Delayed outputs fractional-order hyperchaotic systems synchronization for images encryption

Abstract

Communication networks, play a major role in keeping us connected and sharing the details of our lives with each other. The main driver of multimedia data transmission is to enhance security protection of this multimedia data (speech, image and video). In this paper we addresses the synchronization problem of the masterslave type via a static error feedback. Sufficient conditions expressed by means of linear matrix inequalities (LMI) for the synchronization of fractional-order hyperchaotic systems with a known time delay between them is presented. The delay-dependent criterion is given based upon a Lyapunov function. We take the fractional-order hyperchaotic Liu system as an illustrative example to demonstrate the effectiveness of the proposed synchronization scheme. Moreover, as an application an images cryptosystem is presented. We considered two scenarios corresponding to the transmission channel, under occlusion attack and under noisse addition respectively. Results for the studied scenarios are presented and compared. The extensive simulation results prove that proposed cryptosystem has clear advantage over other proposed in the litterature.

1 Introduction

The dynamic response of a chaotic system with its broadband power spectra that is very similar to noise, is an ideal candidate for combating narrow-band disturbances in communication systems [38]. The dependence of the initial conditions is another intersting feature of chaotic signals, which makes it very difficult to know the structure of the generator and predict the signal over the distant future. Moreover, they are aperiodic, in that the same state is never repeated twice which make them very suitable for image encryption [1, 19]. A chaotic system can caracterized by one positive Lyapunov exponent, on the other hand a hyperchaotic system have more than one positve Lyapunov exponent, in fact higher dimensional chaotic systems with more one Lyapunov exponent clearly exhibit more complex dynamics[57]. One of the major problems in implementing chaotic based communication systems is the synchronization [4, 27, 34, 48].For this last decade, numerous contributions on the chaos synchronization have been made by researchers from different disciplines. In 2000 [8] was the first report on propagation delay in master-slave configurations. The authors called this a phase sensitivity and the existence of it can destroy the synchronization. On the other hand, in the area of control theory time-delay systems have been investigated and the delays often affect the stabilities. Therfore investigating time-delay systems stability becomes an important subject [49].

On the other hand, it is worthy noticing that the propagation delay may exist in the remote communication systems. For the time-delay chaotic systems, the propagation delay might not be equal to the system time delay. The propagation delay problem was reported and studied theoretically in [7, 29, 49]. In [49] the authors initiated the master-slave synchronization scheme for Lur’e type chaotic systems with propagation delay. The same problem was further investigated in [29] where some improving results were presented. In [7], the authors recast it as a more general synchronization problem based on the delayed feedback control scheme where the controller time delay is the propagation delay.

Fractional calculus, are powerful mathematical tools that handles with the differentiation and integration of noninteger orders. In past decades, fractional-order systems have represented an emerging research because of their capability to better exemplify real-world applications. In fact fractional-order systems are feature by an innite memory. Many literatures have demonstrated that some fractional order systems can have a chaotic regime. Several fractional-order chaotic systems have been reported in many studies [16,17,18]. More recently, the synchronization problems of fractional-order chaotic systems becomes an attractive challenge, and many schemes have been proposed [12, 33, 35, 50]. In [35] fractional-order chaos synchronization is investigated where the authors applied a linear controller to the slave system. The authors in [50] based on the comparaison principales studied a lag projective synchronization of fractional-order chaotic systems with time delay. In [12] a function cascade synchronization method for achieving complete synchronization, modified projective synchronization and anti-phase synchronization of two fractional-order hyperchaotic systems was proposed.

These days, one can easily transmit countless of informations through communication network. However, because of the openness of the network, there is a higher concerns of concealment of the informations. The digital image is an important information vector of multimedia communication, thus how to protect the image information becomes high a concern problem, and fascinate a lot of researchers. Cryptosystems have been developed for such reasons to prevent the information from the attackers. However, most conventional cryptosystems such as DES, IDEA and AES are not suitable for image transmission in a noissy channel, because of the inherent features of image such as the strong correlation among adjacent pixels. To cater this problem and due of the features chaotic signals have a lot of chaos-based cryptosystems have been developed up to now [10, 13, 21, 30, 39,40,41, 43, 51, 53]. The main objective of the cryptosystem proposed in [43], is to improve the permutation efficiency and reduce the computation cost, the authors use a heterogeneous bit-permutation and then use a XOR operation. By DNA sequences operation, the authors in [53] shuffled the pixels value of the plain image, and by using the generated sequences from a fractional order hyperchaotic system a pemutation operation on the pixels position is performed. In [40], a cryptosystem is presented, which a SHA-2 function is used to generate the initial conditions for the chaotic systems, after a DNA exclusive-OR is perform for pixel substitution. In [30], based on the region of interest the authors propose a multidimensional chaotic image cryptosystem. The region is messed up by using the improved Henon and Joseph sequences and diffused it using the unified chaotic sequence to hide the sensitive information in the image. Furthermore, they used the improved logistic map to hide the edge information of the target image to achieve a balance between the secrecy of information and the complexity of the encryption. A cryptosystem based on fractional-order chaotic systems has a higher security level, in addition has an improvement on the encryption key space [24, 47, 48, 56]. More recently, many images cryptosystems built from fractional-order chaotic systems have been proposed [20, 28, 32, 33, 46, 52]. An anti-synchronization scheme for fractional order reverse butterfly- shaped hyperchaotic systems is investigated via active control technique, then a cryptosystem is presented in [33].

Motivated by the above discussions, in this paper a synchronization scheme between two identical fractional-order hyperchaotic systems via a static error feedback is presented. In an application point of view, the synchronized systems are applied in a new images cryptosystem. In the confusion (scrambling) stage, two permutations processes are designed to break inter-intra correlation of the plain image. The first is done by cyclic shift operations and according to the selected chaotic sequences indexes we perform the other one. In the diffusion stage, we performs the XOR and expanded XOR operations for each components of the scrambled image, to obtain the encrypted image. Compared with the existing works in the literature, the principal contributions of this study can be summarized as:

• Propagation time-delay has been considered.

