# Knowledge of discrete log is needed in the proof of Cramer-Shoup public key scheme?

In the proof of the Cramer-Shoup public key scheme [1], I understand that the adversary's view can be seen as equations such as $$\log c = x_1 + w x_2, \log d = y_1 + w y_2$$ and so on (equation 1 and 2 in [1]), where $$\log = \log_{g_1}$$ and $$w = \log g_2$$. Does this mean the adversary knows how to solve discrete log? If that's the case, why is it a reasonable assumption? Otherwise, how does the adversary know $$\log c$$, $$\log d$$ and $$w$$?

Of course, it doesn't mean that adversary knows how to solve discrete log. We just wanted to say that adversary only knows that the point $$P$$ lies somewhere on a plain $$\cal{P}$$ of such form. We don't suppose that adversary knows actual parameters of the plane.

But it's enough for us to deduce that $$\cal{P}$$ intersects with another plane $$\cal{H}$$ (which also has a specific form, though we don't know exact parameters) only by a line, so this intersection is negligible (actually, a probability for $$P$$ to lie exactly on the line is negligible).