  One-dimensional Floquet topological superconductors possess two types of
degenerate Majorana edge modes at zero and $\pi$ quasieneriges, leaving more
room for the design of boundary time crystals and quantum computing schemes
than their static counterparts. In this work, we discover Floquet
superconducting phases with large topological invariants and arbitrarily many
Majorana edge modes in periodically driven Kitaev chains. Topological winding
numbers defined for the Floquet operator and Floquet entanglement Hamiltonian
are found to generate consistent predictions about the phase diagram, bulk-edge
correspondence and numbers of zero and $\pi$ Majorana edge modes of the system
under different driving protocols. The bipartite entanglement entropy further
show non-analytic behaviors around the topological transition point between
different Floquet superconducting phases. These general features are
demonstrated by investigating the Kitaev chain with periodically kicked pairing
or hopping amplitudes. Our discovery reveals the rich topological phases and
many Majorana edge modes that could be brought about by periodic driving fields
in one-dimensional superconducting systems. It further introduces a unified
description for a class of Floquet topological superconductors from their
quasienergy bands and entanglement properties.
