Quantum algorithms for hidden nonlinear structures

Attempts to find new quantum algorithms that outperform classical computation
have focused primarily on the nonabelian hidden subgroup problem, which
generalizes the central problem solved by Shor's factoring algorithm. We
suggest an alternative generalization, namely to problems of finding hidden
nonlinear structures over finite fields. We give examples of two such problems
that can be solved efficiently by a quantum computer, but not by a classical
computer. We also give some positive results on the quantum query complexity of
finding hidden nonlinear structures.