• The derived criterion is a sufficient condition for the stability of the error dynamics between the master and the slave systems.

• A new cryptosystem for color images.

The rest of this paper is organized as follows: Section 2 is devoted to present some basic preliminaries. In Section 3, we present the master-slave synchronization scheme of fractional-order hyperchaotic systems and a delay-dependent criterion for the asymptotic stability based on a Lyapunov function is derived. A new cryptosystem is illustrated in Section 4. In Section 5 simulations results for images encryption and decryption with some robustness analysis are presented and discussed. Finally we provide conclusions in Section 6.

2 Preliminaries

The most used definitions for fractional differential equations, given by the first pioneers Grünwald—Letnikov, Riemann–Liouville and Caputo are detailled [37]. As compared with the Caputo fractional operator, the main advantage of the Riemann—Liouville fractional operator lies in the composition properties of its fractional derivatives and integrals. Therefore, our consideration in this paper is the fractional-order systems with the Riemann—Liouville fractional operator.

Definition 1

The Riemann—Liouville fractional integral of a function f(t) with respect to time is defined as follows [37],

$$ {D}_{t}^{-\alpha}f(t)=\frac{1}{{\varGamma}(\alpha)}{{\int}_{0}^{t}}(t-\tau)^{\alpha-1}f(\tau)d\tau $$

(1)

and the fractional derivative also defined as,

$$ {D}_{t}^{\alpha} f(t)=\frac{d}{dt}\left[D_{t}^{1-\alpha}f(t)\right] $$

(2)

where m = [α] + 1 and [α] is the integer part of α and ${\varGamma }(S)={\int \limits }_{0}^{\infty } t^{S-1}e^{-t}dt$, with Γ(S + 1) = S Γ(S) is the well-known Euler’s gamma function.

Property 1 [37].

$$ \begin{array}{lr} Df(t) ~=~ {D}_{t}^{1-\alpha}~{D}_{t}^{\alpha}f(t) \quad where \quad D= \frac{d}{dt}\quad (integer \quad derivation) \end{array} $$

(3)

Lemma 1

[9]

Let v₁ and v₂ be real vectors of appropriate dimensions. For any positive scalar μ, we have:

$$ 2{{v}_{1}^{T}} v_{2} \le \mu {{v}_{1}^{T}} v_{1}+ \frac{1}{\mu} {{v}_{2}^{T}}v_{2}. $$

(4)

Lemma 2

[25]

If α > β > 0, then the following property

$$ {D}_{t}^{\alpha} \left\{ {D}_{t}^{-\beta} f(t) \right\} = {D}_{t}^{\alpha -\beta} f(t) $$

(5)

holds for sufficiently good functions x(t). In particular, this relation holds if x(t) is integrable.

Lemma 3

[31]

Let $x(t): R^{n} \rightarrow R^{n}$, be a vector of differentiable function. Then for any time instant t ≥ t₀, the following relationship holds,

$$ {D}_{t}^{\alpha} x^{T} (t)Px(t) \le 2x^{T} (t) P {D}_{t}^{\alpha} x(t) , \qquad 0<\alpha \le 1 $$

(6)

where P ∈ R^n×n is a constant, square, symmetric and positive definite matrix.

Lemma 4

(Barbalat’s lemma) [23]

Assume that f(t) is a function of time and has a limit when $t \rightarrow \infty$, if $\dot {f}(t)$is uniformly continuous ($\ddot {f}(t)$is bounded), then $\dot {f}(t) \rightarrow \infty$ as $t \rightarrow \infty$.

Lemma 5

[15, 55]

For any positive definite matrix R > 0, a scalar τ > 0, vector function, $f(.):[0,\tau ]\rightarrow R^{n}$ such that the integrations concerned are well defined, the following inequality holds :

$$ \left( {\int}_{0}^{\tau}f(s)ds \right)^{T} R \left( {\int}_{0}^{\tau}f(s)ds \right) \le \tau \left( {\int}_{0}^{\tau}f^{T} (s)Rf(s)ds \right) $$

(7)

Definition 2

(improved expanded XOR operation) [6, 42]

The improved expanded XOR operation introduce an enhancement on the overall security level of the proposed scheme. For two inputs $r1={\sum }_{i=0}^{7} r1_{i}$ and $r2= {\sum }_{i=0}^{8} r2_{i}$, the operator can be described as,

$$ eXOR(r1, r2)=\sum\limits_{i=0}^{7} not(r1_{i} \oplus r2_{i} \oplus r2_{i+1} \times 2^{i}) $$

(8)

where not(r1) flips a single bit r1. The operator has the following property: if eXOR(r₁, r₂) = t, then eXOR(t,r2) = r1. This property can be deduced from Table 1 as,

Table 1 The result of not(r1_i ⊕ r2_i ⊕ r2_i+ 2)

3 Synchronization scheme

Consider the following master—slave synchronization scheme with a time delay τ:

$$ \begin{array}{ll} \mathcal{M}: & \left\{ \begin{array}{ll} {D}_{t}^{\alpha} x(t)&=Ax(t)+Bf(x(t)) \\ p(t)&=Cx(t) \end{array} \right.\\\\ \mathcal{S}: & \left\{ \begin{array}{ll} {D}_{t}^{\alpha} y(t)&=A y(t)+B f(y(t))+u (t) \\ q(t)&=C y(t) \end{array} \right.\\\\ \mathcal{C}: &\left\{ u(t)=F(p(t-\tau)-q(t-\tau)). \right. \end{array} $$

(9)

