1 Introduction

1.1 Research background

With the development of communication and social networking technologies, multimedia data such as images, video and audio are transmitted over the network more conveniently. Consequently, the security of multimedia data becomes more and more important. A huge amount of digital data, which are either private or confidential, need to be protected against the misuse. Therefore, a well secured encryption algorithm is essential for secure communication. The security of multimedia data is receiving more and more attention due to the widespread transmission over various communication networks. Yet a potential risk of information security always exists during the processing and transmission of digital images over an open network. The properties of an ideal encryption scheme for data security such as confidentiality, integrity and authenticity have been drawn more attention by researchers in the field of image encryption. Therefore, designing good image encryption schemes has become a focal research topic. The conventional cryptographic algorithms like RSA (Rivest Shamir Adleman) and DES (Data Encryption Standard) are not effective for encryption of data owing to capacity of data, intrinsic characteristics of images, high redundancy and so on. Due to the main features of chaotic systems like sensitivity to initial conditions, ergodicity, simple analytic description and high complex behavior, cryptographic algorithms using chaotic systems are more effective and secure than traditional cryptographic algorithms. Further, chaotic system based encryption algorithms have several inherent features favorable to data security.

1.2 Literature overview

Over the past decades, several chaos based image encryption techniques have been widely investigated in the field of secure communication. In 1989, Matthews [23] developed the first chaotic stream encryption algorithm. After that, a symmetric image encryption algorithm using the two-dimensional standard Baker map has been proposed by Fridrich [8] in 1998. Lately, the chaotic Boolean bit function has been employed and applied to the image encryption in [14]. In [19], a color image encryption scheme has been designed by using chaos with the help of bijective function. Synchronization of two different six-dimensional hyperchaotic systems [36] has been utilized for image encryption. The total plain image characteristics, crossover operator and chaos have been applied for image encryption respectively in [9, 24]. The statistical properties of image encryption algorithm have been improved by multiple chaotic maps in [40]. A fast color image encryption scheme has been designed in [20] by one-time S-Box, which is generated by the complex chaotic system. DNA sequence and hyperchaotic system have been utilized for image encryption in [11].

In order to improve the security and efficiency performance, several image encryption algorithms have been designed by applying the theory of fractional calculus. The fractional differential equations are generalizations of classical differential equations and it gained popularity in the nonlinear dynamical systems. Many real world systems have been determined by fractional derivatives since they allow more flexibility in the model [15, 21, 42]. The study of chaotic dynamics of fractional order systems has been a hot topic in the field of nonlinear science. Furthermore, applications of control and synchronization of fractional order chaotic systems have been reported in many areas, for instance in medicine [1], telecommunications [32], robotics [7], secure communication and cryptography [3, 25–28, 30, 35]. Several types of control techniques and methodologies have been investigated for synchronizing fractional order systems such as feedback control technique [25], adaptive observer [41], active control method [4], non-fragile control [2], multi-scale synchronization technique [26], fast projective synchronization method [28], Lyapunov based control [17], hybrid phase synchronization [27] and sliding mode control [29]. Apart from synchronization, anti-synchronization is a dominating phenomenon in symmetrical oscillators. The ultimate aim of anti-synchronization is to study the opponent behavior of the master and slave systems so that the sum of their states will converge to zero asymptotically. Due to this reason, different control methods have been utilized for anti-synchronizing chaotic systems in [6, 13, 16, 31, 33].

