I am trying to solve an exercise from a cryptography textbook and I am stuck with these specific subquestion - any kind of suggestion is welcome!
Let $\alpha = 12$ be a generator of the group $(\mathbb{Z}_p)^{*}$ where $p = 12q+1$ is a prime and $q$ is a very large prime. Assume Alice private key is $a$.
Show that it is possible to efficiently (without use of the discrete logarithm on $q$) find an integer $z$ such that $ 12^{qz} \equiv y_A^q \pmod{p}$ where $y_A = \alpha^a$