Secure and Fast Image Encryption Algorithm Based on Modified Logistic Map

1. Introduction

In recent years, multimedia and communications industry have developed rapidly. Many large streams of multimedia data is transmitted over an insecure channel. As the rate of hacking increased with the passage of time, therefore, the security of the data must be increased day by day [1]. Numerous algorithms are developed in the world that are efficient and robust, but are still not sufficient to protect data. In addition, less computationally complex algorithms are required to cater the need to secure high speed data transmissions. A lossless, novel and secure image encryption algorithm based on the modified chaotic logistic map (CLM) that takes less computational time for encryption is presented in this research.

The CLM has many great features such as sensitive dependence on initial conditions, random orbit, pseudorandomness, good ergodicity, better cross-correlation properties, high efficiency, better mixing properties and large keyspace. These features makes CLM a potential candidate in quantum image encryption algorithms [2,3,4]. Therefore CLM is also quantum safe. CLM also has low computational cost and it takes less hardware and it is easy to implement [5]. It is verified that CLM provides high speed with low cost [6].

A good encryption scheme must have a substitution-permutation (SP) network as indicated by C. E. Shanon [7] to increase its security [8]. The pixel level substitution and permutation is used in the algorithm to save computation time and cost. Otherwise, if bit level substitution and permutation is used then the algorithm’s computation time and cost will increase eight times.

Permutation is the property used for the rearrangement of pixels in some pseudorandom order. It means that several pixels of the encrypted image are affected by just changing one pixel of the original image. Therefore, it hides any dependency between the input image and the encrypted image [9]. Any algorithm is more resistant to frequency analysis attacks by using permutation techniques. The permutation techniques are further classifieed into two categories, one is pixel level permutation and the second is bit level permutation. These permutations are achieved by employing various transforms [10,11], chaotic maps [12,13,14,15,16,17], cyclic shifts [18,19,20,21], hash functions [22], sorting techniques [15,23,24,25,26,27] and parallel computing [28]. Although, these transformational techniques based algorithms have many flaws, one of them is that they create high security but increases time complexity which further results in lengthy preprocessing and poor permutation performance. On the other hand, sorting based permutation techiniques gives the best permutation effect but the time complexity will be increased and memory cost will be high. if cyclic shift permutation techniques are utilized, they reduce the computational complexity and therefore reducing memory costs, but weaken the permutation effect. So, there is a need for a secure encryption algorithm which reduces the time complexity and memory cost but not at the cost of reduction in the security.

Substitution is used to obscure the connection between the corresponding pixels of the input image and the encrypted image. This property of substitution makes it ideal to hide the connection between the secret key and the encrypted image. Substitution is also subdivided into pixel level substitution [10,11,12,13,14,15,16,17,24,25,26,27,29,30,31,32] and bit level substitution [18,23,33,34]. In bit level substitution, the substitution can be lengthy and thus time consuming. Therefore in the proposed algorithm, pixel level substitution is used.

Image encryption can be lossy or lossless [35]. Various transformation techniques used for image encryption are lossy [10,11]. In the proposed algorithm, we employ CLM because of its high dependency towards initial conditions. Therefore, we achieved lossless encryption. A novel and secure image encryption algorithm based on CLM that requires less computational time to encrypt an original image is presented. Shannon’s idea of using substitution-permutation and one-time-pad network to achieve ideal secrecy is the backbone of this research [7].

The organization of rest of the paper is as follows: The proposed image encryption algorithm is presented in Section 2. The results alongwith statistical tests are presented in Section 3. The conclusion of the research is presented in Section 4.

