  The Hidden Subgroup Problem (HSP) is a computational problem which includes
as special cases integer factorization, the discrete logarithm problem, graph
isomorphism, and the shortest vector problem. The celebrated polynomial-time
quantum algorithms for factorization and the discrete logarithm are restricted
versions of a generic polynomial-time quantum solution to the HSP for abelian
groups, but despite focused research no full solution has yet been found. We
propose a generalization of the HSP to include arbitrary algebraic structures
and analyze this new problem on powers of 2-element algebras. We prove a
complete classification of every such power as quantum tractable (i.e.
polynomial-time), classically tractable, quantum intractable, and classically
intractable. In particular, we identify a class of algebras for which the
generalized HSP exhibits super-polynomial speedup on a quantum computer
compared to a classical one.