The state vectors of the master and slave systems are x ∈ Rⁿ and y ∈ Rⁿ, respectively. The matrices A ∈ R^n×n, $B\in R^{n \times n_{h}}$, $C \in R^{n_{h} \times n}$ and H ∈ R^l×n are known real constant matrices. and q(t) ∈ R^m are the outputs of the master and the slave systems, respectively. The main objective of this scheme is to synchronize the master system ${\mathscr{M}}$ and the slave system S by applying a linear state error feedback to the slave system with control signal u(t) ∈ Rⁿ with feedback matrix F ∈ R^n×l. Defining the synchronization error as e(t) = x(t) − y(t), the error dynamic can be obtained as,

$$ \mathcal{E}: \left\{ D_{t}^{\alpha} e(t)=Ae(t)+B(f(x(t))-f(y(t)))-FCe(t-\tau) \right . $$

(10)

Assumption 1

For robustness, we will assume that f(x(t)) satisfy the following Lipschitz property:

$$ ||f(x(t))-f(y(t)) )|| \le l_{e} ||x(t)-y(t) ||, \quad \forall x(t), y(t) \in R^{n} $$

(11)

where l_e is an appropriate positive constants. We note that the Lipschitz properties are satisfied locally if f(x(t)), is differentiable with the respect of x(t).

3.1 Delay-dependent synchronization criteria

In this section, the stability of the error system $\mathcal {E}$ given in (10) is explored in order to obtain a synchronization criteria.

Theorem 1

for a given τ > 0 and l_e > 0, the error system described as (10) is asymptoticly stable if there exists the matrices P = P^T > 0, $R_{1}={R}_{1}^{T}>0$, $R_{2}={R}_{2}^{T} >0$ and $R_{3}={R}_{3}^{T} >0$, a matrix F with appropriate dimensions and constant positive scalars μ₁ and μ₂ there is a solution of the following optimization problem:

$$ \begin{array}{@{}rcl@{}} &&\underset{P,~X,~R_{i=1, 2, 3},~\mu_{j=1,~2}}{min} \\ &&Z= \\ &&\begin{array}{lr} \left[\begin{array}{ccc} 2PA+\frac{{l_{e}^{2}}\phi_{max}\left( PBB^{T}P \right)}{\mu_{1}}I +\mu_{1}I -\mu_{2}I + R_{1}+\tau R_{2} & I&R_{3}\\ I & -\frac{\phi_{max} \left( XCC^{T}X^{T} \right)}{\mu_{2}}I-R_{1} &-R_{3}\\ R_{3}&-R_{3}&\frac{-1}{\tau}R_{2} \end{array}\right] \end{array}\\ &&< 0 \end{array} $$

(12)

where F = P^− 1X, I is the n × n identity matrix and ϕ_max is the maximum eigenvalue function.

Remark 1

The matrix inequalites (12) include information on the delay. Therefore this result is a delay–dependent stability criterion for synchronization.

Proof Proof of theorem

Let us construct a Lyapunov function in the following form:

$$ \begin{array}{@{}rcl@{}} V(t)=&&{D}_{t}^{-(1-\alpha)} \left[e^{T}(t) P e(t)\right]+{\int}_{-\tau}^{0} e^{T}(t) (t+\sigma) R_{1} e(t+\sigma) d\sigma+{\int}_{t-\tau}^{t}{\int}_{t+\sigma}^{t}e(\psi)^{T}R_{2} \\ & &e(\psi)d\psi d\sigma+ \left[ {\int}_{-\tau}^{0} e(t+\sigma) d\sigma \right]^{T}R_{3}\left[ {\int}_{-\tau}^{0} e(t+\sigma) d\sigma \right], \quad 1 \geq \alpha >0 \end{array} $$

(13)

where in case α = 1, this function is reduced to a classical Lyapunov-Krasovskii function in the integer-order calculus (see Property 1), and in the case when 1 > α > 0, the term $D_{t}^{-(1-\alpha )}\left [e^{T}(t) P e(t)\right ]$ is constructed as a Rieman-Liouville fractional integral, thus according to Definition 1 and integral property the positive defitiveness is guaranted. □

An application of Property 1 and Lemma 2, we get the time derivative of (13) as,

$$ \begin{array}{@{}rcl@{}} \dot{V}(t) =&&{D}_{t}^{\alpha}\left[e(t)^{T}Pe(t)\right] + e^{T}(t) R_{1} e(t)-e^{T} (t-\tau) R_{1} e(t-\tau)+\tau e^{T}(t) R_{2} e(t)- \\ & &{\int}_{-\tau}^{0} e^{T}(t) (t+\sigma) R_{2} e(\sigma) d\sigma+\left[e(t)-e(t-\tau)\right]^{T}R_{3}\left[{\int}_{-\tau}^{0} e(t+\sigma)d\sigma\right] \\ &&+\left[{\int}_{-\tau}^{0} e(t+\sigma)d\sigma\right]^{T} R_{3}\left[e(t)-e(t-\tau)\right] \end{array} $$

(14)

Applying Lemma 3 to the right side of (14) results we get,

$$ \begin{array}{@{}rcl@{}} \dot{V}(t)\le &&2e^{T}P {D}_{t}^{\alpha} e(t)+ e^{T}(t) R_{1} e(t)-e^{T} (t-\tau)R_{1} e(t-\tau)+\tau e^{T}(t) R_{2} e(t)- \\ & &{\int}_{-\tau}^{0} e^{T} (t+\sigma) R_{2} e(t+\sigma) d\sigma+\left[e(t)-e(t-\tau)\right]^{T}R_{3}\left[{\int}_{-\tau}^{0} e(t+\sigma)d\sigma\right] \\&&+\left[{\int}_{-\tau}^{0} e(t+\sigma)d\sigma\right]^{T} R_{3}\left[e(t)-e(t-\tau)\right] \end{array} $$

