Parrondo paradox in quantum image encryption

Abstract

We present a quantum image encryption protocol that harnesses discrete-time quantum walks (DTQWs) on cycles and explicitly examines the role of the Parrondo paradox in security. Using the NEQR representation, a DTQW-generated probability mask is transformed into a quantum key image and applied via CNOT to encrypt grayscale images. We adopt an efficient circuit realization of DTQWs based on QFT-diagonalization and coin-conditioned phase layers, yielding low depth for \(N=2^n\) positions and \(t\) steps. On \(64\times 64\) benchmark images, the scheme suppresses adjacent-pixel correlations to near zero after encryption (e.g., \(|C_H|, |C_V|, |C_D| \approx 10^{-2}\)), produces nearly uniform histograms, and achieves high ciphertext entropy close to the 8-bit ideal. Differential analyses further indicate strong diffusion and confusion: NPCR exceeds \(99\%\) and UACI is around \(30\%\), consistent with robust sensitivity to small plaintext changes. Crucially, we investigate the impact of the Parrondo paradox on encryption quality and demonstrate that our fully unitary protocol remains robust even in paradoxical regimes. We show that while the paradox can introduce biases in simpler measurement-based schemes, our integrated approach which incorporates spatial diffusion and position-color entanglement, effectively leverages the complex interference patterns of the Parrondo walk to enhance substitution, maintaining high entropy and low correlations. Our results provide a performant DTQW-based quantum image cipher and confirm the suitability of paradoxical dynamics for secure quantum image processing. We discuss implications for hardware implementations and extensions to higher-dimensional walks.