The world's increasing reliance upon technology and the exchange of sensitive information across unsecured channels require a standard of security to remain effective. This paper serves as a concise resource for information on the Buchmann-Williams public-key cryptographic system. The system is based on the Diffie-Hellman protocol but its security relies on the presumed difficulty of solving the discrete logarithm problem "efficiently" in ideal class groups over imaginary quadratic fields. Efficiently here means via a polynomial time algorithm. This thesis includes the necessary algebraic and number theoretic background, the description of the protocol, and its implementation via Magma
