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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 ?
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.
Similar to finite field has no "negative" element, elliptic groups don't have "negative points" either.
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:
$-3^{2} = 3^{2}$
