Diffie Hellman: Why is $(\mathbb{Z}_{8191}^*, \times)$ an insecure group? I noticed that all the binary bits of $8191$ are $1$'s. Could this have something to do with it?
Besides 8191 being too small for real world use, the factorization of 8190 ($p-1$) has all small values. It's factorization is $2\cdot 3^2\cdot 5 \cdot 7\cdot 13$. Having a group whose order is smooth makes it vulnerable to an attack using the Pohlig-Hellman.