Double color image encryption based on fractional order discrete improved Henon map and Rubik’s cube transform

Highlights

• A double color image encryption method based on DNA computation and chaos is proposed.

• A FODIHM is used as a pseudo-random number generator.

• A Rubik’s cube transform method is proposed.

• The CAT transform based on fractional discrete Logistic map is used to improve the security performance.

Abstract

A double color image encryption method based on DNA (deoxyribonucleic acid) computation and chaos is proposed. Differently from the conventional algorithms, double color images are encrypted at the same time so that we can save information of each other, which makes the encryption more safe and reliable. In addition, a new chaotic fractional order (FO) discrete improved Henon map (FODIHM) is proposed as a pseudo-random number generator. To ensure the plain-image sensitivity of the encryption algorithm, the initial value of FODIHM is calculated from the hash value of the color image (SHA-256) and from the three additional keys entered by the user. Furthermore, a Rubik’s cube transform scrambles the pixels in each color component of the two images. Then, each pixel in each color component of the two images is diffused by means of different DNA coding rules. Finally, the CAT transform, based on FO discrete Logistic map and the classic XOR, is used to further improve the security performance. The key space size of the proposed algorithm is of order $10^{135}$, which is about 30 orders of magnitude higher than those available in the literature. The information entropies are 7.9974 and 7.9973, which are very close to the ideal entropy value of 8. The values of the unified average changing intensity (NPCI) are 99.630 and 99.623, while the number of pixels change rate (UACR) are 33.473 and 33.553, which are also close to the ideal NPCR and UACR value of 99.6094 and 33.4635, respectively. The numerical results and security analysis prove that the algorithm has good resistance to several classic attacks.

Introduction

With the rapid development of the Internet of Things (IoT) and Information and Communication Technology (ICT), the human society is increasingly oriented towards communications using the Internet. As text, images and videos are important carriers of information, the security in the transmission process is a key research issue. In the past 10 years, with the advances of technology, the proportion of color image information in multimedia became popular. Differently from text information, images have the characteristics of containing data capacity and strong correlation between pixels. However, traditional encryption methods are not suitable for image encryption and, therefore, various image-oriented algorithms have been advanced. We find in the literature image encryption schemes based on compressive sensing [1], [2], DNA computing [3], [4], [5] and chaos [6], [7], [8], [9], with the algorithms satisfying confusion and diffusion requirements [10], [11].

Chaotic systems have attracted the attention due to their sensitivity to initial values and the complex dynamics behavior. A number of image encryption algorithms based on chaotic systems were proposed [12], [13], [14], [15], [16]. In this case the chaotic system is used as a pseudo-random sequence generator to improve the security performance. However, traditional low-dimensional integer-order chaotic systems lead to a reduction in the key space of the algorithm. To overcome this shortcoming, fractional-order discrete chaotic systems were tested in image encryption because of their larger parameter selection space and dynamic characteristics [8], [9], [17]. Recent results show that the image encryption based merely on a chaotic system reveals loopholes [18], [19], [20]. Therefore, combining other methods to improve the security of encryption algorithm is currently an on-going research topic.

The DNA technology has high information density and high parallelism [21], [22]. Therefore, several researchers combined the DNA technique and chaos to design image encryption algorithms [23], [24], [25], [26]. The core of this technology is the DNA encoding, decoding and operation, so that the binary pattern of image information is transformed into the DNA sequence through coding rules. However, the DNA encryption algorithms still have shortcomings, such as the use of the same encoding rule for all pixels [27], [28]. This means that the encoding rule can be discovered by hackers, resulting in a limited security. On the other hand, the algorithm given by the DNA encoding is not sensitive to the plain image and secret key. On this issue, Guesmi et al. [29] proposed an encryption scheme based on DNA coding and SHA-2, by combining the initial value of the chaotic system with the HASH value of the plain image. This approach solved the problem of the algorithm’s insensitivity to the plain image.

During the transmission of digital image, multiple images need often to be encrypted. If we encrypt each image separately, then the efficiency is limited since it does not take advantage of the multiple images. Situ and Zhang proposed the multiple-image encryption algorithm [30]. As a special case of multiple-images encryption, double-image encryption is also of interest. The existing double-image encryption algorithms are mostly aimed at gray images. Zhou and Jiang [31] proposed a double-image compression-encryption scheme based on a co-sparse analysis model and random pixel exchanging. Mohammed and Saadon [32] developed a optical double-image encryption based on sparse phase information. Other relevant encryption algorithms can be found in [33], [34], [35]. Compared to the gray images, the colored ones are more common in information transmission. Nonetheless, due to the strong correlation between the three color channels, the above double-image encryption algorithm may not be able to be used directly.

Based on these ideas, a new double color image encryption algorithm is developed. First, a FODIHM is used as a pseudo-random number generator. The fractional-order discrete chaotic system has a parameter selection range far exceeding the one of the integer-order chaotic system, which greatly improves the key space of the encryption. Second, a Rubik’s cube transform method is proposed. The six color components of the double color image are used to compose a spatial structure similar to the Rubik’s cube. Based on this idea, the scrambling operation is performed to break the strong correlation between pixels while allowing both parts to save information of each other. Therefore, even if a certain image is completely lost, the plain image can be partially restored. Third, a CAT transform based on the discrete fractional Logistic map is implemented. The choice of parameters is related to the hash value of the plain image, which can effectively improve the security performance and better resist the chosen and known plain text attacks.

The rest of the paper is organized as follows. In Section 2, some important definitions and lemmas are introduced, and the novel FODIHM is formulated. In Section 3, the double color image encryption scheme based on the FODIHM is proposed. In Section 4, several simulation results and security tests are discussed. Finally, in Section 5 the main conclusions are given.