From here,
The Computational Diffie-Hellman problem: Given $y_1 = g^{x_1}$ and $y_2 = g^{x_2}$ (but not $x_1$ and $x_2$), find $y = g^{x_1·x_2}$.
- What happens if I knew one of the $x_1$, would it still be hard?
- Considering the CDH, is the problem still hard for $g^{x_1x_2}/g^{x_1}$ or $g^{x_1x_2}*g^{x_1}$ or even just $zg^{x_1x_2}$ for some integer $z$. (For cases when I don't know $x_1$ and $x_2$ and the other case when I do know one like $x_1$?)
My intuition for no.1 is that if it would reduce to only needing to solve $y_2 = g^{x_2}$ which is a hard discrete log problem. And we know that if CDH is hard, DL is hard [2].