Given $P=xG$, prove in zero knowledge that I know $sqrt(x)$

Question

Assuming that x has a sqrt.

Given $P=xG$ is it possible to prove that I know the $sqrt(x)$ in zero knowledge?

Answer 1

My answer simply extends the comment by SEJPM.

Since the group has prime order (as you said in the comment), and since you assume that it is known that $x$ has a square root, you can simply prove knowledge of $x$ such that $xG = P$, using the standard Schnorr protocol for demonstrating knowledge of a discrete logarithm (see e.g. the wikipedia page, or my description here for a simplified security analysis of this protocol).

Now, since knowing $x$ is equivalent to knowing $\sqrt{x}$ in a group of prime order (each can be computed from the other in polynomial time), convincing the verifier that you know $x$ does also convince him that you know $\sqrt{x}$. Since the proof leaks nothing about $x$, it leaks nothing about $\sqrt{x}$.