1 Introduction

Transmission of digital images (personal and professional) over public data networks have become a necessity in the existing era. The current infrastructure allows us to transmit huge amount of data due to technological advancements and towering bandwidth support. The data flowing through these channels can be accessed by unauthorized users like ransomware. Figure 1 present an approximate total of ransomware family from 2004 till 2016 with a projected total for year 2017 [43]. Therefore, researcher keeps on searching new algorithms in order to make data confidential. Most of the initial algorithms for providing confidentiality focused on encrypting small chunks of data via symmetric [4, 5, 30, 39] and asymmetric key algorithms. These conventional schemes are vulnerable to attacks by unauthorized users [6, 49]. Also, the initial schemes turn out to be ineffective for encryption [10, 42, 46, 48] since, the image possesses high redundancy and correlation among adjacent pixels which motivated various researchers to branch out and construct efficient algorithms for encrypting digital images.

Fig. 1
figure 1

Annual number of ransomware families

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Since 1989, various image encryption [21, 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51] algorithms are proposed. These algorithms are based on various techniques like chaotic systems with Kolmogorov flow [38], or with total shuffling [50], vector quantization [8], and iterative random encoding [25]. The chaotic based systems attracted a lot of attention of researchers due to their high dependency on the initial condition, simplicity, computational efficiency and confusion/diffusion process to make the system more non-linear in nature and hence harder to predict [3, 19, 44, 45] for intruders/hackers.

In this paper, a novel encryption/decryption based on intertwining chaotic maps and RC4 stream cipher has been proposed. The proposed scheme can be used to encrypt/decrypt colored images and provides a high interlinking between the Red (R), Green (G), and Blue (B) planes of the image. So, a minute change in the components of any plane will provide an encrypted image significantly different from the originally encrypted image. Moreover, the RC4 cipher is well known for providing a high key sensitivity and the scheme is not breakable by any attack in a sensible amount of time. All this indicates high security levels for using the scheme in chaotic implementations. Also, the scheme is significantly fast and it ensures high protection against the brute force attack because of its large key spaces available.

The second section is a review of various encryption schemes present in literature. This section contains some of the most popular conventional encryption schemes and a few of modern chaos-based encryption schemes. Third section explains the proposed chaos-based algorithm. The section details about the various components of the scheme which are key generation using intertwining chaotic map, random sequence generation using the RC4 cipher, diffusion and confusion. Also, the algorithm is explained in this section. Fourth section contains the simulation setup parameters used. These metrics include visual assessment, statistical analysis by histogram and correlation analysis, differential attack analysis, key space analysis, quantitative and time complexity analysis. The fifth section contains the results based on these performance metrics. And finally, the sixth section summarizes and concludes the study.

2 Literature Review

Modern cryptography is shaped by a series of efficient algorithms each one of which had their pros and cons. This section is an effort to summarize the most popular conventional cryptography techniques used for encryption of textual data along with some modern chaos-based techniques used for encrypting images. Table 1 summarizes the cryptography techniques.

Table 1 Summary of popular convention and modern chaos-based cryptography techniques
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The conventional cryptography techniques were originally developed to encrypt textual information that was not suitable for encrypting images due to high correlation between adjacent pixels and high time complexity. These shortcomings were removed by chaos-based encryption. The salient features of these schemes [15, 47] are as follows:

  • Simplicity

  • Large key space

  • Sensitivity to key change

  • Resist against differential and statistical attacks

  • Low computational time

The chaos based mechanisms are used efficiently in both one-dimensional [31] and multi-dimensional implementations [9, 13]. Some chaotic encryption schemes were designed only to encrypt gray scale images [23, 24, 33], while many others can encrypt color RGB images too [20]. Though, chaos based encryption schemes are better but are still vulnerable to cryptanalysis [41, 51]. This occurs when improper confusion or diffusion is employed or a bad key is used. We are of the opinion that the salient features of chaos based schemes can still be improved by choosing proper confusion and diffusion process and by keeping the process as simple as possible. In addition, for the process to be practically applicable, it should be fast. This paper is an effort for the same. To prove the efficacy of the proposed scheme, it is compared with most of the popular schemes in literature mentioned below:

  • Scheme Based on a Peter De Jong Chaotic Map and a RC4 Stream Cipher, referred here as Scheme 1.