A short overview of the recently proposed image encryption schemes [10, 12, 18, 26, 37, 39] build from fractional order dynamical systems are given hereafter. A color image encryption algorithm by using coupled-map lattices and a fractional order chaotic system has been proposed to enhance the security and robustness of the encryption algorithms with a permutation-diffusion structure in [37]. The scrambled image has been encrypted once again by the pseudorandom sequences generated from the combined fractional-order hyperchaotic systems in [10]. An encryption algorithm has been constructed in [12] by the fractional order hyperchaotic system which can effectively enhance the cryptosystem security. In [18], a color image encryption algorithm by combining the reality-preserving fractional DCT with chaotic mapping in HSI space has been presented. A new cryptosystem has been proposed for an image encryption by using synchronized fractional order King Cobra chaotic systems with the supports of multiple cryptographic assumptions in [26]. In [39], an image encryption algorithm has been presented where the original image is encoded by a nonlinear function of a fractional chaotic state. Further, these encryption algorithms are experimentally demonstrated which includes correlation analysis, histogram analysis, and key sensitivity analysis to verify the security level of the encryption scheme. Compared to integer order systems, the fractional order systems are found to have more complex dynamics because the fractional derivatives have complex geometrical interpretation due to their nonlocal character and high nonlinearity. Further, the derivative orders can be also used as secret keys as well, which will increase the key space of the cryptosystem. To the best of authors knowledge, few more encryption techniques are available in the literature using the fractional order chaotic systems. Therefore, for the purpose of high security, the construction of new image encryption algorithm by applying fractional order chaotic systems is very essential.

1.3 Our contribution

Based on the aforesaid studies, the anti-synchronization scheme for fractional order reverse butterfly-shaped hyperchaotic systems is investigated via active control technique. The necessary conditions are derived to achieve the anti-synchronization between two systems. Apart from existing image encryption algorithms, a new image encryption-decryption algorithm is introduced by utilizing anti-synchronized fractional order hyperchaotic systems and encryption (decryption) of encryption (decryption) techniques, which is entirely different from other existing image encryption techniques. Further, we have shown that the new algorithm has higher level security by various experimental analysis tests and comparison results.

In Section 2, some basic theories of fractional calculus are given. In Section 3, the fractional order reverse butterfly-shaped hyperchaotic system is described. The process of anti-synchronization between two identical fractional order reverse butterfly-shaped hyperchaotic systems using active control technique is elaborately studied in Section 4. Section 5 contributes to the applications: a new image encryption algorithm is described by the anti-synchronized scheme. The experimental analysis of the proposed algorithm is presented in Section 6. The performance analysis and the security of the proposed algorithm are compared in Section 7. The conclusions of this paper are drawn in Section 8.

2 Preliminaries

In this paper, we have used the Caputo fractional differential operator since the Caputo’s derivative of a constant is zero and it has conventional initial conditions.

Definition 1

[5] The Caputo fractional derivative is defined as

$$\begin{array}{@{}rcl@{}} D^{\alpha}f(t) = \frac{1}{\Gamma(n-\alpha)}{{\int}_{a}^{t}} (t-\tau)^{-\alpha+n-1}f^{(n)}(\tau)d\tau, \end{array} $$
(1)

where n = [α] + 1, [α] is the integer part of α, D α is called the α-order Caputo differential operator, Γ is the usual Gamma function given by and

$$\begin{array}{@{}rcl@{}} {\Gamma}(z) = {\int}_{0}^{\infty} e^{-t}t^{z-1} dt. \end{array} $$
(2)

Further it is noted that Γ(z + 1) = z Γ (z).

Theorem 1

[22] The following autonomous fractional order system

$$\begin{array}{@{}rcl@{}} D^{\alpha}x(t) = Ax(t), \ x(0) = x_{0}, \end{array} $$
(3)

where \(0<\alpha \leq 1, \ x\in \mathbb {R}^{n}\) is asymptotically stable if and only if

$$\begin{array}{@{}rcl@{}} |\arg(eig(A))|>\frac{\alpha\pi}{2}. \end{array} $$
(4)

Also, the system (3) is stable if and only if \(|\arg (eig(A))|\geq \frac {\alpha \pi }{2}\) and those critical eigenvalues that satisfy \(|\arg (eig(A))|=\frac {\alpha \pi }{2}\) have geometric multiplicity one.

