Image encryption based on three-dimensional bit matrix permutation

Highlights

• The problems of pixel- or bit-level permutation are analyzed.

• A 3D bit level permutation is proposed.

• A random visiting to plain-image is proposed.

Abstract

Lately, a number of image encryption algorithms that are either based on pixel level or bit level encryption have been proposed. However, not only pixel level permutation, but also bit level permutation has its intrinsic drawbacks. This paper proposes a new cryptosystem to address these drawbacks. Different kinds of permutation algorithms are first comprehensively analyzed and compared. Because, from a bit level perspective, an image can be considered as a natural three-dimensional (3D) bit matrix (width, height, and bit length), a new 3D bit matrix permutation is proposed, in which the Chen system is used to develop a random visiting mechanism to the bit level of the plain-image. By combining aspects of the Chen system with a 3D Cat map in the permutation stage, a new mapping rule is developed to map one random position to another random position (that is, double random position permutation) in the 3D matrix rather than using traditional sequential visiting to the plain-image. Simulations are carried out and the results confirm the security and efficiency of our new cryptosystem.

Introduction

The development of Internet technology and Social Networking Services (SNS) has led to an increase in online sharing of digital images and videos. Meanwhile, unauthorized access to or modification of private multimedia information has become a serious issue in the digital world. Although problems of this nature can be addressed by using information encryption, traditional encryption schemes, e.g., DES, and AES, are inappropriate for multimedia information encryption applications due to the intrinsic features of this kind of information, including a high correlation among the data in adjacent pixels, and the need for real time transformation [1]. Therefore, a new encryption mechanism capable of addressing the special demands of multimedia information is required. In this regard, recent years have seen significant developments in image encryption technology that use chaos theory together with permutation–diffusion architecture.

In his masterpiece, Shannon first introduced the idea of using permutation and diffusion to protect digital information [11]. In 1997, Fridrich proposed a similar architecture that uses two-dimensional (2D) dynamic systems to facilitate image encryption, thereby achieving a good balance between security and efficiency [12]. A number of image encryption algorithms have subsequently been proposed [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], and these algorithms can be classified into two categories according to the means by which pseudorandom numbers are generated, namely algorithms that are based either on chaos theory or on non-chaos theory. The first of these two categories, chaos-based image encryption, is based on the similarity between chaos dynamic systems and cryptosystems, including pseudorandom behavior, sensitivity to the initial condition, and ergodicity. In this category, one-dimensional (1D), high-dimensional, or hyper-chaotic systems are employed to generate pseudorandom number sequences to enable symmetric cryptosystems to be constructed. For example, in [6], one logistic map and one sine map are used to form a newly coupled 2D chaotic map in terms of which 2D-SLMM is employed to shuffle the plain-image, which can achieve good confusion and diffusion effects. In [14], a new 1D chaotic system is obtained by coupling the tent map and the sine map, which is then utilized to realize a pixel insert algorithm to permute the plain-image. A high-dimensional chaotic map has a more complex structure and pseudorandom properties, such as those used in Lorenz and Chen systems, and they are usually used to change the locations and values of the pixel set in image encryption algorithms. The Chen system, as applied in [10], is used to confuse and diffuse the three-color channel of the image, whereas in [9], the Chen system is employed as a pseudorandom number generator, and a temporary value feedback strategy, based on both confusion and diffusion, is used to encrypt an image with 256 Gy levels. The second of the two image encryption categories, non-chaos-based encryption, involves methods in which a non-chaotic dynamic system is used to generate a pseudorandom number sequence, which is then used to encrypt the image. For example, in [15], a new strategy based on balanced cellular automata is used to facilitate image encryption, whereas in [21], quantum Fourier transformation and a double random-phase encoding algorithm are used for color image encryption.

Bit level permutation (BLP) is a new approach that is employed at the confusion stage of image encryption and was first proposed in 2011 [1]. BLP considers an image to consist of a bit matrix to which all the encryption effects are applied; hence, the cipher-image should ultimately reflect the bit distribution of these encryption effects. In contrast, previous image encryption algorithms performed the encryption process at the pixel level, whereas BLP operates directly on each bit in the plain-image, rather than on a bit group at the pixel level. Recent times have seen the publication of a number of image encryption algorithms, most of which perform several rounds of 2D permutations among the different bit planes of the plain- image, following which the different permuted bit planes are combined to produce the confused (encrypted) image. For example, in [1], the four higher bit planes are permuted independently and the four lower bit planes are permuted together. In this particular case, the statistical information for each bit plane was not modified after the bit levels were confused. However, bit distribution plays a significant role in determining the extent to which an image encryption algorithm is able to secure an image. As illustrated in [2], there is a strong correlation between adjacent bit planes, and in this case, the BLP algorithm was used to restrict the destination location of one of the bits after confusion, which means that, one bit would only be able to move to another position within the same bit plane. The authors addressed this issue by employing a new “Expand and shrink” approach to partly eliminate this restriction, by allowing a bit that originally belonged to an even-numbered bit plane, to move to a destination that is located on any other even-numbered bit plane and this was also applied to bits located in odd-numbered planes.

However, the problem with this issue is that the bit matrix of an image is a natural 3D matrix, because an image with 256 Gy levels with a size of M×N can be considered a M×N×8 bit matrix, on the basis of which most existing algorithms try to use the combination of several 2D permutations to permute the 3D matrix. Furthermore, permutations using 2D chaotic maps or sorting algorithms experience repeated pattern problems, and the increasing number of elements in the sorting sequence results in a rapid increase in the execution time of the sorting algorithm.

In this paper, a new 3D permutation algorithm based on a coupled Chen system and a 3D chaotic map is proposed in an attempt to address the abovementioned problems. The new algorithm eliminates all the restrictions associated with bit level moving and satisfies the ideal situation of bit level permutation proposed in [2]. In the new permutation algorithm, not only the bit location but also the weight of each bit can be simultaneously modified. Furthermore, a new random visiting mechanism toward the 3D plain-image based on Chen system is employed in the first permutation stage, and after that 3D cat map govern the second the permutation stage. Simulations have been carried out and compared with three other image encryption algorithms, which indicating the security and efficiency of proposed scheme.

The remainder of this paper is organized as follows. Section 2 summarizes recently published permutation algorithms and the problems associated with the different approaches are highlighted and analyzed. In Section 3, the concept of a permutation over a 3D bit matrix is introduced and the superior permutation feature reported. Section 4 describes the diffusion phase. Section 5 summarizes the cryptosystem. Section 6 reports the simulation results of the proposed scheme and the comparable algorithm, following which a conclusion is drawn in the last section.