A companion matrix-based efficient image encryption method

The increased use of multimedia applications to share digital images has raised concerns about their security during transmission and storage as well. Thus, the need for integrating the chaotic system with compressive sensing became an important and potentially successful method for enhancing the image security. However, in the integration of a single One-dimensional (1-D) chaotic system with compressive sensing, there is a significant drawback that is the limited chaotic behaviour and key space, which makes it vulnerable against brute force and statistical attacks. Hence, enlarging the key space to improve security by using a single One-Dimensional (1-D) chaotic system and making it resilient against brute force attacks still needs to be addressed.

In this paper, we propose an encryption method that makes use of the notion of a companion matrix and a single One-Dimensional (1-D) chaotic system to enlarge the key space. This method converts the grayscale image into a sparse representation. Thereafter, this sparse matrix is shuffled by applying the Arnold Cat Map, where the parameters for this map are generated through the usage of a One-Dimensional (1-D) Piecewise Linear Chaotic Map. Furthermore, we construct the key matrix by computing the eigenvalues of the companion matrix, and then we diffuse the cipher image to improve the security against statistical attacks.

Experimental results demonstrate that the proposed method balances the security and image reconstruction quality effectively. The advantage of the proposed method is that even by using a single One-Dimensional (1-D) chaotic system (i.e., faster in implementation), by using the concept of companion matrix, it achieves a significantly larger key space of $2^{400}$ that is larger than the several existing state-of-the-art methods that use hyperchaotic systems.