How do I find the exponent of a mod

Question

I have :

${(g^{XB} \bmod P)}^{RA} \bmod P = 12 $

If : $XB =4 $, $g=9$ , $P =23$

How do I find the value of $RA$ in terms of $g$ ?

I have tried and found this to be correct direction, but I can't seem to actually get it to work with the numbers $ ({g^{XB}}^{RA}\bmod P)^{XB^{-1}} = g^{RA} \bmod P$

Answer 1

Hint:

First compute $\;9^4\bmod23=(9^2\bmod23)^2=(-11)^2\bmod 23=6\bmod 23$.

Then determine the order of $6\bmod 23$: it is a divisor of $\varphi(23)=22$, so it can be $2$, $11$ or $22$. Using the fast exponentiation algorithm mod. $23$, you'll find it has order $11$. So you have to make the list of powers of $6\bmod 23$, possibly up to the exponent $10$, until you obtain $12$. The first two powers are comparatively easy…