  • Scheme based on chaotic function using linear congruences, referred as Scheme 2.

  • Scheme based on mixed transform logistic maps, referred as Scheme 3.

  • Scheme based on intertwining chaotic maps, referred as scheme 4.

The next section gives the proposed encryption scheme

3 Proposed Encryption Scheme

The proposed scheme follows an architecture popular in many of the chaos-based cryptography schemes. It contains the following steps:

  1. 1.

    Confusion

  2. 2.

    Diffusion

Confusion is an iterative step of the process. It is the permutation stage in which the pixels of the plain-image are permuted among each other. So, the intensities of the pixel values do not change but their positions are randomized. Diffusion is another iterative step in which each pixel of the image is modified using different values of the key to provide a high complexity to the process. Basically, these confusion and diffusion steps are repeated iteratively in order to provide randomness. In the proposed work only two iterations were enough to provide a high amount of randomization. The proposed scheme has four major steps. These are:

  • Key generation using Intertwining Chaotic Map

  • Random sequence generation using RC4

  • Confusion

  • Diffusion

3.1 Encryption

Figure 2 shows the block diagram of the encryption end of the proposed scheme. As shown, the keys generated using the intertwined chaotic maps are used for the confusion step and as the input for RC4. The RC4 block uses the keys as input and generates random sequences which are used for the diffusion process. The diffusion process is done is done row-wise and column-wise both in forward and backward directions. Here, a three dimensional image containing M rows and N columns is used. The intensities of Red, Green, and Blue channels in the image ranges from 0 to 255. The detailed explanation of different steps of the algorithm is given in sections below.

Fig. 2
figure 2

Block diagram of the encryption end of the proposed scheme

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3.1.1 Intertwining Chaotic Map and Key Generation

Chaotic maps are the mathematical functions which provide a dynamic nature to the system. Logistic maps, also known as log maps, and modified logistic maps are used by researchers to obtain dynamic encryption algorithms. The log map provides high randomness, unpredictability and has simple mathematical representation which is given as:

$$x_{n + 1} = r*x_{n} \left( {1 - x_{n} } \right)$$
(1)

where xn is the initial value and xnϵ (0, 1), and r ϵ (0, 4]. As the function is dependent on the initial conditions and hence provides security than a similar non-chaotic system. Even after these advantages the log maps have an uneven distribution of sequences, stable windows, and a week key [16], hence are not practical choices for modern encryption schemes. Modified logistic maps, like intertwining chaotic maps, are aimed to make the scheme more practical by removing the mentioned weakness. Mathematically, intertwining logistic maps are given as:

$$x_{n + 1} = mod\left( {\left( {\mu *k_{4} *y_{n} *\left( {1 - x_{n} } \right) + z_{n} } \right),1} \right)$$
(2)
$$y_{n + 1} = mod\left( {\left( {\left( {\mu *k_{5} *y_{n} } \right) + \left( {z_{n} *\frac{1}{{1 + \left( {x_{n} + 1} \right)^{2} }}} \right)} \right),1} \right)$$
(3)
$$z_{n + 1} = mod\left( {\left( {\mu *\left( {x_{n} + y_{n} + k_{6} } \right)*sin\left( {z_{n} } \right)} \right),1} \right)$$
(4)

where µ ϵ (0,3.999], \(\left| {{\text{k}}_{4} } \right|\) > 33.5, \(\left| {{\text{k}}_{5} } \right|\) > 37.9, \(\left| {{\text{k}}_{6} } \right|\)  > 35.7.

