Qudit Image Encryption, Space-Filling Curves and Enhanced Baker Map Schemes.

The secure transmission of digital images represents a continuing challenge in the age of increasingly sophisticated cyber threats, and quantum key distribution offers a potential solution, though practical implementation remains complex. Researchers are now exploring methods to enhance image encryption utilising the principles of quantum mechanics, specifically leveraging ‘qudits’ – quantum digits with a dimensionality greater than the standard qubit, allowing for increased information density. Claire Levaillant, an independent researcher, details a novel approach to multi-image encryption and decryption using qudits in a recently published work, proposing a system based on space-filling curves to optimise storage and a generalised Baker map – a mathematical function used to create chaotic systems – to scramble quantum information. The research introduces concepts of mixed scrambling and diffusion, offering adaptable schemes for varying security requirements.

Quantum image encryption receives considerable attention due to the potential for enhanced security and adaptability offered by novel methods, representing a significant development in secure communication. Researchers currently investigate innovative techniques predicated on the availability of qudit-based computing resources, where the dimension d is a power of 2, to address the increasing need for robust image protection. This work centres on a representation of multi-image colour data utilising space-filling curves, notably the Peano curve, to minimise storage requirements and facilitate efficient processing within a quantum computational framework. Space-filling curves map multi-dimensional data onto a single dimension, allowing for streamlined data handling.

Researchers generalise the Baker map, a discrete dynamical system exhibiting chaotic behaviour, to operate on two n-qudits, exponentially expanding its parameter space and enhancing its scrambling capabilities, forming a core component of the proposed encryption schemes. The Baker map, in its original form, transforms coordinates in a way that stretches and folds space, creating a chaotic mixing effect. This modified version provides a robust mechanism for diffusing and permuting image data, ensuring the confidentiality and integrity of the information. Diffusion and permutation are standard cryptographic techniques used to obscure the relationship between the plaintext and ciphertext.

The study introduces two new concepts, mixed scrambling and mixed diffusion, which are integrated into a variety of encryption schemes designed to accommodate diverse user needs and security requirements, providing a flexible and adaptable solution for image protection. These techniques aim to improve the robustness of the encryption by combining different scrambling and diffusion methods, making it more difficult for attackers to break the encryption.

The investigation draws heavily on the mathematical foundations of topological quantum computation, particularly the properties of anyons, and applies these principles to the development of practical image processing algorithms, demonstrating a rigorous mathematical underpinning for the proposed techniques. Anyons are quasiparticles that exhibit exotic exchange statistics, differing from bosons and fermions, and are considered promising candidates for building fault-tolerant quantum computers. References to Temperley-Lieb recoupling theory, a mathematical framework for describing angular momentum in quantum mechanics, and Chern-Simons theory, a topological quantum field theory, further solidify the theoretical basis of the research. The incorporation of hyperchaos – chaotic behaviour in higher-dimensional systems – generated from systems like the Lorenz system, suggests an exploration of chaotic dynamics for enhancing the randomness and security of the encryption process. The Lorenz system is a set of three coupled differential equations that exhibit chaotic behaviour, often used as a model for atmospheric convection.

Recent publications from 2024 and 2025, alongside the consistent contributions of C. Levaillant, demonstrate an active and evolving research landscape, indicating ongoing progress in the field. Future research will focus on implementing these algorithms on quantum hardware and evaluating their performance against known attacks, paving the way for secure quantum communication networks.