(15)

Substituting (10) in (15) yields to,

$$ \begin{array}{@{}rcl@{}} \dot{V}(t)&&\le e^{T}(t) 2PAe(t)+2 e^{T} PB(f(x(t))-f(y(t)))-2 e^{T}(t) P F Ce(t-\tau) \\ & &\quad +e^{T}(t) R_{1} e(t)-e^{T} (t-\tau) R_{1} e(t-\tau)+\tau e^{T}(t) R_{2} e(t)-{\int}_{t-\tau}^{t} e^{T} (\sigma)R_{2} e(\sigma)\\&& d\sigma +\left[e(t)-e(t-\tau)\right]^{T}R_{3}\left[{\int}_{t-\tau}^{t} e(\sigma)d\sigma\right]+\left[{\int}_{t-\tau}^{t} e(\sigma)d\right]^{T}R_{3}[e(t)- \\&&e(t-\tau)] \end{array} $$

(16)

In view of Assumption 1 and Lemma 1, we obtain,

$$ \begin{array}{@{}rcl@{}} 2 e^{T}(t) P B(f(x(t))-f(y(t)))&&\le \mu_{1} || e(t)||^{2} + \frac{{l}_{e}^{2}}{\mu_{1}} ||B P e(t)||^{2} \le \mu_{1} || e(t)||^{2} +\frac{{l}_{e}^{2}\phi_{max}(PBB^{T}P )}{\mu_{1}} \\ &&||e(t)||^{2} \end{array} $$

(17)

$$ \begin{array}{@{}rcl@{}} &&||e(t)||^{2} \end{array} $$

(18)

and

$$ \begin{array}{@{}rcl@{}} 2 e^{T}(t) P F C e(t-\tau)&&\le\mu_{2} ||e(t)||^{2}+ \frac{1}{\mu_{2}} ||C^{T} F^{T} Pe^{T} (t-\tau)||^{2} \le \mu_{2} ||e^{T} (t)||^{2}+ \\ & & \quad \frac{\phi_{max} \left( PFCC^{T}F^{T}P \right)}{\mu_{2}} ||e (t-\tau)||^{2} \end{array} $$

(19)

where $\phi _{max}\left (\bullet \right )$ represnt the maximum eigenvalue of a matrix.

By using the inequality (7) we obtain the following :

$$ \left[ {\int}_{t-\tau}^{t} e(\sigma)d\sigma\right]^{T}R_{2}\left[ {\int}_{t-\tau}^{t} e(\sigma)d\sigma\right]\le \tau {\int}_{t-\tau}^{t} e^{T}(\sigma)R_{2}{\int}_{t-\tau}^{t} e(\sigma)d\sigma $$

(20)

and

$$ {\int}_{t-\tau}^{t} e^{T}(\sigma)R_{2}{\int}_{t-\tau}^{t} e(\sigma)d\sigma \geq \frac{1}{\tau} \left[ {\int}_{t-\tau}^{t} e(\sigma)d\sigma\right]^{T}R_{2}\left[ {\int}_{t-\tau}^{t} e(\sigma)d\sigma\right] $$

(21)

Thus, we have

$$ \begin{array}{@{}rcl@{}} \dot{V}(t) \le &&e^{T}(t) 2PAe(t)+ \frac{{{l}_{e}^{2}}\phi_{max}\left( PBB^{T}P \right)}{\mu_{1}} ||e(t)||^{2}+\mu_{1} || e(t)||^{2}-\mu_{2} \\ & & \quad ||e(t)||^{2}- \frac{\phi_{max} \left( PFCC^{T}F^{T}P \right)}{\mu_{2}} ||e^{T} (t-\tau)||^{2} + e^{T} (t) R_{1} e(t)-e^{T} (t-\tau) R_{1} \\&&e(t-\tau)+\tau e^{T}(t) R_{2} e(t)\frac{-1}{\tau} \left[ {\int}_{t-\tau}^{t} e(\sigma)d\sigma\right]^{T}R_{2}\left[ {\int}_{t-\tau}^{t} e(\sigma)d\sigma\right]\\&& +\left[e(t)-e(t-\tau)\right]^{T}R_{3}\left[{\int}_{t-\tau}^{t} e(\sigma)d\sigma\right]+\left[{\int}_{t-\tau}^{t} e(\sigma)d\sigma\right]^{T} \\&&R_{3}[e(t)- e(t-\tau)] \end{array} $$

(22)

Therefore,

$$ \dot{V}(t) \le \xi(t)^{T}Z \xi(t)<0 $$

(23)

where $\xi (t)=\left [e(t);e(t-\tau );{\int \limits }_{t-\tau }^{t} e(\sigma ) d\sigma \right ]$ and Z =

$$ \left[\begin{array}{ccc} 2PA+\frac{{l}_{e}^{2}\phi_{max} \left( PBB^{T}P \right)}{\mu_{1}}I+\mu_{1}I -\mu_{2} I + R_{1}+\tau R_{2} & I&R_{3}\\ I & -\frac{\phi_{max} \left( XCC^{T}X^{T} \right)}{\mu_{2}}I-R_{1} &-R_{3}\\ R_{3}&-R_{3}&\frac{-1}{\tau}R_{2} \end{array}\right] $$

with X = PF. Then if Z < 0, then $\dot {V}(t)\le 0$. To derive asymptotical stability we use the Barbalat’s lemma. Form $\dot {V}(t)<0$ it is obtain that V (t) < V (0). To verify the boudennes of $\ddot {V}(t)$, it needs to show that ξ(t) ∈ ℓ₂. Note that V is a non-increasing and positive definite function then:

$$ -{{\int}_{0}^{t}} \dot{V}(t) = V(0)-V(t)<\infty $$

(24)

then

$$ \begin{array}{@{}rcl@{}} &&-{{\int}_{0}^{t}} \dot{V}(t)d \tau <\infty \\ & & \Rightarrow {{\int}_{0}^{t}}\left[\lambda_{min}(-Z)||\xi(t)||^{2}\right]d\tau < \infty\\ && \Rightarrow \sqrt{{{\int}_{0}^{t}}\left[\lambda_{min}(-Z)||\xi(t)||^{2}\right]d\tau }< \infty \\ &&\Rightarrow \sqrt{{{\int}_{0}^{t}}||\xi(t)||^{2} d\tau }< \infty \end{array} $$

