Given two unrelated generators $G_1$ and $G_2$, and a third with $H = G_1 + G_2$. Is it hard to compute $xG_1$ from $xH$?

Question

Given some group in which both discrete logarithms and the computational Diffie-Hellman problem are hard. Furthermore, two random, unrelated group generators $G_1, G_2$, and a third generator defined by $H = G_1 + G_2$. Can you compute $xG_1$ if you know only $G_1$, $G_2$, and $xH$?

My guess: I would assume it's hard, because otherwise it would be easy to compute $xG$ knowing only $xH$ for any two unrelated generators $G$ and $H$.

Answer 1

Computing $xG_1$ knowing only $G_1$, $G_2$, and $xH$ is as hard as the computational Diffie-Hellman problem (CDH).

For some given $xH$ we can choose, for example, $G_1 = yH$ and $G_2 = H - G_1$. This choice satisfies $H = G_1 + G_2$.

Computing $xG_1$ from $xH$ is equivalent to computing $xyH$, which breaks the CDH assumption.