Theorem 2

[34] A necessary condition for the system (3) to remain chaotic is keeping at least one eigenvalue λ in the unstable region. This means

$$\begin{array}{@{}rcl@{}} \alpha>\frac{2}{\pi}\ \arctan \left( \frac{|Im(\lambda)|}{Re(\lambda)}\right). \end{array} $$
(5)

3 Description of fractional order hyperchaotic system

Consider the fractional form of the reverse butterfly-shaped hyperchaotic system described in [38],

$$\begin{array}{@{}rcl@{}} \begin{array}{lll} D^{\alpha}x_{1}&=a(x_{2}-x_{1})+x_{4},\\ D^{\alpha}x_{2}&=bx_{1}+kx_{1}x_{3},\\ D^{\alpha}x_{3}&=-cx_{3}-hx_{1}x_{2},\\ D^{\alpha}x_{4}&=x_{1}x_{3}-dx_{2}, \end{array} \end{array} $$
(6)

where 0 < α ≤ 1, \(x=(x_{1},x_{2},x_{3},x_{4})^{T}\in \mathbb {R}^{4}\) is the state variable and a, b, c, d, h, k are the parameters of the system (6). D α is the α-order differential operator in the sense of Caputo [5]. The authors in [38] have been shown that the integer order (α = 1) system (6) behave hyperchaos for the parameters a = 10, b = 40, c = 2.5, d = 2, h = 1 and k = 16.

Throughout this manuscript, we have fix the same parameters a = 10, b = 40, c = 2.5, d = 2, h = 1 and k = 16 for the fractional order system (6). According to Theorem 1, the fractional order system is stable for every α ≤ 0.9606. The system (6) exhibit chaos with two positive Lyapunov exponents when the fractional order α > 0.9606 according to Theorem 2. Therefore, the system (6) is called as a fractional order reverse butterfly-shaped hyperchaotic system and the corresponding hyperchaotic attractors when α = 0.97 are depicted in Fig. 1.

Fig. 1
figure 1

Hyperchaotic attractors of the system (6) when α = 0.97

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4 Anti-synchronization of two identical fractional order hyperchaotic systems

In this section, an active control technique is applied to achieve anti-synchronization between two identical fractional order reverse butterfly-shaped hyper chaotic systems.

Consider the fractional order system (6) as a master system and the following identical system of (6) as a slave system

$$\begin{array}{@{}rcl@{}} \begin{array}{lll} D^{\alpha}y_{1}&=a(y_{2}-y_{1})+y_{4}+u_{1},\\ D^{\alpha}y_{2}&=by_{1}+ky_{1}y_{3}+u_{2},\\ D^{\alpha}y_{3}&=-cy_{3}-hy_{1}y_{2}+u_{3},\\ D^{\alpha}y_{4}&=y_{1}y_{3}-dy_{2}+u_{4}, \end{array} \end{array} $$
(7)

where 0 < α ≤ 1, \(y=(y_{1},y_{2},y_{3},y_{4})^{T}\in \mathbb {R}^{4}\) is the state variable, u = (u 1, u 2, u 3, u 4)T is the active control function to be determined later so that both systems (6) and (7) are anti-synchronized successfully.

To investigate the anti-synchronization between the systems (6) and (7), we define the synchronization error states as e i = y i + x i for i = 1, 2, 3, 4. The ultimate aim is to select the active control function u such that

$$\begin{array}{@{}rcl@{}} \lim\limits_{t\rightarrow\infty}\|e_{i}(t)\|=\lim\limits_{t\rightarrow\infty}\|y_{i}(t)+x_{i}(t)\|=0, \ i=1,2,3,4. \end{array} $$
(8)

Then, the fractional order error dynamical system between the systems (6) and (7) is described by

$$\begin{array}{@{}rcl@{}} \begin{array}{lll} D^{\alpha}e_{1}(t)&=a(e_{2}-e_{1})+e_{4}+u_{1},\\ D^{\alpha}e_{2}(t)&=be_{1}+k(y_{1}y_{3}+x_{1}x_{3})+u_{2},\\ D^{\alpha}e_{3}(t)&=-ce_{3}-h(y_{1}y_{2}+x_{1}x_{2})+u_{3},\\ D^{\alpha}e_{4}(t)&=-de_{2}+y_{1}y_{3}+x_{1}x_{3}+u_{4}. \end{array} \end{array} $$
(9)

