Converting a point in a finite field to its real (x, y) coordinate

Question

Let curve $A: y^2 = x^3 + 7$ and curve $B: y^2 \equiv x^3 + 7 \pmod{p}$

Curve $B$ is secp256k1, assume the usual parameters for that curve.

Let $k$ be any private key, and compute the corresponding public key point $[k]G$ in curve $B$.

Now, from this point $[k]G$ in curve $B$, how can we find what the point $[k]G$ in the "real" curve $A$ is? You don't know the value for $k$, but you do know all other parameters.

Point $G$ in curve $A$ is the positive $y$ solution to the curve $A$ with $x$ equal to the $x$ value of $G$ in curve $B$.

Use Discrete Log. I think you were asking X and wanting Y, we call it X/Y? DLog has complexity $\mathcal{O}(\sqrt{n})$ where $n$ is the group order and for this curve, it has 13 points! — kelalaka, Jan 22 '21 at 08:19
@kelalaka I'm not exactly sure how that could be helpful. I'm asking for the equivalent $Gr$ point in the curve over all real numbers given the $Gr$ point in the curve over the finite field. — enriquejr99, Jan 22 '21 at 08:37
That is scalar multiplication, then use the addition laws, with double-and-add algorithm. — kelalaka, Jan 22 '21 at 08:40
The general answer is that you can't since the modulo operation is destructive. Since you don't know about $k$ then you don't know how many times the scalar multiplication uses the $\mod p$. There are cases that one can lift the curve $B$ and solve the DLog problem easily there, however, secp256k1 has no such known lifting. — kelalaka, Jan 22 '21 at 10:33
With the question as it stands now, $G$ has the same $x$ coordinate in both $A$ and $B$, and $y$ coordinates equal modulo $p$ and perhaps sign. That's not directly compatible with what I hinted by linking to this question, where point $G$ in $A$ would have different coordinates, and (rather amazingly) knowing $G*r$ and $G$ in $A$ allows to find $r$ much faster than brute force. — fgrieu, Jan 22 '21 at 14:41
I’m voting to close this question because it is cross-posted with Math. This is not considered good in Is cross-posting a question on multiple Stack Exchange sites permitted if the question is on-topic for each site? — kelalaka, Jan 22 '21 at 16:08

Converting a point in a finite field to its real (x, y) coordinate

0 Answers0