Figure 3 shows the key generation scheme. The keys k1 to k6 were the defined inputs for required for key generation. These keys were used by three paths, namely X path, Y path and Z path, to create three different intertwined chaotic maps, namely xlog_diff, ylog_diff, and zlog_diff. Each path of the chaotic maps used two keys as input parameters and generate sequences which were further used for the confusion and RC4 processes. All these sequences are 1-dimensional vectors.

Fig. 3
figure 3

Key generation using intertwining chaotic map

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The sequences generated from the X path and the Y path combined was used for the confusion stage. The element values in both of these sequences range from 0 to M*N. Each of the map has a total M*N elements in it, hence making a successful implementation of the confusion process for any image of M*N size. The sequence generated from the Z path was used as the input for RC4 random sequence generation. As the key length required for the RC4 algorithm ranges from 1 to 256 bytes and does not depend on the size of the image, so the sequence generated from the Z path has 256 elements in it, with values ranging from 0 to 256.

3.1.2 RC4 and Random Sequence Generation

The RC4 is a symmetric key, stream cipher and known for its speed and simplicity. It was initially developed in 1987 as a trade secret but in 1994 was leaked to public. During the encryption process, the cipher is fed with a key whose length can vary from 1 byte to 256 bytes. This cipher generates a randomized array which is obtained by feeding the key to a pseudorandom byte generator. The output of the pseudorandom generator is known as the random sequences. The random sequences generated in the proposed algorithm had a length of 2*M*N and were fed as the inputs of the diffusion process. The decryption process follows similar steps as that of encryption process. Figure 4 shows the block diagram of RC4 cipher encryption end of the proposed algorithm.

Fig. 4
figure 4

Block diagram of RC4 cipher

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There are numerous variant of the RC4 cipher. These variants provide a better resistance for the attacks on RC4 and/or are modified for application specific performance of RC4. The current study uses a RC4 in which a 256 byte key is passed directly to pseudorandom generator which is provided by the intertwined chaotic maps hence eliminating the need of key scheduling step. Algorithm for RC4 used is given below.

3.1.3 Diffusion

Diffusion is one of the two important stages of the proposed algorithm stages which adds randomness in the pixels of the plain-image. The diffusion property is responsible for spreading the effect of change in a pixel intensity value throughout the entire image. So any change in the intensity of one pixel value will change the encrypted image drastically and both the images formed will have no or very low resemblance with each other. Figure 5 shows hierarchy of the diffusion process.

Fig. 5
figure 5

Hierarchy of the diffusion process

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As shown in figure above, the diffusion step can be branched into row-wise and column-wise diffusion and both diffusion steps are done in forward and backward directions. In row-wise diffusion, the image is scanned row-wise in alternate directions. This scanning is used to turn the M*N image into an array form before the actual diffusion steps are used. Then the generated array is diffused in forward followed by backward direction. Once the row-wise diffusion is done, the image is turned back into the M*N form and then it is scanned column-wise to form another similar array followed by similar forward and backward diffusion steps. Finally, the image is again turned back into the M*N form and passed to the confusion block.

3.1.4 Confusion

The confusion stage is another step which adds randomness in the pixels of the plain-image. The confusion stage used in the proposed algorithm is a permutation of pixel positions of the original image. The stage shuffles the position of the pixels present in the image while making no changes in the respective pixel intensity values. This stage ensures that an unauthorized user who tries to access the data present in the image will get no useful data about the image as the pixels are moved from their original positions. The permutation of pixel position helps to achieve lower values of correlation coefficients as the correlation present in the adjacent pixels in original image is removed by the movement. Figure 6 shows the confusion stage of the proposed scheme.

Fig. 6
figure 6

Confusion stage of the proposed scheme

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3.2 Decryption

Decryption process follows steps similar to the encryption process. Here, the encrypted image is taken as input and the steps used by the encryption process are used in reverse order.

