A novel image encryption based on Lorenz equation,Gingerbreadman chaotic map and S 8 permutation

Abstract

Internet is used as the main source of communication throughout the world. However due to public nature of internet data are always exposed to different types of attacks. To address this issue many researchers are working in this area and proposing data encryption techniques. Recently a new substitution box has been proposed for image encryption using many interesting properties like gingerbread-man chaotic map and S₈ permutation. But there are certain weaknesses in aforesaid technique which does not provide sufficient security. To resolve the security issue an enhanced version of existing technique is proposed in this paper. Lorenz chaotic map based confusion and diffusion processes in existing technique are employed. Lorenz map is used to remove strong correlation among the plain text image pixels. In diffusion stage a random matrix is generated through lorenz chaotic map and XORed with shuffled image. It the end, existing gingerbread-man chaotic map based S-box is applied to extract the final cipher text image. The proposed enhanced scheme is analysed by statistical analysis, key space analysis, information entropy analysis and differential analysis. In order to ensure the robustness and higher security of proposed scheme, results via Number of Pixel Rate Change (NPRC) and Unified Average Change Intensity (UACI) tests are also validated.

Keywords

Lorenz equation S-Box Gingerbreadman chaotic map confusion diffusion.

1 Introduction

Internet is considered as the most cheaper, faster and easier way to share data and information throughout the world. However the insecurity and open nature of Internet, information can be easily accessed, modified and interrupted by eavesdroppers. Multimedia application produced different types of data such as audio, videos, images which used the insecure wireless and wired IP networks for propagation. But both source and destination are not satisfied with from the security of public networks. They reported certain security issues during transmission and processing of multimedia contents. Therefore it is necessary that multimedia contents must be encrypted in such a way that it can not be accessible to anyone except its final destination. Cryptography is a science, through which one can conceal important multimedia information from illegal access and unauthorized person [1, 2]. Since many decades, cryptographic research community is looking for highly secure image encryption schemes [3–5]. In addition to security, researchers are also working to provide such techniques which can efficiently encrypt data in real-time applications.

Multimedia contents such as images, videos and audio are different from normal text and thereforemany traditional encryption algorithm such as Advanced Encryption Standard (AES), Data Encryption Standard (DES) and Rivest, Shamir, & Adleman (RSA) have a very extensive and complex encryption mechanism, which made it unsuitable for encryption of real time multimedia application [6, 7]. Some intrinsic features of images such as high correlation among pixels and high redundancy play a vital role in motivation of researchers to work for development of new image encryption schemes as per their requirements [8]. There exist a close relationship between chaos and cryptography, through which a highly unpredictable signals are generated which can be further deployed in confusion and diffusion process of encryption. To fulfill the requirements of higher security and minimum latency, chaotic random signals are generated from chaotic maps and has been deployed in many image encryption schemes [9, 10].

There are many inherit properties of chaotic maps such as pseudo-randomness, sensitivity of control parameters and sensitive initial conditions [11–18]. These properties attract many researchers to deploy chaotic maps in image encryption area [19, 20]. In 1989, Mathew et al. [21] were first cryptographers who used the application of chaotic maps in field of cryptography. In [21], authors suggested chaotic map as a replacement of one time pad. Alvarez et al. and Jakimoski et al. verified that chaotic maps are strongly dependent on initial conditions and proved that chaotic cryptosystem can resist statistical and differential attacks [22]. In 2007, Behnis et al. used the application of chaos in the image encryption [23]. A highly secure image encryption scheme was designed via utilizing piece-wise nonlinear chaotic map (PWLCM) [23]. Gao et al. introduced the concept of hyper chaos for image encryption applications [24]. [25], Rhouma et al. found security weakness in Gao et scheme and an enhanced version was presented. Three Dimensional (3-D) baker based image encryption scheme with dynamic compound chaotic sequence was proposed in [26]. Mazloom et al. [27], utilized coupled Nonlinear Chaotic Algorithm (NCA) for color image encryption. In 2010, Yoon et al. introduced the concept of large pseudo-random permutation based image encryption scheme [28]. In the same year Wang et al. propose a novel image encryption scheme based on perceptron model within a neural network [29]. In [30], Hongjun et al. designed a stream-cipher algorithm based on one-time keys and robust chaotic maps, in order to get high security and improve the dynamical degradation by utilizing the piecewise linear chaotic map as the generator of a pseudo-random key stream sequence. In 2011 Hongjun et al. proposes a bit-level permutation and high-dimension chaotic map to encrypt color image [31]. Conversion of 2 - D images into 1 - D using raster and zigzag scanning pattern followed by sub blocks division was proposed in [32]. Zhu et al. presented a modified hyper-chaotic sequences for image encryption applications [33]. In the first stage of Zhu’s scheme, chaotic key stream was generated via hyper-chaotic sequences. In second phase, Zhu et al. uses chaotic key stream and plaintext image to generate high level key and plaintext sensitivity [33]. Zhang et al. encrypted the plaintext image using linear hyperbolic chaotic system based on partial differential equations [16]. To remove high correlation in images, Amir et al. [34] introduced the idea of multiple S-Boxes. However, Jawad et al. found that Amir’s scheme completely fails for highly autocorrelated data i.e., binary images [35]. Jawad et al. enhanced Amir’s scheme via adding XORed diffusion to the existing scheme. In [36], Rehman et al. proposed a new image encryption schemes by using the concept of dynamic S-Boxes. 2 - D Burger chaotic map was employed to shuffle the plaintext image row-wise and column-wise, respectively. In the final stage, shuffled image was divided in blocks and Logistic chaotic map was used to substitute each block of shuffled image with distinct block. Recently, Khan et al. [37] proposed a novel S-Box based image encryption via Gingerbreadman chaotic map and S₈ permutation. In this paper, we have reported the security weakness of Khan’s [37] encryption scheme.