(25)

where λ_min(−Z) represent the minimum eigenvalues of − Z. From (25), it is concluded that ξ(t) ∈ ℓ₂. Then it is derived that $\lim _{t\rightarrow \infty }\dot {V}(t)=0$, the asymptotic stability is concluded. This completes the proof.

3.2 Illustrative example

In this subsection, we will numerically verify the effectiveness of the proposed synchronization scheme between two identical fractional-order hyper-chaotic Liu systems [16].

The master system is given by

$$ \left\{\begin{array}{lr} {D}_{t}^{\alpha_{1}}x_{1}(t)=\eta_{1}(x_{2}(t)-x_{1}(t))+x_{4}(t)\\ {D}_{t}^{\alpha_{2}}x_{2}(t)=\eta_{2}x_{1}(t)-x_{1}(t)x_{3}(t)+\eta_{3}x_{4}(t)\\ {D}_{t}^{\alpha_{3}}x_{3}(t)=4{x_{1}^{2}}(t)-\eta_{4}x_{3}(t)-x_{4}(t)\\ {D}_{t}^{\alpha_{4}}x_{4}(t)=-x_{4}(t)-\eta_{5}x_{2}(t) \end{array} \right. $$

(26)

and the slave system as

$$ \left\{\begin{array}{lr} {D}_{t}^{\alpha_{1}}y_{1}(t)=\eta_{1}(y_{2}(t)-y_{1}(t))+y_{4}(t)+u_{1}(t)\\ {D}_{t}^{\alpha_{2}}y_{2}(t)=\eta_{2}y_{1}(t)-y_{1}(t)y_{3}(t)+\eta_{3}y_{4}(t)+u_{2}(t)\\ {D}_{t}^{\alpha_{3}}y_{3}(t)=4{y_{1}^{2}}(t)-\eta_{4}y_{3}(t)-y_{4}(t)+u_{3}(t)\\ {D}_{t}^{\alpha_{4}}y_{4}(t)=-y_{4}(t)-\eta_{5}y_{2}(t)+u_{4}(t) \end{array} \right. $$

(27)

where η₁ = 10, η₂ = 40, η₃ = 0.5, η₄ = 2.5 and η₅ = 10/15.

Thus the systems matrices in (9) are $A=\left [ \begin {array}{cccc} -\eta _{1}&\eta _{1}&0&1\\ \eta _{2}&0&0&\eta _{3}\\ 0&0&\eta _{4}&-1\\ 0&-\eta _{5}&0&-1 \end {array} \right ]$, $B=\left [ \begin {array}{ccc} 0&0\\ 1&0\\ 0&1\\ 0&0 \end {array} \right ]$, $f(x(t))=\left [ \begin {array}{lr} -x_{1}(t)x_{3}(t)\\ 4{x}_{1}^{2}(t) \end {array} \right ]$ and $f(y(t))=\left [ \begin {array}{lr} -y_{1}(t)y_{3}(t)\\ 4{y}_{1}^{2}(t) \end {array} \right ].$

Initial conditions are selected as: x(0) = [2, − 1, 0.8, 0.8]^T and y(0) = [5, 3, 2, 1]^T respectively. We choose $C=\left [ \begin {array}{cccc}1&~ 0 &~ 0 &~1 \end {array}\right ]$ (it can be easily verified that $\left (C,A \right )$ is an observable pair). The matrices P, X and R_i= 1,2,3 and the scalares μ_{j= 1, 2} are found using Matlab LMI optimization toolbox with ł_e = 5 and τ = 0.163 as :

$R_{1}=10^{-4} \times \left [ \begin {array}{cccc} 0.6403 &~ -0.1115 &~ -0.0701 &~ -0.0761\\ -0.1115 &~ 0.7144 &~ -0.0725 &~ -0.1025\\ -0.0701 &~ -0.0725 &~ 0.6827 &~ -0.1333\\ -0.0761 &~ -0.1025 &~ -0.1333 &~ 0.6584 \end {array} \right ]$, $R_{2}= 10^{-3} \times \\\left [ \begin {array}{cccc} 0.1486 &~ -0.0018 &~ -0.0025 &~ -0.0031\\ -0.0018 &~ 0.1484 &~ -0.0024 &~ -0.0022\\ -0.0025 &~ -0.0024 &~ 0.1486 &~ -0.0021\\ -0.0031 &~ -0.0022 &~ -0.0021 &~ 0.1479 \end {array} \right ]$, $R_{3}=10^{-4} \times \\ \left [ \begin {array}{cccc} 0.6832 &~~~~~~~ -0.0746 &~~~~~~~ -0.0463 &~~~~~~~ -0.0475\\ -0.0746 &~~~~~~~ 0.7393 &~~~~~~~ -0.0454 &~~~~~~~ -0.0691\\ -0.0463 &~~~~~~~ -0.0454 &~~~~~~~ 0.7153 &~~~~~~~ -0.0891\\ -0.0475 &~~~~~~~ -0.0691 &~~~~~~~ -0.0891 &~~~~~~~ 0.6959 \end {array} \right ]$, μ₁ = 2.1032 × 10^− 5, μ₂ = 5.0017 × 10^− 6, $X=10^{-4} \times \left [ \begin {array}{cc} -0.1535 \\ 0.2944 \\ -0.0626\\ 0.0719 \end {array} \right ]$, $P=10^{-5} \times \left [ \begin {array}{cccc} 0.1144 &~ -0.0344 &~ 0.0014 &~ 0.0047\\ -0.0344 &~ 0.1035 &~ -0.0020 &~ -0.0096\\ 0.0014 &~ -0.0020 &~ 0.0935 &~ 0.0010\\ 0.0047 &~ -0.0096 &~ 0.0010 &~ 0.0919 \end {array} \right ]$ and $F=\left [ \begin {array}{cc} -5.5159 \\ 27.5302 \\ -6.1322\\ 11.0624 \end {array} \right ]$. Note that, one can easily find that Z is a negative definite matrix. The fractional orders α_{i(i= 1,2,3,4)} are also set to 0.9 to ensure the existence of chaos [16].

