高维广义Fibonacci变换的量子图像置乱技术

doi:10.6040/j.issn.1671-9352.0.2025.044

摘要/Abstract

摘要： 数字图像加密技术作为信息安全的重要分支,已成为重要的研究热点。目前图像加密多采用低维几何变换,低维几何变换具有小周期性,元素值较为固定,难以抵御穷举攻击,而高维几何变换却不容易构造。本文受二维Fibonacci变换和三维类Fibonacci变换的启发,提出一种基于高维广义Fibonacci变换的量子图像置乱算法。首先,基于等比数列原理,构造两个行列式等于1的特殊高维整数矩阵,使用矩阵操作得到高维广义Fibonacci变换矩阵。其次,由于高维几何变换的大周期性,使用变换的周期性进行解密并不可行,因此构造了高维广义Fibonacci变换的逆变换。最后,基于新型增强量子图像表示模型(novel enhanced quantum representation, NEQR),本文将高维广义Fibonacci变换及其逆变换分别用于量子图像加密和解密过程中。本文算法变换公式多样灵活,可以得到维度很高的加密矩阵。以8位灰度图像加密为例,验证了算法的能行性。仿真实验表明,该算法加密解密效果良好,密钥空间巨大,密钥的随机性较强,具有很好的抗攻击能力。

关键词: 信息安全, 量子图像加密, 量子图像表示模型, 高维几何变换, 等比数列, 广义Fibonacci变换

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

中图分类号:

TP391

邹玮刚,黄江燕,曹锋, 杨火根. 高维广义Fibonacci变换的量子图像置乱技术[J]. 《山东大学学报(理学版)》, 2026, 61(6): 80-94.

ZOU Weigang, HUANG Jiangyan, CAO Feng, YANG Huogen. Quantum image scrambling technology based on high dimensional generalized Fibonacci transform[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2026, 61(6): 80-94.

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