2. Proposed Algorithm

The encryption process is divided into six stages. The block diagram is given in Figure 1. In the first stage, the permutations on the grayscale image is performed. The second stage involves the substitution of pixel intensities of the permuted grayscale image. The pixel intensities are then converted into binary bits in the third stage. In the fourth stage, the random binary bits are collected from the CLM. In the fifth stage, the binary bits from stage three and four are added using an “XOR” operation. In the last and final stage, the resultant bits are converted in to pixel intensities. The details of the six stages are presented as follows:

2.1. Permutation

The permutation process rearranges pixel intensities according to a sequence acquired from CLM. For example, if a permutation matrix [2, 4, 1, 3] obtained from CLM with population growth, is applied to numbers $2,4,1,3$ (Pixel intensities), the resultant sequence after permutation is $4,3,2,1.$ In an image, all the input pixels are shuffled in a random manner. In the proposed algorithm, the new rows and columns of the substituted image are computed by using CLM and its population growth is a part of its key.

2.1.1. Chaotic Logistic Map

The logistic map was first introduced by Robert May [36]. The governing equation for the logistic map is:

$$b_{n+1}=\gamma b_n(1-b_n)$$

(1)

where $\gamma$ is the population growth of the logistic map. The proposed algorithm presents a unique combination of 256 numbers that are required. Therefore, through extensive testing, it is determined that the numbers from $[0,255]$ arranged in ascending order are permuted with the help of CLM. It is necessary for the population growth of the logistic map to be equal to $0.5,1.5,2.5,3.5,4.5,....$. Only on these values, we can achieve a unique and random combination of $[0,255]$ numbers. Therefore, the equation is modified and can be written as:

$$b_{n+1}=(\gamma +0.5)b_n(1-b_n)$$

(2)

where $\gamma \in Z^{+}$ is a set of positive integers in eq. 2. In this way, we permute the arrangement of rows and columns. Consider an image of Lena ($256\times 256$). (The image and histogram are shown in the Figure 2.) Its permuted image and its histogram are shown in Figure 3. It is evident from both histograms (which are identical) that pixels are shuffled in such a manner that the permuted image is not depicting any resemblance with the original image.

2.2. Substitution

The substitution is a process in encryption, where the bits from the original message are substituted with pseudorandom bits. It is used to obscure the relationship between the pixels of input image and the corresponding pixels of the encrypted image [9]. Pixel intensities are used rather than bits to reduce computation time. Therefore, pixel intensities are substituted using enhanced version of CLM, which is explained in Section 2.1.1.

Example of this substitution is shown in Table 1 and Table 2.

The substituted pixels are shown in Table 2 with population growth at $5.5$ of CLM

In comparison to Table 1 and Table 2, pixel intensity 1 is substituted to pixel intensity 245. The permuted image of Lena is shown in Figure 3, is substituted and its image and histogram are shown in Figure 4. The histogram is not similar to uniform distribution, therefore, binary bits are added in the form of one-time pad and it is explained in the latter sections.

2.3. Binary Form of the Image

The data of pixel intensities is now converted as sequence of binary bits of 8-tuple. The range of pixel intensities is from 0 to 255. The pixel intensities from Section 2.2 (each pixel intensity between 0 to 255 ) are now converted as sequence of binary bits of 8-tuple. In the latter section, it will be easier for us to add random binary bits in the pixel intensities to increase the randomness and consequently increasing the security of our proposed cryptosystem.

2.4. Bit Generation

It is a common misconception that one-time pad is breakable. It is contrary to that. If the key used in one-time pad is random and it is kept hidden from all possible hacks (attacks), then the only possible way to hack a one-time pad is only by brute force attack [37].

In this attack, all possible combinations are applied. For Example, if the key consists of 2-bits, then the possible combinations are $2^2=4$. It means that for a two-bit key, hacker must enter the key four time. One of them is the actual key. In this way, brute force attack can occur. In general, if the key is $n-bits$ long then the combinations will be $2^n$. C. E. Shannon [7] in his paper proves that ideal secrecy depends on the randomness of the key.

It is evident from the graph in Figure 5 that if we increase the number of bits then the number of combinations will also increase exponentially. Therefore, the hacker must have to enter more and more combinations, if the key gets longer. The pseudorandom orbit of chaotic logistic map is very high. The secret key space, a chaotic logistic map can offer is more than $2^{302}$ [38]. The random binary bits are generated from the CLM.