Problem statement:

In bijective S-Box, each element must be mapped with a unique symbol. In this context, let us consider message symbols M₀, M₁... M₂₅₅ that maps to S₀, S₁... S₂₅₅ as shown in Figure 2. It can be seen from Fig. 2 similar data symbols are replaced with only single S-Box element. Therefore, in case of a correlated image, the histogram peaks remain constant when single S-Box is deployed in substitution. In [34], Amir et al. found that single S-Box cannot conceal information in a good way and hence they proposed the concept of multiple S-Boxes. In [34], Logistic chaotic map is used as decision maker for symbol selection. However, Jawad et al. [35], highlighted some drawbacks in Amir’s scheme and concluded that Amir’s scheme is not fit for correlated images. In Khan’s proposed scheme [37] a single S-Box was applied for image encryption. The concept of single S-Box can work for color and some gray-scale images but it does not work for highly correlated images i.e., binary images. Issue for binary image is highlighted in Fig. 1. From Fig. 1, it is very straight forward that Khan’s scheme is not secure even by visual inspection and hence an improvement in Khan’s scheme is required.

Fig.1

Encryption results of Khan’s scheme.

Fig.2

Bijective mapping nature of S-box.

Contribution:

As aforementioned that Khan’s scheme cannot work well for binary images and eavesdroppers can apply various attacks on encrypted images. Firstly, we have explained in very detail about the problems may cause by Khan’s scheme. An improved scheme is then presented and analysed against various security tests.

The rest of article is organized as follows: Section 2 describes the proposed scheme. Security analysis and comparison with the existing schemes is illustrated in Section 3. Finally, the research work is concluded in Section 4.

2 The proposed scheme

It has been found that Khan’s scheme cannot encrypt the image information in an effective way, therefore any attacker can easily extract actual information from the ciphertext image. Chaotic confusion and diffusion can be a good solution to the aforementioned problems which can make the scheme more robust and complex.

Flowchart of the proposed modified image encryption scheme is shown in Fig. 3. As explained in [38], Lorenz equations are highly chaotic for a specific range of initial conditions therefore we selected Lorenz equation as a pseudo-random chaotic number generator for confusion and diffusion processes. Mathematical representation of Lorenz equation is shown by Equations 1, 2 and 3.

Fig.3

Flow chart of the proposed scheme.

$\frac{dx}{dt} = - 1 (ax - ay)$ (1) $\frac{dy}{dt} = - 1 (xy + y - bx)$ (2) $\frac{dz}{dt} = - 1 (cz - xy)$ (3)

In above equation t demonstrate time, a, b and c are constant while x, y, and z represent the state of the Lorenz system. Steps involved in the proposed scheme are explained briefly.

Step 1: In order to resist differential attack with chosen plaintext attack, x, y, and z are generated via Secure Hash Algorithm (SHA). We applied SHA algorithm on plaintext image P, as a result 160 bit hash value is produced.

Step 2: Store bits b₁, b₂... b₄₈ in K, b₄₉, b₅₀... b₉₆ in L and b₉₇, b₉₈... b₁₄₄ in M, respectively. Discard last 16 bits.