Figure 1a-d show the state trajectories of the master and slave systems. The synchronization errors are revealed in Fig. 2a-d. As expected, one can observe that the state trajectories of the slave system track those of the master, and the synchronization errors tend to zero.

Fig. 1

The trajectories of the states. a The trajectories of the states x1(t) and y1(t). b The trajectories of the states x2(t) and y2(t). c The trajectories of the states x3(t) and y3(t). d The trajectories of the states x4(t) and y4(t)

Fig. 2

The trajectorie of the synchronization errors. a The trajectorie of the synchronization error e1(t). b The trajectorie of the synchronization error e2(t). c The trajectorie of the synchronization error e3(t). d The trajectorie of the synchronization error e4(t)

4 The proposed cryptosystem

The architecture of the proposed encryption algorithm is shown in Fig. 3.

Fig. 3

An overview of the encryption process

As can be seen, a fractional-order hyperchaotic system is used as pseudo-random generator. The pixel positions of the plain image are scrambled first by circular permutations and secondly by the sorted chaotic sequences index positions. Finally the substitution of the pixels value is done by a XOR and expanded XOR operations. The encryption algorithm can be described as follows.

4.1 Generating and selecting the chaotic sequences

Due that the synchronization between the master and the slave systems, need some time to occur (see Fig. 2). Equation (9) is iterated T_S + 8MN time to generate the chaotic sequences after we discard the first T_S elements to avoid initial synchronization errors.

4.2 Generation of the scrambling and diffusion sequences

In this stage we generate S1, S2, Ind.x_i and $D_{f_{i}}$, where S1, S2 and Ind.x_i are used as scrambling sequences and $D_{f_{i}}$ as diffusion sequences. The following described the generation process.

4.2.1 Scrambling sequences

• Step 01 Sort the first MN elements of the selected chaotic sequences x_1,2,3 in ascendant order according to the following equation:

  $$ [Val.x_{i}(K)~,Ind.x_{i}(K)] = sort (x_{i}(K)) , \quad i=1,2,3 ,\quad K=1,2,\dots,MN $$

  (28)

  where V al.x_i, Ind.x_i are arrays with size 1 × MN which contains the values and the indexs position respectively.

• Step 02 Produce three others arrays $x^{\prime }_{i}$ with size 1 × 8MN by the following formula:

  $$ {x}_{i}^{\prime}= mod\left( \left[(\lceil|x_{i}|\rceil-|x_{i}|) \times 10^{8}\right],256\right),\quad i=1,2,3 $$

  (29)

  where |∙| denote the absolute value, mod(∙) refer to the module operation and ⌈a⌉ refers to get the smallest integer greater than or equal to a.

• Step 03 Combien all there arrays ${x}_{i}^{\prime }$ into an array S with the size 1 × 24MN.

  $$ S=cat({x}_{1}^{\prime}, {x}_{2}^{\prime}, {x}_{3}^{\prime}) $$

  (30)

  Then get two new arrays S1 and S2 with the length M and 24N respectively, using (31) shown as follows,

  $$ \begin{array}{@{}rcl@{}} &&S1= \left[S(1),S(2),\dots,S(M) \right]\\ && S2= \left[S(M+1),S(M+2),\dots,S(24N) \right] \end{array} $$

  (31)

4.2.2 Diffusion sequences

• Step 01 The arrays V al.x_i are processed using (32) as follows,

  $$ Val.x_{i} =mod\left( \left[(\lceil|Val.x_{i}|\rceil-|Val.x_{i}|) \times 10^{8}\right],256\right) $$

  (32)

  then transform them to binary matrices BV_i with the size of M × 8N and combine them into a M × 24N matrix T_V by the following:

  $$ T_{V}=cat(BV_{1},~BV_{2},~BV_{3}) $$

  (33)

• Step 02 Rows circular shift. The rows shift result ${T}_{V}^{\prime }$ is obtained by following rules: the row rw of T_V is moved by the step number S1(rw) and $rw=1,2,\dots ,M$.

• Step 03 Process the matrix ${T}_{V}^{\prime }$ by a circular column shift operation where the column col of ${T}_{V}^{\prime }$ is moved by the step number S2(col) and $col=1,2,\dots ,24N$ resulting $T_{V}^{\prime \prime }$.

• Step 04 Convert the binary values of ${T}_{V}^{\prime \prime }$ into decimal base resulting a matrix with size M × 3N then split it into three sub–matrices $T_{sub_{i=1,2,3}}$ with size M × N and after combine the elements of each sub-matrix $T_{sub_{i=1,2,3}}$ into an array $D_{f_{i=1,2,3}}$ with size 1 × MN respectively.

• Step 7 Substitute the elements of each array $D_{f_i}$ using the XOR operation as follows,

  $$ D_{f_{i}}= D_{f_{i}} \oplus Val.x_{i} $$

  (34)

  where the ⊕ denotes the XOR operation.