Theorem 3

The fractional order mater system (6) and the slave system (7) are globally asymptotically anti-synchronized with the following active control functions

$$\begin{array}{@{}rcl@{}} \begin{array}{lll} u_{1}(t)&=v_{1}-ae_{2}-e_{4},\\ u_{2}(t)&=v_{2}-be_{1}-k(y_{1}y_{3}+x_{1}x_{3}),\\ u_{3}(t)&=v_{3}+h(y_{1}y_{2}+x_{1}x_{2}),\\ u_{4}(t)&=v_{4}+de_{2}-y_{1}y_{3}-x_{1}x_{3}, \end{array} \end{array} $$
(10)

where v i is a linear function of e i such that v i < 0 for i = 1, 2, 3, 4.

Proof

The fractional order error dynamical system (9) together with active control functions u i defined in (10) yields

$$\begin{array}{@{}rcl@{}} \begin{array}{lll} D^{\alpha}e_{1}(t)&=v_{1}-ae_{1},\\ D^{\alpha}e_{2}(t)&=v_{2},\\ D^{\alpha}e_{3}(t)&=v_{3}-ce_{3},\\ D^{\alpha}e_{4}(t)&=v_{4}. \end{array} \end{array} $$
(11)

Since by hypothesis, without loss of generality, we assume that v i (t) = −l i e i where l i > 0 is the gain of v i as well as active control functions u i for i = 1, 2, 3, 4.

Then, the system (11) can be written as

$$\begin{array}{@{}rcl@{}} \begin{array}{lll} D^{\alpha}e_{1}(t)&=-(l_{1}+a)e_{1},\\ D^{\alpha}e_{2}(t)&=-l_{2}e_{2},\\ D^{\alpha}e_{3}(t)&=-(l_{3}+c)e_{3},\\ D^{\alpha}e_{4}(t)&=-l_{4}e_{4}. \end{array} \end{array} $$
(12)

The Jacobian matrix J of fractional order error dynamical system (12) is

$$\begin{array}{@{}rcl@{}} J=\left( \begin{array}{cccc} -(l_{1}+a) & 0 & 0 & 0 \\ 0 & -l_{2} & 0 & 0 \\ 0 & 0 & -(l_{3}+c) & 0 \\ 0 & 0 & 0 & -l_{4} \end{array} \right) \end{array} $$
(13)

The eigenvalues λ i , i = 1, 2, 3, 4 of (13) are λ 1 = −(l 1 + a), λ 2 = −l 2, λ 3 = −(l 3 + c) and λ 4 = −l 4. Since l i > 0 for every i, a = 10 and c = 2.5, then all eigenvalues are less than zero. Further, the value of |arg(λ i )| is equal to π for i = 1, 2, 3, 4. Thus, the asymptotically stable condition (4) is satisfied for the fractional order error dynamical system (12). Consequently, the error states e i are tend to zero as t → ∞. Therefore, the proposed active control function is fulfilled the requirement (8). Hence, anti-synchronization between the master system (6) and the slave system (7) is achieved successfully. □

4.1 Numerical simulations

Consider the initial values of the master system (6) and the slave system (7) respectively by (x 1(0), x 2(0), x 3(0), x 4(0)) = (0.1, 0.1, 0.1, 0.1) and (y 1(0), y 2(0), y 3(0), y 4(0)) = (−1, −1, 1, 1). For convenience, we fix the fractional order α = 0.98 and the controller gain l i is selected as l i = l > 0 for i = 1, 2, 3, 4.

In simulations, the state trajectories between the systems (6) and (7) are depicted in Figs. 2, 3 and 4 for l = 0.5, l = 5 and l = 15 respectively. Further, the corresponding time response of anti-synchronization error states are depicted in Figs. 5, 6 and 7 respectively. From Figs. 5–7, we observed that the convergence rate of anti-synchronization errors are gradually decreased in the fractional order α = 0.98 by increasing the value of controller gain l. Thus, we conclude that anti-synchronization is achieved faster by increasing the control gain.

