With the wide development of internet technology and social networking services (SNS), more and more digital images, videos are transmitted and stored online, therefore the security of multimedium data has received much attention. Image encryption is an effective method to secure the digital images. Traditional encryption schemes, for example, data encryption standard (DES), advanced encryption standard (AES) and Ron Rives Adi Shamir Leonard Adleman (RSA) are not employed to effectively encrypt images because of the intrinsic features of the images, such as high correlation between adjacent pixels, bulk data capacity and high redundancy. Chaotic systems are characterized by sensitivity to initial conditions and parameters, ergodicity, pseudo randomness and topological transitivity, thus they are very suitable for image encryption.^[1–3]

Since Matthews presented the first chaos-based encryption algorithm in 1989,^[4] many chaos-based image encryption methods have been introduced.^[4–10] For example, a fast image encryption algorithm based on only blocks in cipher text was given in Ref. [5], and Logistic map and skew tent map were used for constructing chaotic sequences. Zhou et al. ^[6] introduced a general chaotic framework named as cascade chaotic system (CCS), generated many new chaotic maps by using two one-dimensional (1D) chaotic maps as seed maps based on this framework, these newly generated chaotic maps have more unpredictable and better chaotic performance, and they have been applied to image encryption. Ye^[7] presented a block-based image encryption algorithm by using wave function and chaotic system, and the random sequence produced by Chen chaotic system was employed to obtain the source point in the wave and give a diffusion matrix for modular operation. Recently, a novel chaos-based image encryption scheme based on compressive sensing was introduced,^[8] and one-dimensional skew tent map was used to generate the measurement matrix. In view of the secure problems of low-dimensional chaos in image encryption and cycle degradation caused by computer finite precision, Zhang and Wang^[9] put forward a novel image encryption algorithm by using a mixed linear-nonlinear coupled map lattice, and Tong et al. ^[10] designed a perturbed high-dimensional chaos according to Devaney and topological conjugate definition, and employed it for image encryption as pseudo-random sequence generators.

However, a number of encryption algorithms have illustrated security problems and proved to be unsafe.^[11–25] In 2000, an efficient hierarchical chaotic image encryption (HCIE) algorithm was presented,^[11] and it divided the plain image into blocks with the same size, and then operated position permutation at intra-block and inter-block levels. Recently, this algorithm has been cracked, and the security of HCIE against ciphertext-only attack is much lower, and it should be combined with confusion to provide high security.^[12] Li et al. ^[13] gave a novel double-image encryption scheme by using chaos-based local pixel scrambling technique in gyrator transform (GT) domains, and Chen et al. ^[14] found that there is serious cross-talk disturbance in the decrypted phase-based image when the cipher image undergoes noise perturbation or occlusion attack. Liu et al. ^[15] evaluated the security of the encryption algorithm in Ref. [16], and found that it can be broken with only two pairs of known plain image and corresponding cipher images, pointing out that it has some security issues, such as insensitivity to the change of the plain image and secret keys. Zhang and Liu^[17] proposed a novel image encryption method by using the permutation-diffusion architecture and skew tent chaotic map. Wang and He^[18] found that this method had two flaws including that the key was fixed regardless of the plain image, and the permutation vector was constant for different original images, and it was vulnerable for chosen plaintext attack. Subsequently Eslami and Bakhshandeh^[19] gave an improved method based on Ref. [17], and recently Akhavan et al. ^[20] cryptanalyzed this improved algorithm by using differential attack and found that it was not sensitive enough to the plain image. Recently, Wang et al. ^[21] analyzed a parallel sub-image encryption scheme combined with high-dimensional chaotic system in Ref. [22], and found that the produced keystream was unchanged for different original images, and the plain image may be recovered through chosen-plaintext attack. Li et al. ^[23] reevaluated the security of a novel image encryption algorithm in Ref. [24] and found that the scheme can be broken with only two known plain images, and they also cryptanalyzed the Fridrich’s chaotic image encryption scheme.^[25] By analyzing these papers, the main flaw is that these encryption schemes have not enough sensitivity to the plain image, “one original image, one secret key” cannot be achieved, and they have less resistance to chosen-plaintext and known-plaintext attacks. Therefore, more secure and efficient image encryption methods are needed to be introduced and employed in secure communication field.