4.3 Image scrambling

The proposed scrambling of the plain image is performed by the following steps

• Step 01 Decompose the RGB image P into P_R, P_G, P_B components, then transform them into binary matrices R, G, B with the size of M × 8N and combine them into a M × 24N matrix T_E as:

  $$ T_{E}=cat(R, G, B) $$

  (35)

• Step 02 Rows circular shift. The rows shift result ${T}_E^{\prime }$ is obtained by following rules: the row rw of T_E is moved by step number S1(rw).

• Step 03 Process the matrix ${T}_E^{\prime }$ by a circular column shift operation, where the column col of ${T}_E^{\prime }$ is moved by step number S2(col) resulting ${T}_E^{\prime \prime }$.

• Step 04 Convert the binary values of ${T}_E^{\prime \prime }$ into decimal base resulting a matrix with size M × 3N, then split it into three sub–matrices of size M × N after combining the elements of each sub–matrices into 1-dimensional arrays $I_{D_{i=1,2,3}}$ with size 1 × MN then rearrange the elements of each $I_{D_{i=1,2,3}}$ according to Ind.x_i= 1,2,3 respectively to get the scrambled image as shown in the following equation

  $$ I_{D_{i}}(K)=I_{D_{i}}(Ind.x_{i}(K)) , \quad i=1,2,3 ,\quad K=1,2,\dots,MN $$

  (36)

4.4 Image diffusion

In this stage we proceed as follows,

• Step 1 Diffuse the elements of each array $I_{D_{i}}$ as follows:

  $$ I_{E_{i}}=I_{D_{i}} \oplus D_{f_{i}} $$

  (37)

• Step 2 Use the XOR and expanded XOR operations to perform another substitution by the following equation

  $$ \begin{array}{@{}rcl@{}} I_{E_{i}}(K)=&&eXOR\left( \left( I_{E_{i}}(K), \left( mod \left( \left( (4 \times x_{i}(K)-2 \times x_{i+1}(K)) \times 10^{8}\right),256\right)\right)\right) \right) \oplus (mod (\\ & & \left( x_{i+1}(K)\times 10^{8}\right),256)) , \quad i=1,2,3 ,\quad K=1,2,\dots,MN \end{array} $$

  (38)

• Step 3 Convert the arrays $I_{E_{i}}$ into matrices with size M × N which are separately the red, green and blue components of the final encrypted image.

4.5 Decryption process

The decryption is done by an inverse of encryption process, using the synchronized sequences $y_{j(j=1,2,3,\dots ,n)}$. The decryption is done by a inverse of encryption process, using the synchronized sequences $y_{j(j=1,2,3,\dots ,n)}$.

5 Numerical simulation and cryptanalysis

For the experimental setup a Matlab version 9, operating system Windows 7, processor Core i5-3320M and 4 GB memory were used. In the simulations the fractional order hyperchaotic Liu system is used. The cryptosystem is applied on sevral different images named Lena, Panda, Vegetables, Baboon, Peppers, Girl, Black and also images from USC-SIPI database set. The plain images are shown in Fig. 4a-c while corresponding encrypted and decrypted images are shown in Fig. 4d-i, respectively.

Fig. 4

Encryption and Decryption output of the proposed cryptosystem. a plain-image of Lena. b plain-image of Vegetables. c plain-image of Panda. d encrypted-image of Lena. e encrypted-image of Vegetables. f encrypted-image of Panda. g decrypted image of Lena. h decrypted image of Vegetables. i decrypted image of Panda

The following evaluation parameters, were taken into consideration to test the performance of the proposed cryptosystem.

5.1 Key space

A good cryptosystem, should have a large enough key space to resist an exhaustive. Here, the fractionbal order derivatives α_i(i = 1 : 4) and the parameters η_i(i = 1 : 5) are used as the secret key, therefore, the secret key-set is as (η₁, η₂, η₃, η₄, η₅, α₁, α₂, α₃, α₄), where each key independent of others. In the simulations (repeated over 100 times) we find that the precision of each secret key is approximately 10^− 15, then the key space size is about 10¹³⁵, so it can resist brute force attacks and in comparison with the references given in Table 2 it larger than [11, 14, 36, 40, 44].

Table 2 Comparison of Key space

5.2 Differential analysis

It is well known that a good algorithm can also withstand a brutal differential attack. To test it resistance, two known tests known as Changing Pixel Frequency Rates (NPCR) and Unified Average Intensity Change (UACI) which introduce by [2, 3] to investigate a cryptosystem against differential attacks. The NPCR test is given by (39) and the UACI by (40) as follow

$$ NPCR(E_{1},E_{2})= \left[ \sum\limits_{i=0}^{M} \sum\limits_{j=1}^{N}\frac{D(i,j)}{ M \times N} \right] \times 100<percent> $$

(39)

$$ UACI(E_{1},E_{2})= \left[ \sum\limits_{i=0}^{M} \sum\limits_{j=1}^{N}\frac{|E_{1}(i,j)-E_{2}(i,j)|}{ L \bullet M \times N} \right] \times 100<percent> $$

(40)

where E₁ and E₂ are two encrypted images generated from inputs which differ in one-bit only. M, N represent the height and width of the encrypted images and L denotes the largest intensity allowed in the image for any pixel respectively. D(i,j) is defined like follows

$$ D(i,j)=\left\{ \begin{array}{lr} 1 \qquad if \quad E_{1}(i,j)\neq E_{2}(i,j)\\ 0 \qquad else \end{array} \right. $$

(41)

The NPCR and UACI scores for Lena, Panda, Vegetables and Peppers encrypted images are listed in Table 3. The NPCR and UACI scores are compared to [13, 40, 53].