Step 3: Covert K, L, M into decimal format to get K₁, L₁, M₁. The value of x, y and z is obtained using Equations 4 to 6. $x = \frac{K_{1}}{2^{48}}$ (4) $y = \frac{L_{1}}{2^{48}}$ (5) $z = \frac{M_{1}}{2^{48}}$ (6)

Step 4: Iterate Lorenz chaotic map 300 times and store result of Equations 1, 2 and 3 in X, Y and Z respectively. Where X = X₁, X₂, X₃, . . . . . X₃₀₀, Y = Y₁, Y₂, Y₃ . . . . . Y₃₀₀ and Z = Z₁, Z₂, Z₃, . . . . . Z₃₀₀, respectively. To avoid transient effects, initial 34 values obtained in X, Y and Z are discarded and rest of all random values are stored in α, β and γ.

Step 5: Convert all values of α, β and γ to finite precision discrete format and store results in α_F, β_F and γ_F.

Step 6: Multiply α_F, β_F and γ_F with 10¹⁴ and store results in α_M, β_M and γ_M. Apply modulus (256) operation on α_M, β_M and γ_M to get Δ, Γ and Ω.

Step 7: Employ Δ in shuffling rows of plaintext image P and store results in matrix P′. Permute all columns of P′ via Γ to get P_shuffled.

Step 8: Divide image P_shuffled into B = B₁, B₂, B₂, . . . B₂₅₆ (256) blocks with each block size 16 × 16. An initial block B₁ is bit-wise XORed with Ω random values. Every Bth (B ≥ 2) block of shuffled image P_shuffled is XORed with (B-1)th block. The resultant image is stored as diffused image D.

Step 9: To get the final ciphertext image C, the existing scheme outlined in [37] is applied on diffused image D.

For decryption purpose, apply all the steps in reverse order to get the plaintext image P from ciphertext image C.

3 Experimental and Statistical interpretation

Experimental results of the proposed scheme for color, gray and binary pepper images are shown in Fig. 4. To show robustness and effectiveness of the designed scheme, the proposed image encryption scheme is tested via entropy, histogram, correlation, Mean Square Error (MSE), energy, contrast, homogeneity, Peak Signal to Noise Ratio (PSNR), key sensitivity test and encryption qualityanalysis.

Fig.4

Encryption results of the proposed scheme.

3.1 Entropy

In image encryption scenario, entropy is an important parameter used for the computation of randomness in an encrypted image [39]. We assume an image encryption algorithm which deals with 2⁸ symbols. In such case, the ideal value of entropy must be 8 bits. For a secure image encryption algorithm, the entropy value should be close to the ideal value. If the entropy value is closer to ideal value, the scheme is resistant against entropy attacks. Entropy is calculated as: $E = \sum p (m_{g}) \log_{2} \frac{1}{p (m_{g})},$ (7) where is the probability of random variable m at gth index. Entropy tests is applied on the proposed scheme and results are shown in Table 1. One can confirm from Table 1 that the proposed image encryption scheme possess higher entropy values as compared to Khan’s scheme. In order to prove the robustness of the proposed scheme, it is also compared with Amir’s and Jawad’s scheme as outlined in [34] and [35], respectively. The proposed encryption algorithm can resist entropy attack better than all three schemes i.e., Amir’s [34], Jawad’s [35] and Khan’s schemes [37].

Table 1

Entropy analyses of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme

Image	Plain Image	[34]	[35]	[37]	Proposed
Colour image	7.3318	7.7734	7.9823	7.4341	7.9959
Gray-scale image	7.5201	7.9221	7.9934	7.5178	7.9981
Binary image	0.9838	2.5241	7.8203	0.9928	7.9885

3.2 Histogram test

For a quick visual analysis, histogram test is an easy method. Histogram analysis shows the number of pixel distribution in an image [40–42]. A flat histogram for the ciphertext image indicates uniform pixels distribution. The histogram plots of all schemes are shown in Fig. 5. From histogram plots, it is evident that the proposed scheme has flat histogram in all cases. For colour and gray scale images, the proposed and Jawad’s scheme has almost similar histograms. However, in case of binary image, the proposed scheme is consistent to flat histogram while Jawad’ scheme losses consistency. [37].

Fig.5

Histogram plots ofof Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme.

3.3 Correlation analysis

Correlation coefficients is the method to measure similarity between two variables or images [43]. To carry out correlation analysis, one thousand pairs of neighbouring pixels of plaintext image and its corresponding ciphertext image are selected in three directions, i.e., vertical, horizontal and diagonal directions. Mathematically, correlation coefficient is calculated as: $Cov (g, h) = E (g - E (g)) \times (h - E (h))$ (8) $C . C = \frac{Cov (g, h)}{\sqrt{D (g)} \sqrt{D (h)}}$ (9) where

$E (g) = \frac{1}{1000} \sum_{i = 1}^{1000} g_{i}$

$D (g) = \frac{1}{1000} \sum_{i = 1}^{1000} (g_{i} - E (g))^{2}$

Cov (g, h) is covariance at the adjacent pixels values g and h of plaintext and ciphertext image, respectively. Correlation coefficients for plaintext and ciphertext images are shown from Table 2 to Table 6. One can see from these tables that the ciphertext images have less correlation values. From each table, it is evident that the proposed scheme has lower correlation coefficient than Amir’s scheme [34], Jawad’s scheme [35] and Khan’s scheme [37].

