Computational Diffie-Hellman problem

Question

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 $y = g^{x_1·x_2}$.

1. What happens if I knew one of the $x_1$, would it still be hard?
2. 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].

Answer 1

The hardness of CDH arises from the fact that we do not know how to obtain $x$, given $g^x$, efficiently.

1) If one knows $x_1$, then one can easily compute $g^{x_1 x_2}$ by computing $y_2^{x_1} = (g^{x_2})^{x_1} = g^{x_1 x_2}$ using Square and Multiply!

2) For this case, suppose that computing $k = g^{x_1 x_2} / g^{x_1}$ wasn't hard. Then one could easily obtain $g^{x_1 x_2} = k \, y_1 = k \, g^{x_1}$ and CDH would not be hard which is a contradiction. You can make a similar argument for the other case. And similarly for $z \, g^{x_1 x_2}$ (assuming I didn't misunderstand what you asked), if we can compute this then multiplying it by $z^{-1}$ (which is easy to compute) will solve CDH.