  Topological phases of matter are distinct from conventional ones by their
lack of a local order parameter. Still in the quantum Hall effect, hidden order
parameters exist and constitute the basis for the celebrated composite-particle
approach. Whether similar hidden orders exist in 2D and 3D symmetry protected
topological phases (SPTs) is a largely open question. Here we introduce a new
approach for generating SPT groundstates, based on a generalization of the
Laughlin wavefunction. This approach gives a simple and unifying picture of
some classes of SPTs in 1D and 2D, and reveals their hidden order and flux
attachment structures. For the 1D case, we derive exact relations between the
wavefunctions obtained in this manner and group cohomology wavefunctions, as
well as matrix product state classification. For the 2D Ising SPT, strong
analytical and numerical evidence is given to show that the wavefunction
obtained indeed describes the desired SPT. The Ising SPT then appears as a
state with quasi-long-range order in composite degrees of freedom consisting of
Ising-symmetry charges attached to Ising-symmetry fluxes.