Table 2

Correlation coefficient analyses of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme: Red Channel

Direction	Plain Image	[34]	[35]	[37]	Proposed
Horizontal	0.9767	0.0632	0.0223	0.0745	0.0542
Vertical	0.9688	–0.0517	0.0782	0.0782	–0.0068
Diagonal	0.9378	0.0442	–0.0321	0.0523	0.0319

Table 3

Correlation coefficient analyses of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme: Green Channel

Direction	Plain Image	[34]	[35]	[37]	Proposed
Horizontal	0.9711	0.0745	–0.0355	0.0701	0.0562
Vertical	0.9691	0.0523	0.0816	–0.0667	–0.0186
Diagonal	0.9311	–0.0305	–0.0277	0.0401	0.0205

Table 4

Correlation coefficient analyses of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme: Blue Channel

Direction	Plain Image	[34]	[35]	[37]	Proposed
Horizontal	0.9737	–0.0663	0.0299	0.0702	–0.0331
Vertical	0.9529	0.0629	–0.0705	0.0823	0.0125
Diagonal	0.9452	–0.0308	0.0331	0.0486	–0.0236

Table 5

Correlation coefficient analyses of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme: Gray image

Direction	Plain Image	[34]	[35]	[37]	Proposed
Horizontal	0.9712	0.0689	0.0324	0.0642	–0.0594
Vertical	0.9681	–0.0492	0.0746	0.0690	–0.0074
Diagonal	0.9378	0.0320	–0.0310	0.0467	0.0283

Table 6

Correlation coefficient analyses of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme: Binary image

Direction	Plain Image	[34]	[35]	[37]	Proposed
Horizontal	0.9745	–0.0567	0.0378	0.0905	0.0465
Vertical	0.9738	0.0686	–0.0773	0.0147	–0.0087
Diagonal	0.9239	0.0420	0.0254	0.0429	0.0226

3.4 Mean Square Error Analysis(MSE)

MSE computes average squared difference between the plaintext image and encrypted image. Higher value of MSE reveals that the encryption scheme can strongly resist many attacks. The formula for MSE can be written as [44]:

$MSE = \frac{1}{m \times n} \sum_{i = 0}^{n - 1} \sum_{j = 0}^{m - 1} {[C_{1} (g, h) - C_{2} (g, h)]}^{2},$ (10) In Equation 10, n × m is the size of images, C₁ and C₂ are ciphertext images whose plaintext images are differ by one bit. Higher values of MSE highlights the differences between both ciphertexts. It can be seen from Table 7, that the proposed scheme has MSE value greater than all other traditional schemes. Moreover, it is clear from Table 7 that Khan’s scheme completely fails in MSE analysis as it is based on single S-Box.

Table 7

MSE analysis of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme

Image	[34]	[35]	[37]	Proposed
Colour image	38.2245	40.2278	0	40.4323
Gray-scale image	37.7378	40.3155	0	40.4100
Binary image	36.6710	40.3421	0	40.3965

3.5 Energy analysis

Energy of an image illustrates the information of plaintext image in terms of the sum of squared values. Energy can be written as: $E = \sum X (g, h)^{2},$ (11) where, X (g, h) is pixel value at position g, h. Through energy analysis, disorder in texture for a ciphertextimage can be computed. A lower value of energy indicates, higher encryption quality. The values obtained in Table 8 confirms that the proposed scheme has least energy values as compared with other schemes. Therefore, the proposed scheme has maximum degree of disorder for the ciphertext image.

Table 8

Energy analysis of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme

Image	[34]	[35]	[37]	Proposed
Colour image	0.189	0.165	0.215	0.165
Gray-scale image	0.192	0.178	0.215	0.178
Binary image	0.301	0.181	0.525	0.174

3.6 Contrast analysis

Intensity differences between a pixel and its neighbour pixels is calculated via contrast parameters. Contrast can be given as: $Contrast = \sum | g - h |^{2} X (g, h) .$ (12)

where, X (g, h) is the number of gray level co-occurrence matrices. The contrast values in Table 9 shows that the proposed scheme has considerably higher contrast values as compared to others schemes. Thus, these greater values of contrast shows higher security of the proposed scheme.