3.3 Algorithm Explanation for Encryption/Decryption

The algorithm was executed in three separate channels. The three channels are denoted here as R, G and B which are used for the Red, Green and Blue planes. The plane constitutes the red, green, and blue components of the image respectively. Both, the encryption and decryption algorithms are executed for all three channels separately keeping the same keys and other algorithm variables, i.e., the algorithm is executed three times with the only change of the channel used. The three channels were merged again before providing the encrypted or decrypted images. The steps used for key generation, encryption, and decryption processes are given.

3.3.1 Encryption Algorithm

  1. 1.

    Step 1 Divide the image into Red, Green and Blue channels.

  2. 2.

    Step 2 Generate xlog_diff, ylog_diff, and zlog_diff using intertwining chaotic maps by using the respective keys as defined above.

  3. 3.

    Step 3 For Red channel, generate a random sequence of length 2*M*N by using zlog_diff, following the process define above in the respective section.

  4. 4.

    Step 4 Scan the image row-wise to generate a 1-dimensional array and apply forward diffusion followed by backward diffusion.

  5. 5.

    Step 5 Convert the array back to M*N form.

  6. 6.

    Step 6 Repeat steps 4 and 5 for column-wise diffusion.

  7. 7.

    Step 7 Apply the confusion process.

  8. 8.

    Step 8 Repeat steps 3–7 + for the Green channel and after it repeat the same for the Blue channel.

  9. 9.

    Step 9 Recombine the three channels together to get the encrypted image.

3.3.2 Decryption Algorithm

  1. 1.

    Step 1 Divide the encrypted image into the channels.

  2. 2.

    Step 2 For Red channel, apply the reverse confusion step.

  3. 3.

    Step 3 Apply the RC4 decryption stage.

  4. 4.

    Step 4 Scan the image column-wise to perform reverse diffusion, first in backward then in forward directions for the generated array, and turn the array back into image.

  5. 5.

    Step 5 Repeat step 4 for row-wise diffusion.

  6. 6.

    Step 6 Repeat steps 2–5 for green and blue channels.

  7. 7.

    Step 7 Recombine the three channels to get the decrypted image.

The next section gives the detail of setup parameters used to evaluate the efficacy of our proposed scheme.

4 Simulation Setup Parameters

4.1 Setup Parameters

The proposed algorithm, along with the other algorithms used in the study, was implemented on a personal computer. Table 2 provides the specification of the machine, test plain-images and initial and modified parameters used for the algorithms in the study.

Table 2 Machine and image specifications, initial and modified parameters
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4.2 Performance Metrics Used

4.2.1 Visual Assessment

Visual assessment is one crucial parameter which ensures that no information about the original image can be extracted by looking at the encrypted image. A high pixel scrambling in the encrypted image can distort the visual data of original image and hence can secure a high visual security.

4.2.2 Statistical Attack Analysis

The relation between the pixels of the images can be found by performing their statistical attack analysis. These include histogram and correlation analysis and are described below in detail.

4.2.2.1 Histogram Analysis

For a digital image, a histogram is a representation of frequency distribution of pixels of the image in graphical form. In general, an original RGB image has a non-uniform histogram distributed with occasional spikes present in it. On the other hand, the histogram of an ideally encrypted image will be evenly distributed and hence will has no resemblance with the histogram of original image. Histogram analysis is plays a crucial role as the highly secure data encryption schemes, like AES, are vulnerable by histogram based cryptanalysis [1, 17].

