Image Encryption Using Chaotic 3-D Arnold’s Cat Map and Logistic Map

Abstract

Image encryption is different from that of traditional texts or binary data because of some inherent properties of images such as large data capacity, i.e., enormous size and high redundancy (statistical and psycho-visual), making them difficult to handle by traditional methods. In recent years, chaos theory has been explored to find efficient ways to develop secure image cryptosystems. Due to the desirable properties of mixing and sensitivity to initial conditions and parameters (butterfly effect), chaotic systems have found great deal in the domain of image encryption. In this paper, the 2-D chaotic cat map is generalized to its 3-D map counterpart for designing a real-time secure and reliable symmetric encryption design. This new scheme deploys the 3-D Arnold’s cat map to shuffle the positions of image pixels and uses the other chaotic logistic map to perplex the relationship between the plain-image and the cipher-image, thereby significantly enhancing the robustness to differential and statistical attacks. Experimental tests are carried out with comprehensive analysis, demonstrating the high security of the scheme.

1 Introduction

In recent years, in the world of digital image processing and encryption, the transmission of images over an unprotected channel has been a major source of concern. The images being transferred may be of varying importance ranging from a blueprint of a city, digital signatures, satellite images, etc. Therefore, there is a need of encryption of the image to prevent eavesdropping and tampering in order to protect the integrity and information of the images. Various traditional encryption algorithms such as Advanced Encryption Standard (AES), Data Encryption Standard (DES), Rivest, Shamir and Adleman (RSA), International Data Encryption Algorithm (IDEA) have been incorporated for encryption of text data. But due to the high redundancy and bulk capacity of images, these algorithms have not proven to be suited for encryption of images. A new encryption approach termed as Chaotic encryption has found its way into the thick of things and has become a new cryptographic paradigm. The desirable properties for a good encryption algorithm are sensitivity to keys, confusion, and diffusion. All these properties found in chaotic maps which possess ergodic properties are susceptible to initial conditions and achieve this by iterating them. The main difference between these two techniques is that encryption operations are operated upon finite sets, whereas chaos is defined on real numbers.

The approach used in this paper is to deploy a chaotic map for permutation of the pixels and use Chen’s diffusion approach [2] with the help of 1-D logistic map [10], finally making the image completely unrecognizable. The 2-D chaotic Arnold Map has desirable properties of confusion, but we have gone a step forward and used its 3-D counterpart that mathematically has greater mixing properties due to higher dimensionality and larger Lyapunov exponent. The diffusion approach uses the logistic map for generation of chaotic points. Both the processes work in tandem in the sense that the output from the permutation phase is passed on to the substitution phase every time the iteration increases, thus adding to the novelty of the algorithm and thereby strengthening the algorithm. The encryption scheme proposed by [13] also uses 3-D chaotic maps but the implementation is different from our scheme. They have used Arnold 3-D map on the blocks of pixels and used the matrix formed by operating the map for the purpose of diffusion process along with the original image. We have applied the 3-D map directly on the image and obtained a scrambled image. Then the diffusion step is applied with the help of logistic map on the scrambled image only.

1.1 3-D Cat Map

Arnold’s cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat, hence the name [1, 2, 4].

$$\begin{aligned} \varGamma \begin{bmatrix} i_{n+1}\\ j_{n+1} \end{bmatrix}=\left( \begin{bmatrix} 1&1 \\ 1&2\\ \end{bmatrix}\begin{bmatrix} i_n\\ j_n \end{bmatrix}\right) \mod 1 \end{aligned}$$

(1)

The determinant of the matrix is equal to 1 that renders it area preserving. The eigenvalues $\lambda _1$ and $\lambda _2$ of the matrix in Eq. 1 are the Lyapunov characteristic exponents of the map.

$$\begin{aligned} \lambda _1=\frac{3+\sqrt{5}}{2},\quad \lambda _2=\frac{3-\sqrt{5}}{2} \end{aligned}$$

(2)

Since the larger eigenvalue is greater than 1 in magnitude, the map possesses good mixing properties. The map is extended to the 3-D version [5,6,7] by first performing the 2-D operation in each of the three planes and then combining all the results as given below.

