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.

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• Entanglement entropy
• Entanglement measures
• Entanglement production
• Quantum walks