  Maximally entangled single-particle states (MESPS) are opening new
possibilities in quantum technology as they have the potential to encode more
information and are robust to decoherence compared to their nonlocal
two-particle counterparts. We find that a single coin can generate MESPS at
recurrent time steps (periodically) via discrete-time quantum walks on both $4$
and $8$ site cyclic graphs. This scheme is resource-saving with possibly the
most straightforward experimental realization since the same coin is applied at
each time step. We also show that recurrent MESPS can be generated on any
arbitrary $k$ site cyclic graph, $k\in\{3,4,5,8\}$ via effective-single
(Identity and arbitrary coin) or two coin evolution sequences. Beyond their use
in fundamental research, we propose an application of the generated MESPS in
quantum cryptography protocols. MESPS as cryptographic keys can strengthen
quantum-secure communication.
