<p>Point addition on an elliptic curve involves two formulas (the addition formula and the doubling formula), but adding a point to itself is <em>defined</em> as equal to the doubling. This is because in case the normal addition formula is used when applied to two identical points, you get a division by zero.</p>
<p>So there is nothing surprising about this. The actual addition law is something like:</p>
<ul>
<li>If the two points are equal, use the doubling formula.</li>
<li>Otherwise, use the normal addition formula.</li>
</ul>
<p>So when computing the addition of a point P with itself, what is actually invoked is the doubling formula. No wonder that it gives the same result as performing a doubling directly!</p>
<p>(There are more edge cases to deal with, which I've left out for simplicity)</p>

CC BY-SA 4.0
2023-11-06T12:53:30.883Z
2023-11-06T16:11:15.170Z
2023-11-06T16:11:15.170Z
Pieter Wuille
<p>I am working with the secp256k1 elliptic curve and I have a point Q on the curve. I am trying to determine whether this point Q was derived using the point addition formula or the point doubling formula.</p>
<p>I have tried calculating the point addition of Q with itself (Q + Q) using the point addition formula and also calculating the point doubling of Q (2Q) using the point doubling formula. However, these calculations always result in the same point.</p>
<p>Is there any way to determine the derivation method of a given point on the secp256k1 curve? Are there any additional steps or techniques that can help me differentiate between point addition and point doubling?</p>
<p>Any guidance or insights would be greatly appreciated. Thank you!</p>

CC BY-SA 4.0
2023-11-06T12:02:52.113Z
That's like asking whether it's possible to know whether someone calculated the number `8` by doing `2+2+2+2` or by doing `2*2*2`. The result is the same, the process is unknowable.
bca-0353f40e
CC BY-SA 4.0
2023-11-06T14:14:00.533Z
@bca-0353f40e Perhaps an even better analogy is asking whether you can distinguish `5+3` from `5-(-3)`. Both expressions simplify to the exact same operation.
Pieter Wuille
<secp256k1>
How can I determine if a point on the secp256k1 curve was derived using point addition or point doubling?
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# How can I determine if a point on the secp256k1 curve was derived using point addition or point doubling?

(Asked by: username_0 on 2023-11-06)

I am working with the secp256k1 elliptic curve and I have a point Q on the curve. I am trying to determine whether this point Q was derived using the point addition formula or the point doubling formula.

I have tried calculating the point addition of Q with itself (Q + Q) using the point addition formula and also calculating the point doubling of Q (2Q) using the point doubling formula. However, these calculations always result in the same point.

Is there any way to determine the derivation method of a given point on the secp256k1 curve? Are there any additional steps or techniques that can help me differentiate between point addition and point doubling?

Any guidance or insights would be greatly appreciated. Thank you!

**Tags:** secp256k1

> **Comments:**
> **username_1**: That's like asking whether it's possible to know whether someone calculated the number `8` by doing `2+2+2+2` or by doing `2*2*2`. The result is the same, the process is unknowable.  
> **username_2**: @username_1 Perhaps an even better analogy is asking whether you can distinguish `5+3` from `5-(-3)`. Both expressions simplify to the exact same operation.  

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## 1 answers

### Answer by username_2 on 2023-11-06. Score: 4

Point addition on an elliptic curve involves two formulas (the addition formula and the doubling formula), but adding a point to itself is *defined* as equal to the doubling. This is because in case the normal addition formula is used when applied to two identical points, you get a division by zero.

So there is nothing surprising about this. The actual addition law is something like:

* If the two points are equal, use the doubling formula.
* Otherwise, use the normal addition formula.

So when computing the addition of a point P with itself, what is actually invoked is the doubling formula. No wonder that it gives the same result as performing a doubling directly!

(There are more edge cases to deal with, which I've left out for simplicity)


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