The classical encryption process consisting of confusion and diffusion was first introduced by Shannon,^[26] and afterwards many image encryption schemes were designed based on it.^{[1,2,4–8,27–41]} There are two methods employed in the confusion process. One is pixel level permutation^[27–29] and the other is bit level confusion.^[30–32] The former can change the positions of image pixels while the values of pixels are unchanged, and the latter can not only permute the positions of pixels, but also modify the values of the image pixels. Thus, bit level confusion is considered to be a perfect permutation method for the better encryption efficiency,^[33] and a great many of image encryption schemes using bit level permutation have been introduced. For example, a new permutation method to confuse and diffuse the grayscale image at the bit level was proposed in Ref. [34], Arnold cat map was used to permute the bits, and logistic map was adopted for diffusing the permuted image. In view of the limits of Arnold cat map, it requires that the image must be square. Besides, logistic map has a dense set of periodic windows for a certain range of parameters, which restricts its application in image encryption. To alleviate the problem, Liu and Wang^[35] presented a color image encryption method using bit-level permutation and high-dimensional chaotic system. Firstly, a color image (M × N) was converted into a grayscale image (M × 3N) through R, G, B splitting, and next a binary matrix was obtained by bit plane decomposition; secondly chaotic sequence generated from piecewise linear chaotic map (PWLCM) was employed to permute the two-dimensional (2D) bit matrix; thirdly, Chen’s chaotic system was used for diffusion. Recently, Martin et al. ^[36] gave a color image encryption algorithm based on three-dimensional (3D) cellular automata and chaotic maps, and 24 2D bit planes could be obtained from the bit plane decomposition of the color image, the 2D Cat map was used to permute the positions in each bit plane, and 3D cellular automata was employed for diffusion. In the bit level permutation process, after bit plane expansion of the original image, the 2D image is converted into a 3D bit matrix (the width, height and bit length), and each bit plane has its own weight. A majority of bit level confusion algorithms were manipulated on the 2D bit planes,^[37–40] they decomposed the plain image into many bit planes. The confusion process of the plain image was transformed into that of the bit plane, more bit planes arose and more confusion steps were done. Moreover, some scholars have presented heterogeneous bit level confusion methods in view of the different importance of bit planes.^[41] But in the final analysis, it was the confusion scheme of the 2D matrix or 1D vector from the 2D matrix. Few image encryption schemes confuse the original image as a 3D bit matrix, and thus bits cannot move to any position and any bit plane and the movement range of bits are limited. Thus, we may consider the original image as a 3D bit matrix, and then study the method to make the bit elements randomly move to any position in the matrix, thereby enhancing the permutation effect and upgrading the security level of the encryption scheme.

Based on the above analyses, a new image encryption algorithm using 3D Brownian motion and chaotic systems is introduced in the paper. Brownian motion is a random movement form of particles, generally existing in the liquid and gas, particles move fast in the 3D space, and the movement traces distribute in the whole space. Our contributions are as follows: first, we employ the 3D Brownian motion to confuse the bits in the 3D bit matrix, where the bits are regarded as Brownian motion particles, and use the Monte Carlo method to simulate Brownian motion. Due to the increase of the image size, the 3D bit matrix is split into many sub blocks to save the execution time, a new block confusion based on 3D Brownian motion (BCB3DBM) is given to confuse the positions of the bits within the blocks, and logistic-tent system (LTS) is employed to generate the direction of particle movement. Besides, block confusion based on position sequence group (BCBPSG) is introduced to confuse the positions of the sub blocks, random chaotic sequences are produced by a four-order memristive chaotic system and sorted, a proper position sequence group is picked out according to the plain image and utilized to change the position of the sub block. Our confusion method can fully change the positions of the bits and their weights, the bits in one bit plane can appear at any position and in any bit plane theoretically, and it greatly increases the encryption performance. Furthermore, diffusion is adopted to increase the security of the algorithm. In order to improve the algorithm sensitivity to the plain image and the ability to resist the known-plaintext and chosen-plaintext attacks, the initial values and parameters of chaotic systems are obtained by the SHA 256 hash value of the original image.