Keeping $i_n$ untouched and applying $2-$D map on the $j-k$ plane.

$$\begin{aligned} \begin{bmatrix} i_{n+1}\\ j_{n+1}\\ k_{n+1} \end{bmatrix}=\begin{bmatrix} 1&0&0\\0&1&a_i\\ 0&b_i&a_ib_i+1 \end{bmatrix}\begin{bmatrix} i_{n}\\ j_{n}\\ k_{n} \end{bmatrix} \end{aligned}$$

(3)

Keeping $j_n$ untouched and applying $2-$D map on the $i-k$ plane.

$$\begin{aligned} \begin{bmatrix} i_{n+1}\\ j_{n+1}\\ k_{n+1} \end{bmatrix}=\begin{bmatrix} 1&0&a_j\\ 0&1&0\\ b_j&0&a_jb_j+1 \end{bmatrix}\begin{bmatrix} i_{n}\\ j_{n}\\ k_{n} \end{bmatrix} \end{aligned}$$

(4)

Keeping $k_n$ untouched and applying $2-$D map on the $i-j$ plane.

$$\begin{aligned} \begin{bmatrix} i_{n+1}\\ j_{n+1}\\ k_{n+1} \end{bmatrix}=\begin{bmatrix} 1&a_k&0\\ b_k&a_kb_k+1&0\\ 0&0&1 \end{bmatrix}\begin{bmatrix} i_{n}\\ j_{n}\\ k_{n} \end{bmatrix} \end{aligned}$$

(5)

Combination of these three maps gives a 3-D cat map. The three maps are combined by multiplying the matrices in Eqs. 3, 4, 5.

$$\begin{aligned} \begin{bmatrix} i_{n+1}\\ j_{n+1}\\ k_{n+1} \end{bmatrix} =\left( A\begin{bmatrix} i_{n}\\ j_{n}\\ k_{n} \end{bmatrix}\right) \mod 1 \end{aligned}$$

(6)

$$\begin{aligned} \begin{bmatrix} i_{n+1}\\ j_{n+1}\\ k_{n+1} \end{bmatrix} =\begin{bmatrix} 1+a_ia_kb_j&a_k&a_j+a_ia_k+a_ia_ja_kb_j\\ b_k+a_ib_j +a_ia_kb_jb_k&a_kb_k+1&a_ja_k+a_ia_ja_kb_jb_k+a_ia_kb_k+a_ia_jb_j+a_i\\ a_ib_ib_k&b_i&a_ia_jb_ib_j +a_ib_i+a_jb_j+1 \end{bmatrix}\begin{bmatrix} i_{n}\\ j_{n}\\ k_{n} \end{bmatrix} \end{aligned}$$

(7)

1.2 Logistic Map

The logistic map is a polynomial mapping of degree 2, which is an equivalently recurrence relation. This is a popular example of explaining the complex, chaotic behavior that can arise from very simple nonlinear dynamical equations. The credit of popularization of this map goes to the biologist Robert May [8]. The mathematical definition of logistic map is given below:

$$\begin{aligned} x_{n+1}=\lambda x_n\left( 1-x_n\right) \end{aligned}$$

(8)

Various chaotic encryption algorithms based on the chaotic properties of this map have been designed [11]. It has been found that for $\lambda =4$, the map is chaotic [3, 10], and hence, this chaotic property is utilized in generating chaotic points in the diffusion phase. For initial seed value within $\left( 0,1\right)$, the iterated values stay in $\left( 0,1\right)$.

2 Encryption Scheme

The proposed encryption scheme is completed in three phases—permutation phase, key generation phase, and diffusion phase. A detailed description of each of the three phases is given in the following subsections.

2.1 Permutation Phase

Permutation is performed on the image considered for encryption in the following steps:

Step1: Consider a grayscale image of size $M \times N$. Total number of pixels in the image is MN. Convert the image into 3-D lattice of pixels with each side R pixels wide such that

$$\begin{aligned} R^3=MN \end{aligned}$$

(9)

If in case the number of pixels in the MN is not a perfect cube, try to do zero padding to the image to make it into a perfect cube. For example, the image used in this scheme is of size 512 $\times$ 512. So to convert it into a 3-D cube, the dimensions should be 64 $\times$ 64 $\times$ 64.

Step2: After extraction of all R slices from the image, pile them up to form a cube and operate the 3-D cat map described above on the cube. This is the first part of the initial rounds of iteration. The 6 parameters of the 3-D map also serve as the key for encryption which will be evaluated in the key generation mechanism.