2.5. XOR Operation

The binary of the pixel intensities and binary bits from CLM are added in the form of “XOR” operation. In the “XOR” operation same bits result in the output 0 and unlike bits give the output 1.

2.6. Conversion of Binary Bits to Encrypted Image

The resultant binary bits are then converted into pixel intensities. The acquired pixel intensities represent the encrypted image of Lena from the proposed algorithm. As an example, the substituted image of Lena shown in Figure 4 is then XORed. The encrypted image and its histogram are shown in Figure 6. The histogram is resembles uniform distribution, as shown in Figure 6.

3. Results

We perform numerous statistical tests on the proposed encryption algorithm. These statistical tests include: key space analysis, sensitivity analysis, strict avalanche criteria (SAC), histogram analysis, entropy analysis, mean of absolute deviation (MAD) analysis, correlation analysis, contrast analysis and homogeneity. The images used in the testing are taken from the University of Southern California - Signal and Image Processing Institute (USC - SIPI) database [39].

3.1. Keyspace Analysis

It is a well-known fact that a large key space is essential for an encryption algorithm to be resistant against brute force attack [40]. The proposed algorithm uses the key at four different stages. First, the key is broken into four parts. First two parts are used for permutation of the image. Its distinct combinations are $(256\times 256)!$. The third part is used in the s-box for substitution. Its distinct combination are $256!$. The fourth part is used in obtaining the random bits from the CLM. It ranges from $[0,1]$. Therefore, it accumulates a huge key space, resultantly increase the security of the encryption scheme.

3.2. Complexity Analysis

Complexity analysis is used to check how much memory and time is used to run a certain algorithm on a machine.

The permutation performance of the proposed algorithm is compared with some of the classical encryption algorithms as shown in Table 3. The proposed algorithm shows best performance.

3.3. Sensitivity Analysis

The initial conditions of any algorithm play a key role in its security. Therefore, the security of the algorithm is dependent over its initial conditions. For this, two common measures are used, [41,42], i.e., the number of pixels change rate (NoPCR) and unified average pixel changing intensity (UAPCI) are used.

3.3.1. Number of Pixel Change Rate (NoPCR)

It is used to check how a minute change in the input image can affect the output image. For this purpose, an input image is acquired, only one-bit change is performed in it. In image processing, one-bit change means one intensity change in pixel value. The original input image and the one-bit changed image are processed through the algorithm, two separate ciphered images are acquired. After that, the following relationship is applied to both of the images.

$$D(x,y)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}{C}_{x,y}^1=C_{x,y}^2\hfill \\ 1,\hfill & \mathrm{if}{C}_{x,y}^1\ne C_{x,y}^2\hfill \end{array}\right.$$

(3)

$$NoPCR={\displaystyle \frac{{\sum }_{x,y}D(x,y)}{B\times H}}\times 100\%$$

(4)

where

• $C^1=$ Cipher Image of Input Image
• $C^2=$ Cipher Image of Input Image with one-bit change
• $B=$ Breadth of the Image
• $H=$ Height of the Image
• x and y represents the positions of pixel in the horizontal and vertical direction respectively
• In this manner, a percentage is computed to check how many pixels are change in both of the ciphered images i.e., between $C^1$ and $C^2$. In this way, we are checking the security of our proposed algorithm. It is applied on various images, some of the results are given in Table 4.

In Table 4, it is evident that a one-bit change in the input image can result more than 99% change in the ciphered image.