Table 9

Contrast analysis of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme

Image	[34]	[35]	[37]	Proposed
Colour image	7.9854	8.6032	7.2245	8.6448
Gray-scale image	8.2561	8.6103	7.4733	8.6258
Binary image	3.5387	8.5002	0.7734	8.6577

3.7 Homogeneity analysis

The homogeneity analysis returns the closeness of the distribution in the Gray Level Co-occurrence Matrix (GLCM). The GLCM measures the combinations of pixels brightness values in a tabular form. Mathematically, homogeneity can be determined as: $HOM = \sum \frac{X (g, h)}{1 + | g - h |} .$ (13) Homogeneity analysis for Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme is carried out and results are shown in Table 10. The lower value of homogeneity for the proposed scheme reflects a better image encryption.

Table 10

Homogeneity analysis of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme

Image	[34]	[35]	[37]	Proposed
Colour image	0.4366	0.4149	0.4582	0.4110
Gray-scale image	0.4310	0.4087	0.4412	0.4040
Binary image	0.5678	0.4163	0.9734	0.3728

3.8 Key analysis

Two different tests are carried out for key analysis. One is key space analysis and other is key sensitivity test. Key space analysis of an encryption is checked so that the feasibility of brute force attack is checked [45]. As specified by IEEE standard format that minimum key space must be 2¹⁰⁰ [46]. The available key space for Khan’s scheme [37] and the proposed scheme can be computed as:

$\begin{matrix} {KS}_{1} & = (10^{15} \times 10^{15}) \\ = 10^{30} \\ \approx 2^{99}, \end{matrix}$ (14)

$\begin{matrix} {KS}_{2} = (10^{15} \times 10^{15} \times 10^{15} \times 10^{15} \\ \times 10^{15} \times 10^{15} \times 10^{15} \times 10^{15}) \\ = 10^{120} \\ \approx 2^{398}, \end{matrix}$ (15) where KS₁ and KS₂ are key spaces for Khan’s and the proposed schemes, respectively. The key spaces for Amir’s and Jawad’s scheme are 10¹⁵ and 10³⁰, respectively. Hence the proposed scheme can resist brute force attack better than all schemes.

The key sensitivity test can be applied by making a small change to the initial conditions. Let C_K1 represent the ciphertext image with an initial condition ψ and C_K2 represent the ciphertext image with initial condition ψ + 10^-15. This test is applied on gray scale image only. The resultant difference images are shown in Fig. 6. It is evident that Khan’s scheme completely fails with slight change in initial conditions.

Fig.6

Key sensitivity analysis.

3.9 Encryption quality analysis

Encryption quality analysis judges an image encryption scheme in terms of pixel deviation of plaintext image from encrypted image. Two parameters are used to compute encryption quality i.e., Maximum Deviation (M_D) and Irregular Deviation (I . D). Mathematically M_D and I_D can be calculated as: $M_{D} = (\frac{D_{0} + D_{n - 1}}{2}) + \sum_{i = 1}^{m - 2} D_{g} .$ (16) $I_{D} = \sum_{i = 0}^{N - 1} H_{Dg},$ (17) where $H_{Dg} = | H_{g} - A_{h} |$ D_g represents the amplitude of difference between plaintext image and ciphertext image histograms at index g. N shows the total number of pixels. H_i shows the amplitude of histogram at index i while A_h demonstrates the average sum of histogram values. One can see from Table 11 that higher values of M_D is obtained for the proposed scheme. The lower values of I_D from Table 12 highlights the uniformity of pixels distribution for the proposed scheme.

Table 11

M_D analysis of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme

Image	[34]	[35]	[37]	Proposed
Colour image	54421	54720	53003	58691
Gray-scale image	37928	38847	36667	41293
Binary image	1264	13388	13972	13110

Table 12

I_D analysis of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme

Image	[34]	[35]	[37]	Proposed
Colour image	54823	44562	78832	43506
Gray-scale image	52830	42467	71283	41045
Binary image	13601	11543	14876	09846

3.10 Number of Pixels Change Rate (NPCR) and Unified Average Change Intensity (UACI)

Two most common measures, NPCR and UACI are used to check the possibility of differential attacks. Detail mathematical description of NPCR and UACI can be found in [35, 39]. As outlined in [35, 39], higher the values of NPCR/UACI, higher the security level is. Tables 13 and 14 are obtained after NPCR and UACI randomness tests which highlights that the proposed scheme is more secure against differential attack.

