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Is it possible to prove that a point on an elliptic curve falls within a given range of another point, without revealing the distance between them. For example:

Let's say $X$ and $Y$ are two points on an elliptic curve such that $X = x \cdot G$ and $Y = y \cdot G$, where $G$ is the generator. The proover knows values of $x$ and $y$, and wants to prove to the verifier that $|x - y| < z$, where $z$ is some value supplied by the verifier.

Another option: prove that $\frac{x}{y} < z$, where $z$ can be a relatively small number (e.g. 2).

Is either of these possible?

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  • $\begingroup$ Two step Yao's Millionaires' problem? $\endgroup$ – kelalaka Dec 23 '18 at 12:40
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The proof showing that, given $G$ and $x \cdot G$ and a range $[A,B]$, $x$ is in the range $[A,B]$ is known as the range proof. In your case, we can convert your statement as the range proof as, given $G$ and $H = xG - yG$, showing $x-y$ is in the range $[-z,z]$.

For example, the following papers (and a lot of papers) proposed efficient range proofs.

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