Parrondo paradox in quantum image encryption

TL;DR

The paper introduces a fully unitary quantum image encryption scheme based on discrete-time quantum walks (DTQWs) on cycles and NEQR encoding, augmented by diffusion and nonlinear confusion to secure grayscale images. It investigates the Parrondo paradox as a potential security risk and demonstrates that, when integrated into a three-layer pipeline (diffusion, confusion, substitution), the paradox enhances complexity without degrading security. On $64\times64$ images, the scheme achieves near-zero directional correlations, ciphertext entropy near the ideal $8$ bits, and high diffusion/ confusion metrics (NPCR $>99\%$, UACI around $30\%$), in both non-paradox and paradox regimes. The results support the viability of coherent, reversible DTQW-based quantum image processing and point to scalable hardware implementations and extensions to higher-dimensional walks.

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 -- incorporating 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.

Figures (6)

• Figure 1: Quantum circuit for a discrete-time quantum walk on a cycle with $n$ position qubits. The quantum Fourier transform, $\tilde{\mathcal{F}}$ and its inverse do not contain the SWAP operations (see QiskitQFT for details). The operations $R_k$ are defined in Eq. \ref{['eq:rk']}
• Figure 2: Example results of the quantum image encryption protocol for the case when the quantum walk does not exhibit the Parrondo paradox. The first row shows the original images, the second row shows the encrypted images, and the third row shows the decrypted images. The images are 64x64 pixels in size.
• Figure 3: Scatter plot of the pixel values of the original and encrypted images for the image of Lena. The original image is shown in Fig. \ref{['fig:no-paradox-examples']}. The scatter plot shows that the pixel values of the encrypted image are uniformly distributed, hence no useful information can be extracted from the encrypted image.
• Figure 4: Histograms of the pixel values of the original and encrypted images. The first row shows the histograms of the original images, and the second row shows the histograms of the encrypted images. The histograms are computed for the 64x64 pixel images. The histograms of the encrypted images are uniform, while the histograms of the original images are not.
• Figure 5: Example results of the quantum image encryption protocol for the case when the quantum walk exhibits the Parrondo paradox. The first row shows the original images, the second row shows the encrypted images, and the third row shows the decrypted images. The images are 64x64 pixels in size.