Table 13
NPCR analysis of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme

Image [34] [35] [37] Proposed

Colour image 0 99.1733 0 99.415

Gray-scale image 0 99.2186 0 99.7643

Binary image 0 99.1312 0 99.6423

Image	[35]	Proposed
Colour image	99.1733	99.415
Gray-scale image	99.2186	99.7643
Binary image	99.1312	99.6423

Table 14

UACI analysis of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme

Image	[34]	[35]	[37]	Proposed
Colour image	0	32.4416	0	33.1273
Gray-scale image	0	32.1564	0	33.3443
Binary image	0	31.3125	0	33.1412

3.11 Contrast Analysis

Contrast analysis computes the intensity differences between a pixel and its neighbour over the whole image [4, 5]. Higher value of contrast reflects that image encryption scheme is good. Contrast parameter can be written as: $C = \sum_{i, j = 1}^{N} | i - j |^{2} p (i, j),$ (18) where p (i, j) is the number of gray-level co-occurrence matrices (GLCM), N is the number of rows and columns. Table 15 shows that the contrast values of the proposed scheme is higher than other schemes and hence the proposed scheme is more secure.

Table 15

Contrast analysis of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme

Image	[34]	[35]	[37]	Proposed
Colour image	7.9532	8.1378	7.8823	10.1098
Gray-scale image	8.0303	8.6834	8.0145	10.6290
Binary image	8.3421	8.6845	8.2238	10.7023

3.12 Chosen plaintext attack

Let us consider that hash value of original plaintext P is Hash_P1. A single bit change in the original plaintext P will generate totally different hash value i.e., Hash_P2. Due to the fact that Hash_P1 ≠ Hash_P2, values x, y and z are different. The ciphertexts C₁ and C₂ are different which proves that theproposed scheme is resistant against chosen plaintextattack.

3.13 Time analysis

The proposed and existing schemes are tested on MATLAB 2012b and the results are shown in Table 16. One can conclude from Table 16 that the proposed scheme has slighter greater time than [34, 35] and [37]. It’s due to the fact that Amir’s and Khan’s scheme only consist of substitution and Jawad’s scheme just added bitwise XOR to Amir’s scheme. In the proposed scheme we added row and column wise confusion and bitwise XOR operation. Such row and column permutation and XOR operation are time consuming parts of algorithm.

Table 16
Time analysis of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme

Image [34] [35] [37] Proposed

Colour image 15.00 15.30 14.16 16.03

Gray-scale image 10.43 10.66 10.33 10.84

Binary image 10.20 10.27 10.09 10.45

Image	[34]	[35]	[37]	Proposed
Colour image	15.00	15.30	14.16	16.03
Gray-scale image	10.43	10.66	10.33	10.84
Binary image	10.20	10.27	10.09	10.45

4 Conclusion

In this paper, we enhanced the existing Gingerbreadman chaotic map and S₈ permutation based image encryption scheme. Firstly, we identified some shortcomings in existing image encryption scheme and then proposed a solution. In our enhanced solution, we employed chaotic confusion and diffusion technique in already existing scheme. The proposed scheme broke the strong correlation among pixels, whereas the Lorenz equations shuffled the plaintext image in row-wise and column-wise, respectively. In our first stage i.e., diffusion stage, a bitwise XORed operation is used to diffused and shuffle the image, which is followed by the existing substitution scheme. The results of our proposed and enhance image encryption scheme is verified through various experiements and tests such as entropy, histogram, correlation, mean square error, peak signal to noise ratio, energy, contrast, homogeneity, key space, key sensitivity, NPCR and UACI. The experimental results are compared with existing Gingerbreadman chaotic map scheme and two other traditional schemes, extensively used for image encryption. The comparative analysis of our proposed/enhance scheme with existing schemes reveals that the proposed/enhanced image encryption scheme is more efficient, robust and highly resistant/secure against many attacks as compared to existing schemes.

Conflict of interest

There is no conflict of interest.

References

Zhang

Y.-Q.

, Wang

X.-Y.

, Liu

and Chi

Z.-L.

, An image encryption scheme based on the mlncml system using dna sequences, Optics and Lasers in Engineering82 (2016), 95–103.

Zhang

Y.-Q.

and Wang

X.-Y.

, A new image encryption algorithm based on non-adjacent coupled map lattices, Applied Soft Computing26 (2015), 10–20.

Wang

and Teng

, An image blocks encryption algorithm based on spatiotemporal chaos, Nonlinear Dynamics67(1) (2012), 365–371.

Wang

X.-Y.

, Gu

S.-X.

and Zhang

Y.-Q.

, Novel image encryption algorithm based on cycle shift and chaotic system, Optics and Lasers in Engineering68 (2015), 126–134.