4.2.2.2 Correlation Analysis

A digital image, in general, has a high level of correlation between the adjacent pixels which provides the effect of image as a whole. This correlation is present in horizontal, vertical and diagonal directions. But for an encrypted image there should be no correlation between the adjacent pixels as an unauthorized user can take advantage of this and can reconstruct the originally image either partially or fully. The value of correlation coefficient typically lies between − 1 ando + 1, with zero representing no linear relation between the adjacent pixels. Mathematically this can be represented as:

$$r_{\alpha \beta } = \frac{{cov\left( {\alpha , \beta } \right)}}{{\sqrt {D\left( \alpha \right)} \sqrt {D\left( \beta \right)} }}$$
(5)
$$E\left( \alpha \right) = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \alpha_{i}$$
(6)
$$D\left( \alpha \right) = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \left( {\alpha_{i} - E\left( \alpha \right)} \right)^{2}$$
(7)
$$cov\left( {\alpha , \beta } \right) = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \left( {\alpha_{i} - E\left( \alpha \right)} \right)\left( {\beta_{i} - E\left( \beta \right)} \right).$$
(8)

4.2.3 Differential Attack Analysis

A secure encryption algorithm should provide totally different encrypted image if any of initial conditions, i.e. pixel or key values, is changed even by a single bit. In order to evaluate the robustness of an algorithm against the differential attack two parameters, namely net pixel change ratio (NPCR) and unified average change in intensity (UACI), are used. These parameters are explained below.

In NPCR,two encrypted images are compared on the basis ofthe pixel difference. The two encrypted images are obtained from the original plain-image and one-pixel modifiedplain-image encrypted using the same encryption algorithm. Let C1 and C2 be the respective encrypted images. Mathematically, NPCR is given as:

$$NPCR = \frac{{\mathop \sum \nolimits_{i = 1}^{H} \mathop \sum \nolimits_{j = 1}^{W} D\left( {i, j} \right)}}{W \times H} \times 100 \%$$
(9)

where H is the height and W is the width of the images. D is a bipolar array of size equal to the images and constitute components 0 and 1 only. Mathematically, each component D(i, j) is given as:

$$D\left( {i, j} \right) = \left\{ {\begin{array}{*{20}c} {0 \quad C1\left( {i, j} \right) = C2\left( {i, j} \right),} \\ {1 \quad C1\left( {i, j} \right) \ne C2\left( {i, j} \right)} \\ \end{array} } \right.$$
(10)

UACI is a representation of average intensity difference between the plain and encrypted images. Mathematically, it is given as:

$$UACI = \frac{1}{W \times H}\left[ {\mathop \sum \limits_{i = 1}^{H} \mathop \sum \limits_{j = 1}^{W} \frac{{\left| {C1\left( {i, j} \right) - C2\left( {i, j} \right)} \right|}}{{2^{L} - 1}}} \right] \times 100 \%$$
(11)

where L is the number of bits of red, green and blue channels.

There are two conditions defining the efficiency of the differential attack analysis. In both the conditions, an original key and a key obtained by minutely changing the original key are used. According to the first condition, if an image is encrypted to get two ciphered images using both the keys individually, then the ciphered images should be completely different. According to the second condition, if an encrypted image is decrypted using the two keys uses separately, then only the image obtained by original key should provide the original image with no data loss. On the other hand, the image obtained by the modified key should not provide any information, partially or fully, about the original image.

4.2.4 Key Space Analysis

The key space analysis is done to determine the robustness of an encryption algorithm against the brute-force attack [22, 32]. The analysis evaluates this on the basis of number of key combinations an algorithm produces.

4.2.5 Quantitative Analysis

Quantitative analysis is comparison of image enhancement obtained by the scheme. The parametersof the analysis included in the study arepeak signal-to-noise ratio (PSNR) analysis andinformation entropy analysis.

4.2.5.1 Peak signal-to-noise ratio (PSNR) analysis

PSNR is the ratio between the maximum power component and noise present with a signal. In case of images, the plain-image is considered as the signal and the encrypted one as the noise. Mathematically, it is given as:

$${\text{PSNR }} = 20 \times \log_{10} \left( {\frac{255}{{\sqrt {\text{MSE}} }}} \right){\text{dB}}$$
(12)

where MSE is mean squared error. Mathematically, it is given as:

$$MSE = \frac{1}{WH}\mathop \sum \limits_{i = 1}^{W} \mathop \sum \limits_{j = 1}^{H} \left[ {I\left( {i,j} \right) - K\left( {i,j} \right)} \right]^{2}$$
(13)

where I represents the pixel values of original and k for the encrypted images and (i, j) is the location.

