Inverse in ElGamal signature
In ElGamal Signature Scheme we have:

$$\beta=\alpha^a \bmod p$$

The values $p,\alpha$ and $\beta$ are public key, and $a$ is private key.

$$\operatorname{sig_k}(x,k)=(\gamma , \delta)$$


  where $\gamma = \alpha^k \bmod p $ and $\delta = (x-a\gamma)k^{-1} \bmod {p-1}$.