Is it possible to prove key-equivalence across elliptic curves of different order? Specifically:
- Suppose I have a key $x$ valid for both curves listed below
- On curve $g$ (for example, Curve25519) it maps to $X_1 = x*G$, where $G$ is the generator point for $g$
- On curve $h$ (for example, Curve1174) it maps to $X_2 = x*H$, where $H$ is the generator point for $h$
Is it possible for me to submit some public proof that $X_1$ and $X_2$ are generated using the same key $x$, but without revealing the value of $x$?