Note: In the 3-D cat map, the variables $i_{n+1}, j_{n+1}, k_{n+1}$ will be the changed coordinates of the pixels after applying the transformation on the initial coordinates $i_n, j_n, k_n$. So all the 6 parameters of the transformation should be integer, because at the end what we will get is a changed coordinate which has to be an integer.

Step3: After performing the permutation above, convert the cube back to a 2-D array, i.e., a matrix.

Now collect the pixel values of the permuted matrix into 1-D array and prepare for the substitution phase.

2.2 Key Generation Mechanism

The key consists of the initial input of 16 characters by the user which is converted into a 128-bit-long key. The steps for generating the key are as follows:

Step1: Take the first eight bits of 128 bit key and XOR it with the last 8 bit of the key. Then take the 9th to 16th bit of the key and XOR it with 113th to 120th bit. Repeat the process till all the bits are exhausted. The computation of first two values for the key is demonstrated below:

$key_1$= $v(1:8) \oplus v(121-128)$

$key_2$= $v(9:16) \oplus v(113-120)$

$key_3$= $v(17:24)\oplus v(105:112)$

$key_4$= $v(25:32)\oplus v(97:104)$

$key_5$=$v(33:40)\oplus v(89:96)$

$key_6$=$v(41:48)\oplus v(81:88)$

$key_7$=$v(49:56)\oplus v(73:80)$

$key_8$= $v(57:64)\oplus v(65:72)$

where v(i : j) means the starting point and the ending point chosen from the 128 bit string entered initially, and $\oplus$ is the bitxor operations between the two operands.

Step2: After getting all the 8 keys from the above procedure, assign the first 6 keys to the parameters of the 3-D map. Setting $a_i=key_1$, $a_j=key_2$, $a_k=key_3$, $b_i=key_4$, $b_j=key_5$, $b_k=key_6$.

2.3 Diffusion

Step1: The diffusion phase starts with the choice of the initial seed to the logistic map. To make the initial seed dependent on the original image and the current round of iteration, use the seed as:

$$\begin{aligned} seed=c_1=\frac{(\sum \limits _{i,j}M_{i,j})+itr}{M \times N \times 256} \end{aligned}$$

(10)

where $M_{i, j}$ is the pixel value at location (i, j), M and N are the dimensions of the image, and itr is the current iteration number. The experiments have been done on square images of size 512, i.e., $M=N=512$, and 256 is the maximum possible graylevels in the image.

Step2: The next step is to provide the seed to the logistic map to generate MN chaotic points, and these chaotic points are stored in the vector c. All the iterated values must fall in the interval (0.2, 0.8), and the value 0 or 1 must not be attained anywhere in the middle as it would imply further iterated values to be 0. To ensure this, after each iterated value is obtained, it is properly digitized by multiplying it with $10^{14}$ and scaled by taking it modulo 255.

$$\begin{aligned} d_i=\lfloor c_i*10^{14}\rfloor mod~255, \quad i=2 \ldots MN \end{aligned}$$

(11)

$\lfloor .\rfloor$ denotes the greatest integer function or the floor function.

$$\begin{aligned} w_1=key_1+key_2+key_3+key_4+key_5+key_6+key_7+key_8 \end{aligned}$$

(12)

Step3: The governing encryption equation is as follows [2]:

$$\begin{aligned} w_i= di_i\oplus \{(v_i + di_i)\mod 255)\}\oplus w_{i-1} \end{aligned}$$

(13)

where $w_i$ = current diffused value, $di_i$ = current digitized value, $v_i$ = current pixel value that has been collected in the $1-D$ array, and $w_{i-1}$ = previously diffused value.

Step4: After all the diffused values have been obtained, convert the 1-D array into a 2-D matrix and replace the original image matrix with this diffused matrix. If more rounds of encryption are needed, just increase the iteration counter and repeat from Step1 of the permutation round.