4.2.5.2 Information Entropy Analysis

For an encryption scheme, entropy is the amount of randomness obtained in the pixels of encrypted image when compared to the pixels of plain-image. Mathematically, it is given as:

$${\text{H}}\left( {\text{S}} \right) = \mathop \sum \limits_{i = 0}^{n - 1} P\left( {S_{i} } \right)\log_{2} \left( {\frac{1}{{P\left( {S_{i} } \right)}}} \right)$$
(14)

where n is the number of symbols, Si is the pixel values and P(Si) represents the probability of occurrence of Si.

4.2.6 Time complexity analysis

Time complexity is the time taken by the system to execute the scheme. This time is the encryption/decryption time of the scheme. It can be calculate manually as well as by using the in-built operations of the software used for implementation. Here, the tic toc function of MATLAB. To get an unbiased comparison, the parameters of Table 2 were kept the same during the study. Any variations in these parameters can affect the time complexity drastically.

5 Results

5.1 Visual Assessment

The visual assessment of the encrypted and decrypted images obtained from the proposed scheme can be done from Fig. 7. The figure shows visual assessment of four test images. From the figure, it can be clearly seen that all the encrypted images obtained are highly scrambled and there is no visual resemblance between them and the original images. So, the information contained in the image, or any part of it, is too complex to be resolved by a human eye. Moreover, it can also be seen that the decrypted images are visually same as the plain-images, hence ensuring the reliability of data obtained after decryption.

Fig. 7
figure 7

Visual assessment of encrypted and decrypted images

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Additionally, a visual assessment comparison of proposed and other implemented chaotic schemes is provided in Table 3. From the table, it is clear that the encrypted image obtained by proposed scheme has same visual scrambling as of other schemes. The decrypted images obtained by the proposed scheme are also as visually acceptable as the ones obtained from other schemes.

Table 3 Visual assessment and histogram analysis of proposed and other schemes
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5.2 Statistical Attack Analysis

5.2.1 Histogram Analysis

The histograms of the original, encrypted and decrypted images are provided in Table 3. The histograms of the proposed and three other schemes contain three colors, representing the frequency distribution of the red, green and blue color plane. The scheme 1 can encrypt the gray-scale images, so the histograms in this scheme do not contain the three planes.

Table 3 also provides the histogram analysis of the proposed and other schemes. In the table, it can be seen that the histograms of encrypted images obtained by the proposed scheme are uniformly distributed and have no resemblance with the histograms of the respective original images. So, the proposed algorithm has a strong resistance for the statistical attacks. Moreover, the histograms of the decrypted images are similar to the histograms of respective original images. Hence there is no measurable data loss during the process.

5.2.2 Correlation Analysis

The correlation analyses of the original and the encrypted images were performed by taking 10,000 random pixel pairs. These pixel pairs were formed by randomly choosing a pixel from the image and pairing it with an adjacent pixel. Table 4 provides the horizontal, vertical and diagonal correlation coefficients for the original images and the encrypted images obtained by proposed algorithm. It can be seen that the encrypted images provided very low values of correlation coefficients in all the three orientations. Hence, there is very less amount of correlation between the adjacent pixels of the encrypted images.

Table 4 Correlation coefficient of original and encrypted images
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Table 5 shows a comparison of correlation coefficients of images encrypted by the proposed algorithm and other implemented algorithms. It can be seen that all the correlation coefficients obtained have very low values. Hence, the proposed scheme provides a correlation value similar to other schemes in literature.

Table 5 Comparison of correlation coefficients of proposed and other schemes
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Table 6 shows the correlation plots of the original and the encrypted images. It shows that the plots obtained from the original images have the elements focused around a central line and doesn’t contains significant number of elements elsewhere in the plots. Hence, the plots have a highly non-uniform distribution for all the three orientations. On the other hand, the plots obtained by the encrypted images are uniformly distributed throughout the plot region. The nature was similarly shown by the proposed and other schemes. As there is no resemblance between the two correlation graphs, no information leakage is present and the scheme can withstand the statistical attacks efficiently.

