I'm trying to prove something and if I can show that there is a simple way to calculate $(g^a \bmod p)^k$ if I know both $g^k \bmod p$ and $g^a \bmod p$, then (I think) it will help me prove it, but I'm not able to see an obvious way to use the two expressions to calculate the other. Is there a property in modular arithmetic that would allow for that? Or does it seem clearly untrue? I think there may just be some gaps in my knowledge about modular arithmetic so if you have a resource for that you'd like to point me to, that might also help.
If it makes a difference, the broader question I'm trying to solve is the following. We consider a modification of the ElGamal cryptosystem (called the basic ElGamal cryptosystem) where we have $p$, a prime, and $q$, a prime divisor of $p-1$. We have $g$, an element of order $q$ in $\mathbb{Z}^*_p$. The private key is an integer $a$ chosen randomly between $1$ and $q-1$. The public key is $h = g^a \bmod p$. And the ciphertext is $c=\langle c_1, c_2\rangle$ where $c_1 = g^k \bmod p$ and $c_2 = m \, h^k \bmod p$, where $k$ is random between $1$ and $q-1$. I need to prove that this is insecure under a chosen ciphertext attack (and whether it is insecure based on total break, one-way security, semantic security).
Since I know that the regular ElGamal cryptosystem is semantically secure if the Decisional Diffie-Hellman (DDH) assumption holds, I thought that if I can prove that it doesn't hold here, I can show that it is not semantically secure. That's why I wanted to ask this question but if this doesn't make sense as a way to show this, let me know so I can try something else.