Fig. 2
figure 2

State trajectories between the master system (6) and the slave system (7) when l = 0.5

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Fig. 3
figure 3

State trajectories between the master system (6) and the slave system (7) when l = 5

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Fig. 4
figure 4

State trajectories between the master system (6) and the slave system (7) when l = 15

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Fig. 5
figure 5

Time response of the anti-synchronization error states when l = 0.5

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Fig. 6
figure 6

Time response of the anti-synchronization error states when l = 5

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Fig. 7
figure 7

Time response of the anti-synchronization error states when l = 15

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In the following section, these anti-synchronized fractional order reverse butterfly-shaped hyperchaotic systems are utilized to develop an encryption and decryption algorithm for digital images.

5 Proposed encryption-decryption algorithm

In this section, a new encryption algorithm for an image without any key exchange is introduced by anti-synchronized fractional order reverse butterfly-shaped hyperchaotic systems with the support of the discrete logarithm problem and it can be described as follows.

Assume that two cryptographic entities Alice and Bob. Let Alice be a sender and Bob be a receiver. Also, assume that the master system (6) as a sender system and the slave system (7) as a receiver system. Both Alice and Bob agree on the fractional order α > 0.9606 and l > 0 at time t > t 0 where t 0 is a time if anti-synchronization errors between the systems (6) and (7) are tend to zero from t 0 onwards for given values of α and l.

Further, assume that I is the original image and D is the dummy image of size M × N. The images (A 1, A 2, A 3) and (B 1, B 2) are the encrypted images computed by Alice and Bob respectively. R is the decrypted image, which is computed by Bob. Alice and Bob agree on a positive integer ρ such that ρ < 256 and \(\gcd (\rho ,256) = 1\). Note that, |X| is the absolute value of X and f l o o r(X) is the largest integer less than or equal to X.

  1. Step 1.

    Alice wants to send a digital image I.

  2. Step 2.

    Alice chooses a real number t 1 > t 0 and finds the solution of the system (6) at t 1.

  3. Step 3.

    She computes the first encrypted image A 1 of I.

    $$A_{1}\equiv I\rho^{r}\ (\text{mod}\ 256), $$

    where \(r\equiv |floor(t_{1}\sum \limits _{i=1}^{4} x_{i}(t_{1}))|\) (mod 256).

  4. Step 4.

    The element r is kept secret by Alice and she sends A 1 to Bob.

  5. Step 5.

    Bob receives A 1, then he chooses a real number t 2 > t 0 and finds the solution of the system (7) at t 2. Then, he chooses a dummy image D with the size of A 1.

  6. Step 6.

    He computes the second encrypted image B 1 of I by using the dummy image D and assign the dummy image D as B 2.

    $$\begin{array}{ll} B_{1}&\equiv(A_{1}\rho^{s} +D)\ (\text{mod}\ 256),\\ B_{2}&\equiv D\ (\text{mod}\ 256), \end{array} $$

    where \(s\equiv |floor(t_{2}\sum \limits _{i=1}^{4} y_{i}(t_2))|\) (mod 256).

  7. Step 7.

    The element s is kept secret by Bob and he sends (B 1, B 2) to Alice.

  8. Step 8.

    Alice receives B 1 and B 2, then she computes the resulting encrypted image A 2 of I and the encrypted dummy image A 3 of D.

    $$\begin{array}{ll} A_{2}&\equiv B_{1}\rho^{-r}\ (\text{mod}\ 256), \\ A_{3}&\equiv B_{2}\rho^{-r}\ (\text{mod}\ 256). \end{array} $$
  9. Step 9.

    She sends the encrypted images A 2 and A 3 to Bob.

  10. Step 10.

    Finally, Bob recovers an original image I by computing

    $$R\equiv (A_{2}-A_{3})\rho^{-s}\ (\text{mod}\ 256). $$

For,

$$\begin{array}{llll} (A_{2}-A_{3})\rho^{-s}&\equiv(B_{1}\rho^{-r}-B_{2}\rho^{-r})\rho^{-s} \ (\text{mod}\ 256)\\ &\equiv(B_{1}-B_{2})\rho^{-r}\rho^{-s} \ (\text{mod}\ 256)\\ &\equiv(A_{1}\rho^{s}+D-D)\rho^{-(r+s)}\ (\text{mod}\ 256)\\ &\equiv(I\rho^{r}\rho^{s})\rho^{-(r+s)}\ (\text{mod}\ 256)\\ {\kern50pt}R&\equiv I\ (\text{mod}\ 256). \end{array} $$

Remark 1

The proposed encryption and decryption algorithm is fully based on the discrete logarithm problem with the backbone of fractional order systems. Obviously, it contains an encryption (decryption) of encryption (decryption) images more than one time. Therefore, this algorithm is called as cascade encryption and decryption or multiple encryption and decryption algorithm.

