  Amongst the most remarkable successes of quantum computation are Shor's
efficient quantum algorithms for the computational tasks of integer
factorisation and the evaluation of discrete logarithms. In this article we
review the essential ingredients of these algorithms and draw out the unifying
generalization of the so-called abelian hidden subgroup problem. This involves
an unexpectedly harmonious alignment of the formalism of quantum physics with
the elegant mathematical theory of group representations and fourier transforms
on finite groups. Finally we consider the non-abelian hidden subgroup problem
mentioning some open questions where future quantum algorithms may be expected
to have a substantial impact.
