The dihedral hidden subgroup problem is equivalent to the
hidden shift problem for a cyclic group, while the hidden shift
problem for an arbitrary abelian group is generally similar.   In
2003, I found a subexponential time algorithm for this problem, more
precisely stretched exponential time.  Later there were two
variations, one due to Regev and the other due to me.   These
algorithms became more interesting when Childs, Jao, and Soukharev
showed that they yield a quantum algorithm to find isogenies between
elliptic curves.  I will discuss my lesser known second algorithm,
which deserves more attention because it supersedes my original
algorithm as well as Regev's algorithm.   The newer algorithm has a
better constant in the exponent, it is expensive only in classical
space and not quantum space, and it is tunable in various ways.   The
algorithm also breaks out of the representation theory of finite
groups and instead uses a novel quantum data structure that can be
called a "phase vector".