Using the ElGamal cryptosystem in $\mathbb{Z}_{p}^{\times}$, the proposed prime is $p = 2^{1947}\cdot 5 + 1$. The exercise asks me to show why this is a poor choice, and I can't quite do it.
In my textbook, the keyspace is $K = \{(p, \alpha, a, \beta) : \beta \equiv \alpha^{a} \bmod p\}$. So $log_{\alpha}\beta = a$, and my initial thought was to somehow solve a system of congruence equations for $a$ with the Chinese Remainder Theorem for $p-1 = 2^{1947}\cdot 5$. I couldn't really find the arguments that allowed me to do this, so I would appreciate any hints, or full answers (my crypto exam is tomorrow, so maybe not extremely subtle hints as I would like to understand this tonight)