  The ElGamal cryptosystem is the most widely used public key cryptosystem. It
uses the discrete logarithm problem as the cryptographic primitive. The MOR
cryptosystem is a similar cryptosystem. It uses the discrete logarithm problem
in the automorphism group as the cryptographic primitive. In this paper, we
study the MOR cryptosystem for finite $p$-groups. The study is complete for
$p^\prime$-automorphisms. For $p$-automorphisms there are some interesting open
problems.