Wang

and Luan

, A novel image encryption algorithm using chaos and reversible cellular automata, Communications in Nonlinear Science and Numerical Simulation18(11) (2013), 3075–3085.

Ahmed

and Hwang

S.O.

, A fast encryption/decryption scheme for biometric images using multiple chaotic maps, IMTIC’15—International Multi-Topic Conference, p. 104.

Zhang

Y.-Q.

and Wang

X.-Y.

, A symmetric image encryption algorithm based on mixed linear–nonlinear coupled map lattice, Information Sciences273 (2014), 329–351.

Khan

M.A.

, Ahmad

, Javaid

and Saqib

N.A.

, An efficient and secure partial image encryption for wireless multimedia sensor networks using discrete wavelet transform, chaotic maps and substitution box, Journal of Modern Optics (2016), 1–10.

Hua

, Zhou

, Pun

C.-M.

and Chen

C.P.

, 2d sine logistic modulation map for image encryption, Information Sciences297 (2015), 80–94.

10.

Zhou

, Cao

and Chen

C.P.

, Image encryption using binary bitplane, Signal Processing100 (2014), 197–207.

11.

Habib

, Khan

J.S.

, Ahmad

, Khan

M.A.

and Khan

F.A.

, Secure speech communication algorithm via dct and td-ercs chaotic map, pp, in Electrical and Electronic Engineering (ICEEE), 2017 4th International Conference on IEEE, 2017, pp. 246–250.

12.

Wang

X.-Y.

, Zhang

Y.-Q.

and Bao

X.-M.

, A novel chaotic image encryption scheme using dna sequence operations, Optics and Lasers in Engineering73 (2015), 53–61.

13.

Wang

, Liu

and Zhang

, A novel chaotic block image encryption algorithm based on dynamic random growth technique, Optics and Lasers in Engineering66 (2015), 10–18.

14.

Wang

and Wang

, A novel image encryption algorithm based on dynamic s-boxes constructed by chaos, Nonlinear Dynamics75(3) (2014), 567–576.

15.

Zhang

Y.-Q.

and Wang

X.-Y.

, Analysis and improvement of a chaosbased symmetric image encryption scheme using a bit-level permutation, Nonlinear Dynamics77(3) (2014), 687–698.

16.

Zhang

, Xiao

, Shu

and Li

, A novel image encryption scheme based on a linear hyperbolic chaotic system of partial differential equations, Signal Processing: Image Communication28(3) (2013), 292–300.

17.

Wang

, Zhao

and Liu

, A new image encryption algorithm based on chaos, Optics Communications285(5) (2012), 562–566.

18.

Wang

, Wang

, Zhao

and Zhang

, Chaotic encryption algorithm based on alternant of stream cipher and block cipher, Nonlinear Dynamics63(4) (2011), 587–597.

19.

Ahmad

, Khan

M.A.

, Ahmed

and Khan

J.S.

, A novel image encryption scheme based on orthogonal matrix, skew tent map, and xor operation, Neural Computing and Applications (2017), 1–11.

20.

Zhou

, Bao

and Chen

C.P.

, A new 1d chaotic system for image encryption, Signal Processing97 (2014), 172–182.

21.

Matthews

, On the derivation of a chaotic encryption algorithm, Cryptologia13(1) (1989), 29–42.

22.

Alvarez

and Li

, Some basic cryptographic requirements for chaosbased cryptosystems, International Journal of Bifurcation and Chaos16(08) (2006), 2129–2151.

23.

Behnia

, Akhshani

, Ahadpour

, Mahmodi

and Akhavan

, A fast chaotic encryption scheme based on piecewise nonlinear chaotic maps, Physics Letters A366(4) (2007), 391–396.

24.

Gao

and Chen

, A new image encryption algorithm based on hyper-chaos, Physics Letters A372(4) (2008), 394–400.

25.

Rhouma

and Belghith

, Cryptanalysis of a new image encryption algorithm based on hyper-chaos, Physics Letters A372(38) (2008), 5973–5978.

26.

Tong

and Cui

, Image encryption scheme based on 3d baker with dynamical compound chaotic sequence cipher generator, Signal Processing89(4) (2009), 480–491.

27.

Mazloom

and Eftekhari-Moghadam

A.M.

, Color image encryption based on coupled nonlinear chaotic map, Chaos, Solitons & Fractals42(3) (2009), 1745–1754.

28.

Yoon

J.W.

and Kim

, An image encryption scheme with a pseudorandom permutation based on chaotic maps, Communications in Nonlinear Science and Numerical Simulation15(12) (2010), 3998–4006.

29.

Wang

X.-Y.

