Permutation-based special linear transforms with application in quantum image encryption algorithm

Abstract

A cryptographic system based on substitution–permutation network involves various linear and affine transforms in order to create diffusion in the ciphertext. In this paper, special linear unitary transforms associated with permutations of n distinct objects are designed. These transforms are composed of n components each of which is a controlled NOT operation. The domain of special linear transforms is the $2^n$-dimensional Hilbert space ${{\mathbb {H}}}^{\otimes n}$, and these transforms define bijection from the set of computational basis onto itself. The latter characteristic of these transforms has enabled us to propose an efficient quantum image scrambling strategy. The application of special linear transform on the quantum state, which represents pixels positional information, results in the scrambled quantum state of positional information. On the other hand, its application on quantum state representing pixels value produces the encrypted quantum state. Accordingly, an efficient quantum image encryption algorithm based on special linear transforms and Chen chaotic dynamical system is presented for the novel quantum representation of color digital images (NCQI) model. Firstly, a selected pair of special linear transforms is employed to scramble the quantum image state of the original image. Then, the scrambled image is processed under the controlled special linear transforms determined by the values of three sequences generated from Chen chaotic system. The objective of this part is to encrypt color information of red, green and blue layers of the image. For an image of size $2^n \times 2^n$, the time complexity of the proposed quantum image scrambling method is 2n while the time complexity of the proposed quantum image encryption algorithm is $O(2^{2n})$, which is linear in the size of image. Finally, the simulation experiments are performed in order to measure the strength of proposed encryption algorithm. It is evident from the analyses of the simulation results that the newly proposed algorithm is fast, secure and reliable.