  In this paper we discuss the problem of performing elementary finite field
arithmetic on a quantum computer. Of particular interest, is the
controlled-multiplication operation, which is the only group-specific operation
in Shor's algorithms for factoring and solving the Discrete Log Problem. We
describe how to build quantum circuits for performing this operation on the
generic Galois fields GF($p^k$), as well as the boundary cases GF($p$) and
GF($2^k$). We give the detailed size, width and depth complexity of such
circuits, which ultimately will allow us to obtain detailed upper bounds on the
amount of quantum resources needed to solve instances of the DLP on such
fields.