3.3.2. Unified Average Pixel Changing Intensity (UAPCI)

In the previous subsection, the change in number of pixels is calculated for the entire image. In this subsection, we compute how much one pixel is changed according to its neighbouring pixel. A unified average value is computed for the whole image. First, two ciphered images are taken whose input image is changed in one-bit.

$$D(x,y)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}{C}_{x,y}^1=C_{x,y}^2\hfill \\ 1,\hfill & \mathrm{if}{C}_{x,y}^1\ne C_{x,y}^2\hfill \end{array}\right.$$

(5)

$$UAPCI=\frac{1}{B\times H}\left[\sum _{x,y}\frac{|C_{x,y}^1-C_{x,y}^2|}{255}\right]\times 100\%$$

(6)

where

• $C^1=$ Cipher Image of Input Image
• $C^2=$ Cipher Image of Input Image with one-bit change
• $B=$ Breadth of the Image
• $H=$ Height of the Image
• x and y represents the positions of pixel in the horizontal and vertical direction respectively

In Table 5, UAPCI of various images with different image sizes along with one-bit change in input image are given. It is proved from the table that an average of 7% change is occurring from pixel to pixel if a one-bit change occurred in the input image. These two measures show us that our proposed algorithm is dependent on input image. If any hacker tries to change one-bit value in the system, it can easily be identified.

3.4. Strict Avalanche Criteria (SAC)

It is performed to check the algorithm’s dependency over its initial conditions. It is performed in such a manner that one bit in $"O"$ creates more than 50% change in $"C"$. A function $g:Z_2^n\to Z_2^m$ exhibits the avalanche effect if and only if

$$\sum _{x\in Z_2^n}wt(g\left(O\right)\oplus g(O\oplus C_i^n))=m{2}^{n-1}$$

(7)

$\forall i\in [1,n]$

where

• $O=$Original Image
• $C=$Cipher Image
• $\oplus =$ Exclusive OR Operation
• Eq. 7 depicts that if one input bit is changed then 50% of the output bits must change [43,44]. Therefore, strict avalanche criteria is applied on the proposed algorithm and it was found that almost 50% bits are inverted. Table 6 gives the comparison of SAC of various algorithms.

3.5. Histogram Analysis

It is performed to check whether the encrypted image represent any resemblance towards the original image or not. If the histogram of the image is equiprobable than it is hard for the attackers to know which original image was transmitted. Equal distribution gives no clue to the hackers and it increases the security of the algorithm. It is shown in Figure 7 and Figure 9 , the histogram is equally distributed. Therefore, it makes it hard for the hackers to retrieve the original message.

3.6. Entropy Analysis

Entropy gives us the measure of randomness and distortion within the gray values of the image. [46]. To achieve ideal distribution, then the entropy should be equal to 8 [47]. In this proposed algorithm, entropy of various images was found and it was close to 8. The entropy of various images are given in Table 7. In Table 8, entropy of the proposed algorithm is compared with AES [48] and one of its variation [48]. It is verified that the average entropy of the algorithm is better than AES and much closer to 8.

3.7. Mean of Absolute Deviation (MAD) Analysis

If the difference between the input image and the encrypted image is high. Then it is more difficult for the hackers and therefore, it gives us more security against them. MAD gives us the quantified value, that how much encrypted image is displaced from the input image [49,50]. MAD is determined to compute the difference between two images. MAD can be mathematically represented as:

$$MAD=\frac{1}{B\times H}\sum _{y=1}^L\sum _{x=1}^L|O_{xy}-C_{xy}|$$

(8)

where

• $O_{xy}=$ pixels of original image at $(x,y)$ position,
• $C_{xy}=$ pixels of the encrypted image at $(x,y)$ position,
• $B=$ Breadth of Image
• $H=$ Height of image.
• MAD analysis of Various Images are performed, and the results are compiled in Table 9

3.8. Correlation Analysis

Correlation is the measure of dependency of one image on another. Every algorithm developer tries to reduce the dependency. It will be harder for any hacker to perform any kind of malicious activity. Therefore, it increases the algorithm’s security. It is computed by the following equation:

$$corr(O,C)={\displaystyle \frac{E\left((O-{\mu }_o)(C-{\mu }_c)\right)}{{\sigma }_o{\sigma }_c}}$$