The purpose of introducing multiple encryption is, no one got fully encrypted image or encrypted trick in the middle of the two parties because the encrypted image is encrypted more than one time. Consequently, nobody decrypts an image from the knowledge of encrypted image between two parties. Hence, the proposed cascade encryption and decryption processes are more efficient than existing encryption algorithms established by chaotic systems.

6 Experimental analysis and results

In this section, the performance of the proposed cascade image encryption algorithm is analyzed and its high level security has been investigated experimentally through various security test measures. These measures are taken as follows: key space analysis, statistical analysis including correlation coefficients of adjacent pixels, information entropy analysis, histogram analysis and test security against differential attack.

The standard image processing color plain image of Lena and the Baboon image with a size of 256 × 256 are utilized for encryption and decryption processes. The parameters of the systems (6) and (7) for experimentation are: a = 10, b = 40, c = 2.5, d = 2, h = 1 and k = 16. The value of the fractional order α, the feedback gain l, a positive integer ρ, the real numbers t 1 and t 2 are taken as 0.98, 5, 5, 7 and 3.85 respectively. Assume that the Lena image is an original image I and the Baboon image is a dummy image D. We implement the proposed algorithm by using Matlab 7.1. The original color image, the dummy image and the encrypted images are displayed in Fig. 8.

Fig. 8
figure 8

a Original image I, b First encrypted image A 1, c Second encrypted image B 1, d Dummy image B 2, e Resulting encrypted image A 2 and f Encrypted dummy image A 3

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6.1 Key space analysis

The size of key space is the total number of different keys that can be applied in the encryption process. A good encryption algorithm should be sensitive to the secret keys and the key space should be large enough to ensure the security of the encryption algorithm against brute-force attacks. In our cascade encryption algorithm, the initial conditions of the master system x i (0) and the slave system y i (0), i = 1, 2, 3, 4, the fractional order α, the parameters a, b, c, d, h, k, the time t 1 and the feedback control gain l are secret keys. If the precision is 10−14, then the size of the initial conditions key space is 1014×8. Additionally the fractional order, the time and the feedback control gain keys can also produce large key spaces. Therefore, the total key space is more than 1014×8, which is greater than 2370 approximately. Hence, the proposed cascade encryption algorithm has a large enough key space to resist all varieties of brute-force attacks.

6.2 Correlation analysis

The correlation coefficient between images can be used to evaluate the quality of the encryption algorithm. In ordinary images having definite visual content, each adjacent pixels are highly correlated. This means that the correlation coefficients of plain image are closer to 1. For the good encryption algorithm, the correlation coefficients among the adjacent pixels of the encrypted image are close to 0. The correlation coefficient of two adjacent pixels in an image is calculated by using the following formula:

$$\begin{array}{@{}rcl@{}} \gamma_{xy}=\frac{cov(x,y)}{\sqrt{D(x)}\sqrt{D(y)}}, \end{array} $$
(14)

where

$$\begin{array}{@{}rcl@{}} cov(x,y)&=&\frac{1}{N}\sum\limits_{i=1}^{N} (x_{i}-E(x))(y_{i}-E(y)),\\ D(x)&=&\frac{1}{N}\sum\limits_{i=1}^{N} (x_{i}-E(x))^{2},\\ E(x)&=&\frac{1}{N}\sum\limits_{i=1}^{n} (x_{i}), \end{array} $$

where x, y are grey scale values of two adjacent pixels in the image, c o v(x, y) is the covariance between x and y, D(x) and D(y) are the variance of x and y respectively, and E(x), E(y) are the expectation of x and y.