, Yang

, Liu

and Kadir

, A chaotic image encryption algorithm based on perceptron model, Nonlinear Dynamics62(3) (2010), 615–621.

30.

Liu

and Wang

, Color image encryption based on one-time keys and robust chaotic maps, Computers & Mathematics with Applications59(10) (2010), 3320–3327.

31.

——, Color image encryption using spatial bit-level permutation and high-dimension chaotic system, Optics Communications284(16) (2011), 3895–3903.

32.

Sathishkumar

and Sriraam

D.N.

, et al., Image encryption based on diffusion and multiple chaotic maps, arXiv preprint arXiv:1103.3792, 2011.

33.

Zhu

, A novel image encryption scheme based on improved hyperchaotic sequences, Optics Communications285(1) (2012), 29–37.

34.

Anees

, Siddiqui

A.M.

and Ahmed

, Chaotic substitution for highly autocorrelated data in encryption algorithm, Communications in Nonlinear Science and Numerical Simulation19(9) (2014), 3106–3118.

35.

Ahmad

and Hwang

S.O.

, Chaos-based diffusion for highly autocorrelated data in encryption algorithms, Nonlinear Dynamics (2015), 1–12.

36.

Rehman

A.U.

, Khan

J.S.

, Ahmad

and Hwang

S.O.

, A new image encryption scheme based on dynamic s-boxes and chaotic maps, 3D Research7(1) (2016), 1–8.

37.

Khan

and Asghar

, A novel construction of substitution box for image encryption applications with gingerbreadman chaotic map and s8 permutation, Neural Computing and Applications, 1–7.

38.

Gao

and Chen

, Image encryption based on a new total shuffling algorithm, Chaos, Solitons & Fractals38(1) (2008), 213–220.

39.

Ahmad

and Ahmed

, Efficiency analysis and security evaluation of image encryption schemes, International Journal of Video & Image Processing and Network Security12(4) (2012), 18–31.

40.

Wang

, Teng

and Qin

, A novel colour image encryption algorithm based on chaos, Signal Processing92(4) (2012), 1101–1108.

41.

Wang

and Jin

, Image encryption using game of life permutation and pwlcm chaotic system, Optics Communications285(4) (2012), 412–417.

42.

Wang

X.-Y.

and Yu

, A block encryption algorithm based on dynamic sequences of multiple chaotic systems, Communications in Nonlinear Science and Numerical Simulation14(2) (2009), 574–581.

43.

Younas

M.B.

and Ahmad

, Comparative analysis of chaotic and nonchaotic image encryption schemes, in Emerging Technologies (ICET), 2014 International Conference on IEEE, 2014, pp. 81–86.

44.

Khan

J.S.

, ur Rehman

, Ahmad

and Habib

, A new chaos-based secure image encryption scheme using multiple substitution boxes, in 2015 Conference on Information Assurance and Cyber Security (CIACS) IEEE, 2015, pp. 16–21.

45.

Zhang

Y.-Q.

and Wang

X.-Y.

, Spatiotemporal chaos in mixed linear–nonlinear coupled logistic map lattice, Physica A: Statistical Mechanics and Its Applications402 (2014), 104–118.

46.

Ahmad

, Khan

M.A.

, Hwang

S.O.

and Khan

J.S.

, A compression sensing and noise-tolerant image encryption scheme based on chaotic maps and orthogonal matrices, Neural Computing and Applications (2016), 1–15.

A novel image encryption based on Lorenz equation,Gingerbreadman chaotic map and S 8 permutation

Abstract

Keywords

1 Introduction

Table 13 NPCR analysis of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme Image [34] [35] [37] Proposed Colour image 0 99.1733 0 99.415 Gray-scale image 0 99.2186 0 99.7643 Binary image 0 99.1312 0 99.6423

3.13 Time analysis

Table 16 Time analysis of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme Image [34] [35] [37] Proposed Colour image 15.00 15.30 14.16 16.03 Gray-scale image 10.43 10.66 10.33 10.84 Binary image 10.20 10.27 10.09 10.45

Conflict of interest

References

Table 13
NPCR analysis of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme

Image [34] [35] [37] Proposed

Colour image 0 99.1733 0 99.415

Gray-scale image 0 99.2186 0 99.7643

Binary image 0 99.1312 0 99.6423

Table 16
Time analysis of Amir’s scheme [34], Jawad’s scheme [35], Khan’s scheme [37] and the proposed scheme

Image [34] [35] [37] Proposed

Colour image 15.00 15.30 14.16 16.03

Gray-scale image 10.43 10.66 10.33 10.84

Binary image 10.20 10.27 10.09 10.45