used to generate the points as well as the generator point P from which the two other points are derived.

Example: Alice creates the points 32P and 64P and hands them over to Bob without showing the scalars used. Can Bob somehow determine the second point (64P) is greater than the first point (32P).

A solution would be to divide the first point by the second point to find out if the value is above or below 1, but as far as I know division only works with scalars; not points.

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  • $\begingroup$ What means greater when it comes to points of ell. curves? $\endgroup$ – Wuestenfux Sep 3 '19 at 17:29
  • $\begingroup$ What I mean with greater is that it required a greater scalar value X to reach that point XP starting from the generator point P. $\endgroup$ – enriquejr99 Sep 3 '19 at 17:31
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    $\begingroup$ Bob takes the points $S=32P$ and $T=64P$ and can immediately check that $2S=T$. But all this makes no sense. Imagine a curve of order $11$. Then $32P=10P=-P$, and $64P=9P=-2P$. It makes no sens to use the "greater" word without a clear definition and a clear argumentation why it is useful. $\endgroup$ – dan_fulea Sep 4 '19 at 2:39
  • $\begingroup$ The problem is points are not always 32P and 64P, they may also be 30P and 66P (they do always add up to 96P though). I see what you mean with order 11. Seems like solving this may not be possible then. $\endgroup$ – enriquejr99 Sep 4 '19 at 2:59
  • $\begingroup$ What is the aim here? just curiosity? $\endgroup$ – kelalaka Sep 4 '19 at 17:29

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