If I have g = primitive root and p = prime number such that:
X = $g^x$ mod p
Y = $X^y$ mod p
I know the values of g, p, X, Y. Can I calculate $g^y$ without knowing x? How do I do that?
For example: Let us say I know that g = 5; p = 23; X = $g^x$ mod p = 8; Y = $X^y$ mod p = 2.
I do not know that x = 6 and y = 15. How do I find the values x = 6 and y = 15?
Even if you are given $x=1$ [and so $g = X$ is known too], there is still no known efficient algorithm at the writing of this post, to find the remaining unknown variable $y$.
Indeed, if you could, then you'd have an algorithm that finds efficiently, for general instances $p,Y,g$, the integer $y$ that solves the equation $Y=g^y \pmod p$; $Y,p,$ and $g$ known. There is no such algorithm known at this time. See discrete logarithm problem.