3 Security Analysis

A good encryption algorithm should be sufficiently robust against several kinds of statistical, cryptanalytic, and brute-force attacks. To prove the robustness of the proposed encryption algorithm against several kinds of attacks, security analysis is performed in this section. In particular, statistical analysis, differential attack analysis, sensitivity analysis with respect to the plaintext and key, etc., has been done. An encrypted image is said to be sufficiently robust to a differential attack, if a minute variation in the plain-image causes a considerable disparity in the cipher-image, i.e., even an iota of change in the original text spreads like an avalanche over the entire encrypted image. The sensitivity of the chaotic maps to initial conditions and parameters is responsible for this kind of effect. This phenomenon is termed as measure of the plaintext sensitivity [9] that can be computed using the following procedures. First of all, a plain-image is encrypted to a cipher-image $C_1$ and then a pixel is chosen randomly in the plain-image. The selected pixel is changed a little, say, a decrement or increment of 1 is made to its decimal value. For example, if the random pixel selected has a intensity value of say 162, then it is changed to 161. The modified image is now encrypted using the same key (used to encrypt the plain-image) to produce a new cipher-image $C_2$. A quantitative comparison is made between the two cipher-images using the number of pixel change rate (NCPR) [12]. A standard benchmark graylevel image ‘Lena’ of size 512 $\times$ 512 has been taken. NPCR is computed between Lena plain-image and its cipher-image using our encryption scheme and found to be more than $99\%$, thus proving the plaintext sensitivity of the proposed algorithm. The quantitative results with respect to different permutation and diffusion rounds are shown in Table 1, with m: permutation round and n: diffusion round.

• 1. Key Sensitivity analysis: The encryption must be highly sensitive to the variation in the encryption keys. Even a one bit difference in the key should cause large amount of diversity in the encrypted images. In our experiment on key sensitivity, key used initially was ‘ASDFGHJKLZXCVBNM’ and the changed key was ‘ASDFGHJKLZXCVBNP’. The change in the encrypted images was above 99%, and in some cases, it was as much as 99.62%. The results of this experiment prove that the proposed encryption scheme is highly sensitive to key change.

• 2. Number of pixels change rate (NPCR): This is a measure of plaintext sensitivity. We take the original image and form a new image from it by randomly selecting a pixel from the original image and increase/decrease its pixel value by 1. Encrypt both the images by the same key and observe the difference in the encrypted images. This test gives a percentage of the count of number of pixels that differ in value at corresponding locations in the two cipher-images. It can be formulated as given below [12]:

  $$\begin{aligned}&\mathbf NPCR =\sum \limits _{j=1}^{N}\sum \limits _{i=1}^M\frac{D(i,j)}{M \times N}\times 100\\&D(i,j)= {\left\{ \begin{array}{ll} 1,\text {if}~C_1(i,j)\ne C_2(i,j) \\ 0, \text {if}~C_1(i,j)\ne C_2(i,j) \end{array}\right. } \end{aligned}$$

  where M and N are the width and height of the cipher-image. The numerical results are shown in Table 1.

• 3. Unified average changing intensity (UACI): This is a measure of the average intensity of differences between the cipher-images $C_1$ and $C_2$, defined as below [12]:

  $$ \mathbf UACI =\sum \limits _{j=1}^{N}\sum \limits _{i=1}^{M} \frac{\left| C_1(i,j)-C_2(i,j)\right| }{L}\times 100 $$

  where L is the number of possible graylevel values. As the experiment is random in nature, i.e., the pixel chosen for modification is random, the experiment should be repeated suitably large number of times with diverse images. The ideal UACI value should be near 33 for the encryption to resist differential attacks. The numerical results for the UACI values are given in Table 2 and are found to be favoring the proposed encryption scheme.

Table 1 NPCR values for different rounds of encryption

Table 2 NPCR values for different rounds of encryption

3.1 Statistical Analysis

An encryption algorithm must be robust to statistical attacks as well. The following measures are defined to evaluate the statistical robustness of the encryption algorithm.

Fig. 1

Encryption results: a original image; b histogram of original image; c encrypted image; d histogram of encrypted image

Fig. 2

Encryption results: a original image; b histogram of original image; c encrypted image; (d) histogram of encrypted image

Fig. 3

Encryption results: a Correlation of pixels in horizontal direction; b Correlation of pixels in vertical direction

• 1. Histogram analysis: The histogram is plotted for the encrypted image and checked for uniformity. The histogram of the encrypted image should be sufficiently uniform to resist statistical attack. A uniform histogram indicates that there is uniform probability of finding any pixel value in the image. The histograms plotted for plain-image and cipher-image are shown in Figs. 1 and 2. It is clearly visible from the figures that the histograms of cipher-images are very much uniform (Fig. 3).