(9)

Where

• $corr(O,C)=$ correlation between the original image and its encrypted image
• $O=$ Original Image
• $C=$ Cipher Image
• ${\mu }_o=$ mean of the original image
• ${\mu }_c=$ mean of the encrypted image
• $E=$ Expected Value Operator
• ${\sigma }_o=$ Standard Deviation of the original Image
• ${\sigma }_c=$ Standard Deviation of the cipher Image
• Table 10 gives the correlation between various original images and their cipher images. It’s evident from Table 10, that the correlation is less than 1%. Also in Table 11, correlation of AES and its variations are compared with our proposed algorithm. It was verified that the correlation of our proposed algorithm is less than 1%. Therefore, it shows that it will be hard for the hackers to determine the original image from the cipher image.

3.9. Contrast Analysis

It gives the user to identify the textures of two images. It allows the user to identify any resemblance of texture between two separate images [51,52]. If the texture of an original image and its encrypted has any closeness of texture between them, it computes contrast using eq.( 10). In this equation, it is clearly visible that a co-occurrence matrix is used to compute contrast value. It basically gives any kind of resemblance between any neighboring pixels of the same image. It is mathematically represented as:

$$C={\displaystyle \frac{{\sum }_{x,y}{|x-y|}^{2}p(x,y)}{B\times H}}$$

(10)

Where

• $p\left(x,y\right)=$ gray level co-occurrence matrix.
• $B=$ Breadth of $p(x,y)$
• $H=$ Height of $p\left(x,y\right)$
• $x,y$ represents the location of elements within $p\left(x,y\right)$

In Table 12, it is evident that our proposed algorithm encrypt any two or more images and those encrypted images has the same contrast value. This shows that it will be harder for any hacker or intruder to comprise the security of our proposed algorithm.

3.10. Homogeneity

It measures the closeness of elements within a specified image. It tells how the neighboring elements of a pixel is related to another. This analysis is based on the distribution of any pixel with respect to its neighboring pixels. It gives the statistical distribution over the whole image.

The homogeneity can be determined using the following relation:

$$\sum _{x,y}\frac{p\left(x,y\right)}{1+\left|x-y\right|}$$

(11)

Where

• $p\left(x,y\right)=$ gray level co-occurrence matrix.
• $x,y$ represents the location of elements within $p\left(x,y\right)$

In Table 3, the proposed algorithm is compared with [15,22,28] techniques, where it reduces space complexity by utilizing only two coefficients and taking minimum permutation time. Therefore, the proposed technique is time efficient and less complex as compared to the other techniques. It means, memory cost for the implementation of the proposed technique is reduced. In table 6, the SAC is compared with various established techniques, the ideal value of SAC is 50%. The proposed techniques gives 49% SAC. In table 8, the average entropy of various images is compared with AES and its variants [48]. The ideal entropy should be 8 for a 8-bit image. This means that the information of the input image is scattered evenly in encrypted image. The AES and its variants can only give entropy upto 7.96 The proposed technique gives 7.9952 entropy. It means, it is harder for the hacker to get scattered information from encrypted image. In table 11, the correlation of the proposed technique is with AES and its variants [48]. AES and its variants can provide 2% to 7% similarity between input and encrypted image. The proposed technique has reduced this similarity measure to 0.3% If we reduce the similarity measure, this proves, we are enhancing its security.

4. Conclusion

In this study, an image encryption algorithm is proposed, which is based on CLMs. The algorithm’s security is verified though various tests, which include key space analysis, complexity analysis, sensitivity analysis, strict avalanche criteria, histogram analysis, entropy analysis, mean of absolute deviation analysis, correlation analysis, contrast analysis and homogeneity. These tests have assured a high level of security in applied mathematical, cryptography and engineering applications. Researchers/practitioners can use the proposed encryption algorithm in different fields such as image encryption, data encryption, audio/video encryption etc.