For the proposed algorithm, the correlation coefficients of two adjacent pixels in the original image and encrypted images are tried out respectively in horizontal, vertical and diagonal directions. Table 1 shows the outcomes of the correlation coefficients in three directions. Figures 9 and 10 show the corresponding distribution of the original and the resulting encrypted image in horizontal, vertical and diagonal directions respectively. From Table 1, one can see that the estimated correlation coefficients of encrypted images in three directions are very close to 0, implying that the ciphered image has been well encrypted. Therefore, the proposed encryption algorithm is secure and robust against correlation attacks.

Table 1 Correlation coefficients of two adjacent pixels in original image and encrypted images
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Fig. 9
figure 9

The correlation of two adjacent pixels in different directions for original image I: (a) Horizontal, (b) Vertical and (c) Diagonal

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Fig. 10
figure 10

The correlation of two adjacent pixels in different directions for resulting encrypted image A 2: (a) Horizontal, (b) Vertical and (c) Diagonal

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6.3 Information entropy analysis

Information entropy defines the randomness and the unpredictability of an information in the image. To measure the value of entropy H(s) of a source s, we have

$$\begin{array}{@{}rcl@{}} H(s) = -\sum\limits_{i=0}^{2^{N}-1}P(s_{i})\log_{2}P(s_{i}), \end{array} $$
(15)

where s i is the i-th gray scale value for an 256 gray level image, P(s i ) is the probability of s i , s i ∈ s. For truly random source emitting 2N symbols, the entropy value of the source is H(s) = N. For a random image with 256 gray levels, the entropy should be H(s) = 8 theoretically. However, a good encryption algorithm should produce an encrypted image with the entropy very close to 8. For the proposed algorithm, the entropy values of encrypted images are calculated and listed in Table 2. The results show that the information entropies of encrypted images are close to 8 and the resulting encrypted image is very close to 8. Hence, the proposed encryption algorithm is secure against the entropy analysis.

Table 2 Information entropy of encrypted images
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6.4 Histogram analysis

For an image encryption algorithm, the histogram analysis is very important because it describes the distribution of the image pixels by plotting the number of pixels at each intensity level. If the histogram of an encrypted image is uniform, then the encryption scheme is more robust against statistical attack and differential attack. Figure 11a and b represents the histograms of the original and resulting encrypted images in red, green and blue color components respectively. It shows that histograms of the resulting encrypted image are uniform and significantly different from the histograms of original image. Hence, it does not provide any clue to employ statistical attack and differential attack on the encrypted image.

Fig. 11
figure 11

Histograms of the color image in red, green and blue components: (a) Original image I and (b) Resulting encrypted image A 2

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6.5 Differential attack analysis

Differential attack means that attacker creates a slight change to the original image, and use the proposed image encryption algorithm to encrypt for the original image before and after changing, to find out the relationship between the original image and the cipher image through comparing two encrypted images. The number of pixel change rate (NPCR) and the unified average changing intensity (UACI) are two most common measures used to assess the strength of image encryption algorithms with respect to differential attacks. The NPCR is applied to measure the percentage of the number of pixels change rate of the ciphered image while one pixel of the original image has changed and it is calculated as:

$$\begin{array}{@{}rcl@{}} NPCR=\frac{1}{M\times N}\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{N}D(i,j)\times 100 \% , \end{array} $$
(16)

where

$$\begin{array}{@{}rcl@{}} D(i,j) = \left\{ \begin{array}{r} 0 \ \text{if}\ I_{1}(i,j) = I_{2}(i,j)\\ 1 \ \text{if}\ I_{1}(i,j)\neq I_{2}(i,j) \end{array} \right., \end{array} $$

where I 1(i, j) and I 2(i, j) are the pixel gray value of two cipher images in the same position. The closer the NPCR comes to 100 %, the more sensitive the encryption algorithm is to the original image and the more effective the encryption algorithm resists differential attack.

