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I'm exploring the secp256k1 elliptic curve in the context of cryptography and encountered the concept of Point negation. I would appreciate clarification on what point negation means in this context.

As I understand it, given a point $P = (x, y)$ on the secp256k1 curve, the negation of this point is represented by $-P = (x, -y)$.

I would like to know how to determine if a point $P = (x, y)$ is negated or not ?

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I would like to know how to determine if a point $P = (x, y)$ is negated or not ?

It appears you are asking: I was just given the value $(x, y)$, and I want to know if the person who gave it to make picked $(x, y)$, or whether they first picked $(x, -y)$ and then negated the $y$ value.

The answer is: unless you have some information on the value they originally picked, you have no way of knowing. Assuming that he picks a uniformly distributed random point (and doesn't decide on whether or not to negate $y$ based on the value he picked), the distribution of the values he gives to you are precisely the same.

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Similar to finite field has no "negative" element, elliptic groups don't have "negative points" either. They only have "additive inverse".

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There is no way you can know that because SECP256k1 uses:

$y^{2} = x^{3}+7$

Therefore multiplication removes negative sign. From my understanding point negation is only used in the purpose of point subtraction by simply prepending a negative sign to the $y$ value of a point then perform point addition with it, Other than than whether $y$ is negative or not it doesn't really matter.

Take it as:

$-3^{2} = 3^{2}$

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