The rest of the present paper is organized as follows. In Section 2, preliminary studies including a four-order memristive chaotic system, LTS and 3D Brownian motion are described. Section 3 gives our encryption scheme in detail. Section 4 presents the simulation results, and security performances of the proposed algorithm are analyzed in Section 5. Conclusions are drawn in the last section.

In this section, several experiments are conducted on multiple images, Lena, Baboon and Peppers of size 512 × 512 are chosen as the original images, and all the experiments are performed on a personal computer with the following hardware environment: 2.5-GHz CPU, 4-GB memory, and Windows 7 operating system, and the compiler software is Matlab 2014a. Some keys we use here are as follows: α = 9, β = 13, γ = 2, p = 0.003, a = 0.8, b = −0.03, c = 0.01, and k = 0.05, the Brownian movement rounds R = 100, the size of every sub block image of 8 × 8 × 8 and the iterating parameter of the memristive chaotic system, l = 500. Figures 6(a), 6(d), and 6(g) are the plain images, and after BCB3DBM and BCBPSG, we obtain the confused images Figs. 6(b), 6(e), and 6(h), and figures 6(c), 6(f), and 6(i) are their corresponding cipher images.

Fig. 6.

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	Fig. 6. (color online) Encryption results. showing (a) original image of Lena, corresponding (b) confused image, and (c) cipher image; (d) original image of Baboon, corresponding (e) confused image, and (f) cipher image; (g) original image of Peppers, corresponding (h) confused image, and (i) cipher image.

From the results, we can conclude that the confused images and cipher images are similar to noise images, any useful information cannot be obtained from them, and thus the proposed encryption algorithm has a good encryption effect and it can be applied to the secure transmission of private information.

Key sensitivity of an image encryption algorithm can be assessed in two ways: (i) when the plain image is the same and the key has a slight change, a completely different cipher image should be obtained; (ii) the plain images cannot be recovered even though there is a slight difference between the encryption key and decryption key. Next, we will test the performance of our algorithm from these two aspects.

Firstly, Lena (Fig. 6(a)) is regarded as the plain image, and its cipher image is shown in Fig. 6(c). The SHA 256 hash value K is “b32171537de519d7c1c83ff25989a559532c6b269a303 ac19d3d0a906bb5e250”, then we modify K into K¹ = a32171537de519d7c1c83ff25989a559532c6b269a303ac19 d3d0a906bb5e250” with a bit change, and the other parameters are the same. The cipher image is shown in Fig. 9(a), and figure 9(b) is the subtraction of Fig. 6(c) from Fig. 9(a), the NPCR is 99.61%, the histogram of Fig. 9(b) is illustrated in Fig. 9(c).

Fig. 9.

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	Fig. 9. (color online) Encryption results with the key changed, showing (a) cipher image with K1 , (b) difference between two cipher images, and (c) the histogram of Fig. 9(b).

Secondly, we use a different decryption key to decrypt the cipher image shown in Fig. 6(c). When K is employed, the recovered image is shown in Fig. 10(a), and it is the same as the plain image. When we use K ₁ to decrypt, we obtain the result presented in Fig. 10(b) and it is something like noise, we cannot obtain any information from it. The subtraction of Fig. 10(b) from Fig. 6(c) is illustrated in Fig. 10(c), and the NPCR is 99.60%.

Fig. 10.

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	Fig. 10. Decryption results with different keys, showing (a) recovered image with K, (b) recovered image with K 1, and (c) subtraction of Fig. 10(b) from Fig. 6(c).

From the above analyses, it is clear that the proposed algorithm is highly sensitive to the secret key and the plain image, and it is secure to withstand statistical attack and brute force attack.

