Abstract: As an important branch of information security, digital image encryption technology has become a major research hotspot. At present, most image encryption schemes adopt low-dimensional geometric transformations, which suffer from small periodicity and relatively fixed element values, making them difficult to resist exhaustive attacks. In contrast, high-dimensional geometric transformations are not easy to construct. Inspired by the two-dimensional Fibonacci transform and the three-dimensional Fibonacci-like transform, this paper proposes a quantum image scrambling algorithm based on high-dimensional generalized Fibonacci transforms. First, according to the principle of geometric sequences, two special high-dimensional integer matrices with determinant equal to 1 are constructed, and the high-dimensional generalized Fibonacci transform matrix is obtained via matrix operations. Second, due to the large periodicity of high-dimensional geometric transformations, decryption by directly using the periodicity is infeasible; therefore, the inverse transform of the high-dimensional generalized Fibonacci transform is constructed. Finally, based on the novel enhanced quantum representation(NEQR)model, the high-dimensional generalized Fibonacci transform and its inverse are applied to the quantum image encryption and decryption processes, respectively. The proposed algorithm features flexible and diverse transformation formulas and can generate high-dimensional encryption matrices. Taking 8-bit grayscale image encryption as an example, the effectiveness of the algorithm is verified. Simulation results show that the algorithm achieves satisfactory encryption and decryption effects, has a large key space and strong key randomness, and exhibits good resistance against attacks.

Key words: information security, quantum image encryption, quantum image representation model, high-dimensional geometric transformation, geometric progression, generalized Fibonacci transformation