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We give an algorithm for the hidden subgroup problem for the dihedral group$D_N$, or equivalently the cyclic hidden shift problem, that supersedes ourfirst algorithm and is suggested by Regev's algorithm. It runs in$\exp(O(\sqrt{\log N}))$ quantum time and uses $\exp(O(\sqrt{\log N}))$classical space, but only $O(\log N)$ quantum space. The algorithm also runsfaster with quantumly addressable classical space than with fully classicalspace. In the hidden shift form, which is more natural for this algorithmregardless, it can also make use of multiple hidden shifts. It can also beextended with two parameters that trade classical space and classical time forquantum time. At the extreme space-saving end, the algorithm becomes Regev'salgorithm. At the other end, if the algorithm is allowed classical memory withquantum random access, then many trade-offs between classical and quantum timeare possible.

Another subexponential-time quantum algorithm for the dihedral hidden subgroup problem