Table 6 Comparison of correlation plots of original and encrypted images
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5.3 Differential Attack Analysis

The effect of one-bit key change is observed on the basis of NPCR and UACI parameters. Table 7 shows the NPCR and UACI values obtained for the test images after implementing the proposed and other schemes. For the test images, an average value of 99.60334% is obtained for NPCR and an average value of 31.841% is obtained for UACI. This shows a high sensitivity of the scheme for even a single bit change in the input parameter. The use of row-wise and column-wise forward and backward diffusion along with RC4 plays a critical role for the same as it ensures a high scrambling production. Hence, the proposed scheme has a high resistance against the differential attacks.

Table 7 NPCR and UACI for one-bit key change
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5.4 Key Space Analysis

Table 8 shows the key space produced by the proposed and other schemes used in the study. As it can be seen, the proposed scheme provides a very large number of possible key solutions. So, it is impractical for an unauthorized user to retrieve the data by guessing the possible key. Hence, the proposed scheme is efficient enough to resist the brute force attacks.

Table 8 Key space analysis of the proposed and other implemented schemes
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5.5 Quantitative Analysis

5.5.1 Peak Signal-to-Noise Ratio (PSNR) Analysis

Table 9 shows the PSNR values for different images encrypted using the proposed and other implemented algorithms. It can be seen that the PSNR values obtained from the proposed scheme are similar to that of other schemes present in literature. An average PSNR value of 8.0684 is provided by the proposed scheme for the images used in the study.

Table 9 PSRN values obtained using proposed and other schemes
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5.5.2 Information Entropy Analysis

Table 10 shows the information entropy values of the original and encrypted images. It can be seen that, similar to the other schemes implemented, the entropy values provided by the proposed scheme are approximating to the ideal value of 8. The average entropy value obtained was 7.9994.

Table 10 Information entropy of the original and encrypted images
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5.6 Time Complexity Analysis

Figure 8 shows the time complexity possessed by the proposed and other schemes implemented for 256*256 images of the SIPI database. It can be seen that the proposed scheme is proposed scheme is faster than other proposed schemes and hence reflects the efficiency to be used in practical cases.

Fig. 8
figure 8

Time complexity of the proposed and other schemes

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6 Conclusion

The following inference can be drawn from the above results:

  • The paper proposes an image encryption scheme based on intertwining map and RC4 stream cipher. The proposed scheme has been evaluated on various performance metrics. The results show that the proposed scheme provides highly scrambled encrypted images which have no visual resemblance with the original images. Whereas, the images obtained after decryption were visually alike as of the original images.

  • The scheme proved to be strong against the statistical attacks. It is clear from the histograms, which are uniformly distributed and have no resemblance with the histograms of the original images. Also, the encrypted images provide very low values of the correlation coefficients and have uniformly distributed correlation graphs for all three orientations with no resemblance to the original correlation graphs. All this ensures no information leakage and hence ensures security against the statistical attacks.

  • The scheme also provides a high resistance against the differential attacks. The NPCR and UACI values obtained are close to the ideal values. An average NPCR of 99.60334% and an average UACI of 31.841% were obtained for a single bit change in the key used. Hence, even for a small change in the value will provide a drastic change in the encrypted/decrypted images and the data in the original image is secured.

  • The proposed scheme has a key space of 2384, which makes it resistive against the brute force attack. The scheme provides the PSNR values similar to other schemes, including the ones implemented. Also, the scheme provides an average value of 7.9994 for the information entropy, approximating the ideal value of 8. Finally, the scheme has a faster implementation as compared to the other schemes implemented because of its lower time complexity and hence provides it with an edge over the other schemes.