A flexible image cipher based on orthogonal arrays
Introduction
In the era of big data, digital image is one of the common data types. Digital images always carry a lot of secret information, so image security has become increasingly important along with the rapid development of computer and Internet technology. Image security contains many aspects, such as image encryption, image authentication, digital fingerprint and copyright protection. Among these techniques, image encryption is an efficient one to protect image information by transforming one image into an unrecognizable form.In the past decades, chaotic systems have played a vitally important role in image encryption [1], [3], [11], [14], [23], [25], [28], [33], [34], [35]. Chaotic systems have sensitivity to initial conditions, unpredictability and ergodicity [8], [26], which are very suitable for cryptography. However, there are also some disadvantages when the chaotic systems are applied to cryptography. An obvious disadvantage is that cryptosystems are defined on finite sets, whereas chaotic systems have meaning only in the real number field [32]. To solve this problem, discretization is necessary. Unfortunately, some problems always arise along with the process of discretization, such as limited precision, periodic behaviors [26], [32]. These problems may affect the security of cryptosystems directly. To enhance the security, many new techniques have been introduced into image encryption [4], [7], [12], [13], [19], [21], [24], [29]. Among these techniques, the utilization of combinatorial configurations presents great superiority. The typical example of combinatorial configurations applied in image encryption is Latin square. A Latin square of order n is an array defined on a set of n distinctive symbols, where each symbol appears exactly once in each row and each column [6]. Latin squares have uniformity (each symbol appears equally often), discreteness (Latin squares are defined over finite sets), and 2D matrix form. With these characteristics, many image encryption algorithms based on Latin squares have been proposed [5], [10], [13], [15], [18], [29], [30]. These algorithms mainly use Latin squares to generate some 1D maps, and then use these maps to scramble the 2D images. Latin cube is the high-dimensional form of Latin square. A Latin cube of order n is an cube defined on a set of n distinctive symbols, where each symbol appears exactly once in each row, each column, and each file [6]. Latin cubes also have uniformity and discreteness. In particular, Latin cubes have the 3D attribute. With the 3D attribute of Latin cubes, Xu and Tian propose an image encryption algorithm in [31]. It firstly uses a group of orthogonal Latin cubes to generate several 3D maps, and then uses these maps to scramble the 3D bit matrices of images. By the aid of Latin cubes, the algorithm achieves a desirable encryption effect.
In this paper, we will introduce another combinatorial configuration, orthogonal array, to image encryption. Orthogonal array (OA) was introduced by Rao over half a century ago [16], [17]. Applications of OAs have arisen in many areas, such as experimental design, software testing and medicine. OAs bear close relations to Latin squares and Latin cubes. In an OA with , any three columns correspond to a Latin square, any four columns correspond to a pair of orthogonal Latin squares; in an OA with , any four columns correspond to a Latin cube, any six columns correspond to a group of orthogonal Latin cubes. Although Latin squares and Latin cubes have been widely used in image encryption, there are few efforts to apply OAs to image encryption. Actually, OAs have some excellent properties which are very suitable for image encryption:
(1) Analogous to Latin squares and Latin cubes, OAs are also defined over finite sets, then they do not need discretization when they are applied to cryptosystems.
(2) An OA has multiple columns, and each element appears equally often in each column. It means that each column has a uniform histogram, then each column can be used for substitutions on images.
(3) OAs have orthogonality. Specifically, in an OA with , any two columns are orthogonal. It means that any two columns can generate a 2D mapping for permutations on images. Based on the above properties, we can choose different columns of OAs for different plaintext images to perform the permutations and substitutions, then the encryption process is highly plaintext-related. In most image encryption algorithms, once the secret key is determined, the sequences for substitutions and the maps for permutations are also determined. In this situation, the same plaintext image corresponds to the same cipher image. To obtain different cipher images from the same plaintext image, the value of secret key has to be changed. However, changing secret keys frequently means increased computation load. In the proposed algorithm, we increase two public parameters to control the encryption and decryption. If the values of public parameters are changed, the substitution and permutation operations on plaintext images will be different, then the same plaintext image will correspond to different cipher images. Since the secret key is not changed in this process, the OA does not need to be reconstructed, then no extra calculation burden is increased. The security and efficiency can both achieve a satisfactory level in the proposed cryptosystem. The rest of this paper is organized as follows: Section 2 presents some basic definitions and conclusions. Section 3 describes the encryption and decryption processes in detail. Section 4 evaluates the proposed algorithm, and Section 5 concludes the paper finally.
Section snippets
Logistic map
Definition 1 Logistic map is a one-dimensional map defined by:where is the system parameter, . is a floating-point number in .[14]
Orthogonal array
Definition 2 An orthogonal array of size n, with q constraints (or of degree q), s levels (or of order s), and strength t, denoted , is an array with entries from a[6]
The proposed image encryption algorithm
To standardize the encryption process, the image processing block is set to be a block. In the following description, P denotes a plaintext image, C denotes its cipher image; O is an orthogonal array OA is the secret key, and are two parameters. According to the Kerckhoffs’ principle [20], a cryptosystem should not contain a secret parameter except for the secret key, so we set the parameters and public.
Simulation results and efficiency analysis
In this section, the simulation results of the proposed algorithm are given. Furthermore, we make comparisons between our algorithm and eight other representative algorithms [1], [3], [11], [22], [23], [24], [29], [33]. All the simulation experiments are conducted under C2010, in a computer with Microsoft Windows 10 operating system, Intel Core i5-3320 M 2.60 GHz processor and 3.22 GB RAM. In the simulation experiments, we have tested a large number of images and secret keys. For ease of
Conclusion
In this paper, we introduce a novel instrument, orthogonal array, to image encryption. Orthogonal array is a typical combinatorial configuration with some excellent cryptographic properties. One orthogonal array with specified parameters can generate multiple substitution sequences and permutation maps for image cipher. To utilize this property, we set two public parameters to control the selection of substitution sequences and permutation maps in the proposed algorithm. When the Internet
CRediT authorship contribution statement
Ming Xu: Methodology, Software. Zihong Tian: Conceptualization.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by the National Natural Science Foundation of China [Grant No. 11871019], Foundation of Hebei Education Department of China [Grant No. QN2019127], Natural Science Foundation of Hebei Province [Grant No. A2018210120].
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