Occlusion attack means that the cipher image is damaged in the transmission process, and the more data it loses, the more distortion the recovered image has. The peak signal-to-noise ratio (PSNR) is employed to calculate the quality of the recovered image after being attacked. For a grayscale image, the PSNR may be expressed as

(27)

(28)

The results are shown in Fig. 11 and Table 6. Figures 11(a)–11(d) give the cipher images of Fig. 6(c) with 1/16, 1/8, 1/4, and 1/2 occlusion respectively, and figures 11(e)–11(h) show the corresponding recovered images whose PSNRs are 37.9059, 34.8570, 31.8821, and 30.5522, and there are 11.45%, 22.56%, 45.12%, and 49.81% difference between the recovered images and the plain image (shown in Fig. 6(a)). In Fig. 11(h), it is evident that the recovered image can be easily recognized though the cipher image lost 50% information. Therefore, the proposed encryption algorithm has a superb robustness to the occlusion attack and is comparable to that in Ref. [51].

Fig. 11

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	Fig. 11 (color online) Occlusion attack analysis results of the cipher images with (a) 1/16, (b) 1/8, (c) 1/4, and (d) 1/2 occlusion, and recovered images with (e) 1/16, (f) 1/8, (g) 1/4, and (h) 1/2 occlusion.

Table 6.

Table 6.

Table 6. Quantitative results of resisting occlusion attack. . Occlusion Ours Ref. [51] MSE PSNR Correlation NPCR UACI PSNR 1/16 10.5316 37.9059 0.6132 0.1145 0.0232 39.23 1/8 21.2510 34.8570 0.4602 0.2256 0.0461 − 1/4 40.1568 32.8821 0.3151 0.4512 0.0922 33.37 1/2 57.2614 30.5522 0.0635 0.4981 0.1432 30.40

Table 6. Quantitative results of resisting occlusion attack. .

In the image transmission, channel noise is inevitable. Gaussian white noises with zero mean and three different variances are added to the cipher image (Fig. 6(c)), the respective images are illustrated in Figs. 12(a)–12(c), the recovered images are shown in Figs. 12(d)–12(f), with the PSNRs of 28.5242, 28.2274, and 28.1247, and other quantitative results of resisting noise attack are given in Table 7. With the increase of the variance of noise, the MSE values increase, PSNR values decrease, and the correlations of adjacent pixels in the recovered images are very low. The NPCR and UACI values are less than those in Ref. [52], which means that our algorithm is better. When the variance of Gaussian white noise is 0.0005, the recovered image (Fig. 12(f)) has the worst image quality, and it can be roughly recognized visually, which means that our algorithm can withstand the noise attack to a certain extent.

Fig. 12.

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	Fig. 12. Cipher and decrypted images under Gaussian noise with different level of noise: cipher images of (a) mean = 0, variance = 0.0001; (b) mean = 0, variance = 0.0003; (c) mean = 0, variance = 0.0005; decrypted images of (d) mean = 0, variance = 0.0001; (e) mean = 0, variance = 0.0003; (f) mean = 0, variance = 0.0005.

Table 7.

Table 7.

Table 7. Quantitative results of resisting noise attack. . Variance Ours Ref. [52] MSE PSNR correlation NPCR UACI NPCR UACI 0.0001 91.3405 28.5242 0.1821 0.9269 0.2050 0.9920 0.2844 0.0003 97.8001 28.2274 0.1220 0.9561 0.2247 0.9961 0.2864 0.0005 100.1408 28.1247 0.1056 0.9656 0.2325 0.9974 0.2880

Table 7. Quantitative results of resisting noise attack. .

Many encryption schemes are broken by known-plaintext and chosen-plaintext attacks. In the present paper, two methods are manipulated to upgrade the ability to resist known-plaintext and chosen-plaintext attacks. On the one hand, the chaotic sequences used in confusion and diffusion processes are generated from the four-order memristive chaotic system and LTS, the initial values and parameters of the chaotic systems are calculated by SHA 256 hash value of the plain image, and thus our algorithm depends on the original image. On the other hand, 24 different position sequence groups are obtained in BCBPSG, a certain group is picked out to confuse the sub blocks, and its choosing method has high relationship with the plain image. Therefore, in theory, our encryption algorithm has strong ability to withstand the known-plaintext and chosen-plaintext attacks.

