A quasigroup is a set of elements with one binary operation whose multiplication table forms a Latin square. Because quasigroups are not required to be associative, multiplying a string of elements together in different orders can produce different results. A message authentication code, or a MAC, is a cryptographical tool used to verify the authenticity of a message. In this dissertation, we create a new message authentication code called QMAC whose security is based on this non-associativity. In order to obtain security against forgeries, a highly non-associative quasigroup of large order must be used. Methods for efficiently creating and representing such quasigroups are also discussed.