  Encryption schemes attempt to provide a means for entities to communicate
confidentially over a public channel. Such schemes have been studied for
centuries, and their use has become widespread. However, developments in the
area of quantum computing indicate that many of the public key cryptosystems
currently in use could easily be broken if large-scale quantum computers become
technologically feasible. Further, encrypted messages captured and stored in
the past could also be decrypted by a future quantum attacker. Since
large-scale quantum computers may one day be developed, We need to prepare for
that eventuality by analysing modern cryptosystems with respect to attacks with
a quantum computer.
  In this thesis, submitted in fulfilment of the requirements for the degree of
Master of Mathematics in Combinatorics & Optimisation at the University of
Waterloo, we examine the quantum strengths and vulnerabilities of several of
the most popular classical public key cryptosystems.