The cryptanalysts often choose special images, such as all black and all white to attack the cryptosystem. They want to obtain the key by analyzing the cipher images. Figure 13 shows the encryption results of all white and black images by using our method. The entropies and correlation coefficients of the plain, cipher images of all white and all black are listed in Table 8. From the figure and the table, we can find that the histograms of the cipher images distribute evenly and uniformly, the correaltions of adjacent pixels decrease to nearly 0, and any infromation cannot be obtained from the ciphers. So our algorithm can resist the known-plaintext and chosen-plaintext attacks.

Fig. 13.

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	Fig. 13. (color online) Experimental results of (a) all white, (b) cipher image of all white, and (c) histogram of the cipher image; (d) all black, (e) cipher image of all black, and (f) histogram of the cipher image.

Table 8.

Table 8.

Table 8. Entropies and correlation coefficients of the plain, cipher images of all white and all black. . Images Entropies Correlation coefficients Horizontal Vertical Diagonal All white 0 – – – Cipher of white 7.9993 0.0140 0.0011 0.0071 All black 0 – – – Cipher of black 7.9992 0.0018 0.0154 –0.0242

Table 8. Entropies and correlation coefficients of the plain, cipher images of all white and all black. .

Speed is an important issue for a cryptosystem. The proposed encryption scheme consists of block confusion based on 3D Brownian motion (BCB3DBM), block confusion based on position sequence group (BCBPSG) and diffusion. Before the confusion process, the original image is transformed into a 3D cubic matrix, the aim is to reduce the iterative times of memristive chaotic systems and improve the encryption speed. In addition, the cubic matrix is divided into many sub blocks. The permutation process is manipulated by blocks, and it can run in parallel, thereby saving time. The size of the sub block image and the test image can affect the encryption speed. For a 512 × 512 gray image, the encryption times for 4 × 4 × 4, 8 × 8 × 8, 16 × 16 × 16, 32 × 32 × 32, 64 × 64 × 64 blocks are tested, and the relationship between execution time and the size m of the sub block image is shown in Fig. 14. It is evident that the encryption time is the shortest when m = 8, and it is 5.69 s. The proposed algorithm has shorter encryption time than Wang’s^[50] algorithm (15.14 s) and Zhou’s^[32] methods (11.42 s). Therefore, our encryption algorithm is more efficient and suitable for the secure communication.

Fig. 14.

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	Fig. 14. (color online) Relationship between block size and encryption time.

A novel chaos-based image encryption algorithm using the three-dimensional (3D) Brownian motion is given in this paper. We present the block confusion based on 3D Brownian motion (BCB3DBM) and block confusion based on position sequence group (BCBPSG). The BCB3DBM is used to confuse the position of the bits within the plain-image sub blocks, and the subsequent BCBPSG is employed to permutate the positions of the sub blocks. In theory, the permutation process can make the bits in one bit plane appear at any position and in any bit plane, and our confusion scheme has a good scrambling result. Moreover, we also utilize confusion to enhance the encryption effect. In the encryption scheme, the four-order memristive chaotic system and LTS are used to generate the chaotic sequences. Besides, SHA 256 hash function of the plain image is employed to produce the initial values and parameters of the chaotic systems, and therefore our encryption scheme has strong sensitivity to the plain image.

In the security analyses, we employ 10 different methods, i.e., key space analysis, histogram analysis, information entropy analysis, correlations between two adjacent pixels, and resistance to differential attacks, key sensitivity analysis, resisting occlusion attack analysis, resisting noise attack analysis, resisting known-plaintext and chosen-plaintext attack analysis and speed analysis to assess the performance of this algorithm. The experiment results show that our method is secure and effective. It has potential applications in multimedia communication.