The UACI is applied to measure the percentage of the the average intensity difference of two ciphered images, whose corresponding original image has only one pixel difference and it is calculated as:

$$\begin{array}{@{}rcl@{}} UACI=\frac{1}{M\times N}\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{N}\frac{|I_{1}(i,j)-I_{2}(i,j)|}{2^{N}-1}\times 100 ~\%, \end{array} $$
(17)

where I 1(i, j) and I 2(i, j) are the pixel gray value of two cipher images in the same position. The value of UACI is very close to 33%. The greater the UACI is, the better the encryption algorithm resists the differential attack.

For a image with 256 gray levels, the expected NPCR and UACI values are 99.6094 % and 33.4635 % respectively. For the proposed algorithm, the NPCR and UACI values are given in Table 3 for resulting encrypted images with one bit difference in original image. Note that the value of NPCR and UACI of resulting encrypted image is higher than their expected values.

Table 3 NPCR and UACI percentage of encrypted images
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7 Comparison with previous work

In this section, we compare the security performance of the proposed encryption algorithm with other existing chaos based image encryption algorithms suggested in Refs. [9, 10, 12, 14, 19, 20, 24, 36, 37, 40].

The total key space of the proposed algorithm is greater than 2370, which is enough to prevent the exhaustive searching. Thus, brute-force attacks on the key are computationally infeasible and the proposed scheme has the large key space size than other encryption algorithms in Refs. [9, 10, 12, 19, 20]. This signifies more number of trials required to crack the proposed encryption algorithm by comparing the other chaos based encryption algorithms.

The comparison performed of the correlation coefficient of the proposed algorithm in Table 4 shows that the proposed encryption algorithm is superior to other methods reported in Refs. [9, 12, 14, 37]. The comparison of correlation coefficients of two adjacent pixels of encrypted Lena image in different directions with some other algorithms is depicted in Fig. 12. It shows that the correlation coefficients of the proposed algorithm in horizontal, vertical and diagonal directions are close to 0. Therefore, The encrypted image using our proposed algorithm has the highest performance in the horizontal, vertical and diagonal directions. Table 5 compares information entropy using the proposed encryption algorithm with those using the existing algorithms mentioned in Refs. [9, 10, 14, 19, 20, 24, 36, 37, 40]. Hence, the entropy obtained using our proposed algorithm is indeed closer to the maximum entropy value of 8, which shows the strength of the proposed encryption algorithm. So, information leakage in the encryption process could be negligible and the proposed algorithm is secure against entropy analysis. Table 6 compares the NPCR and UACI for the proposed encryption algorithm and the existing algorithms in Refs. [9, 10, 12, 14, 19, 20, 24, 40]. From the results, one can easily see that the proposed encryption algorithm achieves a higher performance by comparing the other methods. Therefore, the proposed algorithm is very sensitive with respect to the small changes in the original image and it has a strong power and secures to resist the differential attack.

Fig. 12
figure 12

Comparison graph of correlation coefficients of two adjacent pixels of encrypted Lena image in horizontal, vertical and diagonal directions with some other algorithms

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Table 4 Comparison of correlation coefficients of two adjacent pixels in different directions encrypted Lena image using the proposed algorithm with some other algorithms
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Table 5 Comparison of information entropy of encrypted Lena images with different algorithm
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Table 6 The results of NPCR and UACI with different algorithm
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Remark 2

The experimental and comparison results show that the proposed cascade encryption algorithm has large key space and more secure against the most common attacks such as correlation attack, entropy attack, differential attack, sensitivity to the secret key. Therefore, the proposed encryption algorithm can be applied to encrypt images for transmission over insecure channel.

8 Conclusions

In this paper, anti-synchronization scheme for fractional order reverse butterfly-shaped hyperchaotic system has been suggested by using active control technique . In order to verify the effectiveness of the anti-synchronization, enough numerical investigations have been done by different values of active controller gain. Finally, we conclude that the convergence rate of anti-synchronization errors are inversely proportional to an active controller gain. A novel secure cascade encryption-decryption algorithm for digital images has been presented analytically and numerically. The security and performance analysis of the proposed algorithm have been carried out by several tests. The obtained results prove that the proposed image encryption algorithm preserves good encryption performance than existing algorithms.