• 2. Entropy: The entropy of a system as defined by ‘Shannon’ gives a measure of uncertainty about its genuine structure. Let t be the message source, then the entropy H(t) is defined as below:

  $$ H(t)=-\sum \limits _{i=1}^{S}p(t_i)\times \log _2t_i $$

  where S is the total number of symbols, which in our case is 256, $p(t_i)$ stands for the probability of occasions of symbol $t_i$, and the $\log _2$ is used so that the entropy is measured in bits. An entropy more closer to the figure $\log _2L$ bits implies more uniform histogram that in turn ensures the security of encrypted information. A quantitative evaluation is given in the Table 3. From the table, it is clear that the entropy values obtained are very close to $\log _2 256=8$, which again indicates a uniform histogram and hence encrypted information is secure enough.

• 3. Correlation of adjacent pixels: The correlation between the pixels of an encrypted image should be close to zero. To compute the correlation between two adjacent pixels, select P number of pairs of horizontally/vertically adjacent pixels randomly from the image. Compute the correlation coefficient $\rho$ for each pair using the following equations:

  $$ \rho =\frac{cov(u,v)}{\sigma _u\cdot \sigma _v} $$

  where u and v are the values of the adjacent pixels.

  $$ cov(u,v)=\mathbf E \{(u-\mathbf E (u))(v-\mathbf E (v))\}. $$
  $$ \mathbf E (u)=\frac{\sum _{i=1}^{P}u_i}{P}. $$
  $$ \sigma _u^2=\frac{\sum _{i=1}^{P}(u_i-\mathbf E (u))}{P} $$

  The results are given in Tables 4 and 5 for $P=5000$. Correlation between the horizontal adjacent pixels and correlation between the vertical adjacent pixels have been plotted to visualize the outcome as shown in Fig. 3.

• 4. Parameter sensitivity: The parameters in this algorithm are the entries of the Arnold 3-D map and the seed value to the logistic map. These values in turn depend on:

• (a) The user entered key :If the key entered is altered by even a single bit, then the key scheduling algorithm gives an entirely different digest of the input that is further processed by the maps.

• (b) The size of the image : The size of the image plays an important role since for our encryption, the basic need is that the size of the image should be a perfect cube in order to apply the 3-D map. Secondly, the sum of all the pixels is one more key aspect in our encryption algorithm which in turn depends on size.

• (c) To check the parameter sensitivity, the UACI and the NPCR tests have already been performed, but we try to give a correlation test analysis as well. We encrypt the same image by two different sets of keys and find the correlation between the two images. The results in Table 6 show that the correlation between the two cipher images $C_1$ and $C_2$ is close to zero, where $C_1$ is the original cipher and $C_2$ is the changed cipher, thereby strengthening our claims of parameter sensitivity.

4 Conclusion

As a concluding note, we would like to summarize the proposed algorithm’s pros and some cons that are seen from the numerical results. Firstly, the use of 3-D map makes it more topologically mixing and thereby producing more confusion in lesser number of rounds. The NPCR results show that the sensitivity to plaintext is around $100\%$, i.e., if there is even the slightest change in the plaintext, the final encrypted text will be approximately $100\%$ changed. The UACI results show that there is >33.3% change in the final encrypted image if the original image is changed. The encryption approach shows sensitivity to the key also. Even one bit change in the encryption key can produce a difference over $99\%$ in the cipher-image. The entropy value is close to 8, which shows that in the final encrypted image, each pixel has maximum amount of information content in it, i.e., $\log _2L$ bits. The correlation between the pixels horizontally initially is not close to 0 but as the number of rounds increases, the correlation gets reduced subsequently and tends toward 0, thereby showing improvement of the scheme. Hence, looking over all results, we have got some concrete and substantial proofs to think that our encryption approach can withstand the major cryptographic attacks and can serve as a secure image encryption approach.

Table 3 Entropy values of two different images

Table 4 Horizontal correlation between the pixels in the encrypted image

Table 5 Vertical correlation between the pixels in the encrypted image

Table 6 Correlation of original cipher with incorrect cipher image