  Although iron-based superconductors are multi-orbital systems with
complicated band structures, we demonstrate that the low energy physics which
is responsible for high-$T_c$ superconductivity is essentially governed by one
effective Hamiltonianwith two almost decoupled orbitals near half filling. This
underlining electronic structure is protected by the $S_4$ symmetry. With
repulsive or strong next nearest neighbor antiferromagnetic exchange
interactions, each single-orbital effective Hamiltonian results in a robust
$A_{1g}$ s-wave pairing which can be exactly mapped to the d-wave pairing
observed in cuprates. The classification of the superconducting(SC) states
according to the $S_4$ symmetry leads to a natural prediction of the existence
of two different phases named A and B phases. In the B phase, the
superconducting order has an overall sign change along c-axis between the top
and bottom As(Se) planes in a single Fe-(As)Se trilayer structure, which is an
analogy of the sign change under the $90^\circ$ degree rotation in the d-wave
SC state of cuprates. Our derivation provides a unified understanding of
iron-pnictides and iron-chalcogenides, and suggests that cuprates and
iron-based superconductors share identical high-$T_c$ superconducting
mechanism.
