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Is it possible to get the slope of a public key given its $x$ and $y$ coordinates?

Since all the ECC calculations come from geometry, I thought this calculation might be possible.

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  • $\begingroup$ What is your aim? What do you want to achieve? What is your curve? $\endgroup$
    – kelalaka
    Commented May 22 at 14:49
  • $\begingroup$ no particular aim nor curve in mind just studied ECC and found it quite interesting but is there any curve that does it $\endgroup$
    – Dev Tenji
    Commented May 22 at 15:25
  • $\begingroup$ The slope is usually used in the point addition with the tangent-and-chord method. The formulas work for the finite case too. The single slope is used for the doubling of a point. $\endgroup$
    – kelalaka
    Commented May 22 at 15:55

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Since you are talking about ECC and we are in a crypto stack exchange, I suppose you define your elliptic curve over a finite field, lets say $\mathbb{F}_q$.

Here is a tool that lets you plot elliptic curves over finite fields. As you can observe, over a finite field, an elliptic "curve" is just disconnected points scattered around.

Therefore, (at least to my knowledge), you can't perform any "Calculus" like getting the slope of a point etc.

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    $\begingroup$ Actually. The slope addition formula still works. In addition, the curve might not be non-singular so that some point has not slope at all $\endgroup$
    – kelalaka
    Commented May 22 at 14:49
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    $\begingroup$ @kelalaka: the slope formula still works, in the sense it does gives a number (which is consistent with how point doubling works). Of course, it doesn't give the 'tangent' at that point of the curve (because, as Alex mentioned, tangent doesn't make sense for discontinuous collections of points) $\endgroup$
    – poncho
    Commented May 22 at 18:26
  • $\begingroup$ @poncho, of course, the slope of a point is meaningless, even continuity is not enough, it must be differentiable. I've linked the figures for that point where there are cases for this. $\endgroup$
    – kelalaka
    Commented May 23 at 6:31

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