Is there a simple solution to these 2 equations without trying all possible values?

Question

Now I have two equations and the computation is in a finite field GF(p), where p is a prime.

$x, y$ are unknown, and $a, b$ are known. ($0<y<p-1$, and $0<ax<p-1.$)

$\begin{cases} x^y = a, \\ y^{ax} = b. \end{cases} $

Is there a simple solution to solve $x$ and $y$ other than trying all their possible combinations (which is $p^2$ tries)?

Thanks a lot!

Answer 1

I think one thing we can do is to take log on both sides, so that we have:

$ \begin{cases} y\log{x} = \log{a}, \\ ax\log{y} = \log{b}. \end{cases} $

Since $y = \log{a} / \log{x}$, by substituting it to the 2nd equation we have:

$ax\log{(\log{a}/\log{x})} = \log{b}$.

Then by trying all $x$ we can solve this equation. Then it is not far from solving $y$. At least this reduces the complexity from $p^2$ to $p$.