Table 3 Average NPCR and UACI scores for three-color channels of plaintext sensitivity

5.3 Key sensitivity analysis

An efficient cryptosystem should have a sensitivity to its secret key-set. In anthoer word a very small change in the secret key-set will cause a significant change in the decrypted image. To analyze the sensitivity of the secret key-set we do a litlle modification in initial one (i.e γ₀ = (η₁, η₂, η₃, η₄, η₅, α₁, α₂, α₃, α₄)) we obtaine the other secret key-sets as following γ₁ = (η₁ + 10^− 15, η₂, η₃, η₄, η₅, α₁, α₂, α₃, α₄), γ₂ = (η₁, η₂ + 10^− 15, η₃, η₄, η₅, α₁, α₂, α₃, α₄), γ₃ = (η₁, η₂, η₃ + 10^− 15, η₄, η₅, α₁, α₂, α₃, α₄), γ₄ = (η₁, η₂, η₃, η₄ + 10^− 15, η₅, α₁, α₂, α₃, α₄),γ₅ = (η₁, η₂, η₃, η₄,η₅ + 10^− 15, α₁, α₂, α₃, α₄), γ₆ = (η₁, η₂, η₃, η₄, η₅, α₁ + 10^− 15, α₂, α₃, α₄), γ₇ = (η₁, η₂, η₃, η₄, η₅, α₁, α₂ + 10^− 15, α₃, α₄), γ₈ = (η₁, η₂, η₃, η₄, η₅, α₁, α₂, α₃ + 10^− 15, α₄) and γ₉ = (η₁, η₂, η₃, η₄, η₅, α₁, α₂, α₃, α₄ + 10^− 15). Using the secret key-sets, the rate of difference is calculated for Lena, Vegetables and Panda and shown in Tables 4 and 5. The results of key sensitivity are also compared to [40, 45] and the proposed cryptosystem has clear advantage over it.

Table 4 NPCR score between two encrypted images generated using slightly different key

Table 5 NPCR score between two encrypted images generated using slightly different key

5.4 Statistical analysis

5.4.1 Histogram and uniformity analysis

The image histogram reflects the distribution of the pixels value. A strong cryptosystem should mask the perceptual meaning of the plain image and flatten its histogram (i.e., become near uniform distribution). The histograms before and after encryption are shown in Fig. 5 for the test image Lena. It clear that the encrypted image histograms are sufficiently uniform.

Fig. 5

Histogram analysis of Lena. a Plain Red channel. b Plain Green channel. c Plain Blue channel. d Encrypted Red channel. e Encrypted Green channel. f Encrypted Blue channel

According to the quantitative analysis method in [49], to measure the uniformity of an encrypted image we compute the variance of it histograms. The lower the variance value indicate a higher uniformity of the encrypted image The variance can be calculated as

$$ var= \frac{1}{256^{2}} \sum\limits_{i=1}^{256} \sum\limits_{j=1}^{256} \frac{1}{2}(z_{i}-z_{j})^{2} $$

(42)

where z_i and z_j are the frequencies at ith and jth gray levels respectively. In Table 6, the variances of histograms of encrypted images, for each secret key-set are provided.

Table 6 The variance values of the histograms

In order to further examen the influence of the modification of the secret set-keys on the uniformity of the encrypted images, we compute the percentage of the variance differences between two encrypted images obtained separately by the initial secret key-set g0 and the secret key-sets γ_i(i = 1,2,...,9). The percentage can be computed by (43) as,

$$ PP(var)_{\gamma_{i}}=\frac{|var_{\gamma_{i}}-var_{\gamma_{0}}|}{var_{\gamma_{0}}} $$

(43)

where $PP(var)_{\gamma _{i}}$ is the percentage of variance difference when only one key is changed, $var_{\gamma _{0}}$ and $var_{\gamma _{i}}$ represent the histogram variances of the encrypted image by the secret key-set γ₀ and $\gamma _{i=1,2,\dots ,9}$ respectively. The results are listed in Table 7. The percentage of average variance difference scores by the proposed cryptosystem are better than [45] and comparable to [40].

Table 7 Percentage of variance difference

5.4.2 Correlation of two adjacent pixels

It well known that in an image there is strong interconnected relationship between a adjacent pixels in the horizontal, vertical and diagonal directions, so it is very important for a good encryption process to not preserve those relationships or at least weaken them, to get the ability to face on statistical attacks. Those relationships are elaborated in Fig. 6a-c for plain Lena image Fig. 4a and Fig. 6d-f for the corresponding encrypted Fig. 4d.

Fig. 6

Correlation of adjacent pixels analysis of Lena. a Plain Red channel. b Plain Green channel. c Plain Blue channel. d Encrypted Red channel. e Encrypted Green channel. f Encrypted Blue channel

In the simulations, we select randomly 3000 pairs of adjacent pixels to measure the correlation coefficients for Fig. 4a and Fig. 4d in the three directions using (44),

$$ \rho_{x,y}=\frac{|(cov(x,y))|}{\sqrt{D(x)D(y)}} $$

(44)

where the covariance cov(x,y) is obtainted by

$$ cov(x,y)= \frac{1}{L} \sum\limits_{i=1}^{L}(x_{i}-E(x)) (y_{i}-E(y)) $$

(45)

the variance value D(x) is obtainted by

$$ D(x)=\frac{1}{L} \sum\limits_{i=1}^{L}(x_{i}-E(x))^{2} $$

(46)

the mean value E(x) is obtainted by

$$ E(x)=\frac{1}{L} \sum\limits_{i=1}^{L}x_{i} $$

(47)

and x, y are the gray values of two adjacent pixels and L is the number of samples taken , (in this case L = 3000). The results are listed in Table 8, and it clear that our algorithm has a low correlation as compared with the plain image (i. e. no information leakage from the encrypted images when statistical attacks happen). One can easily see that the results are comparable to the results in [13, 40, 42].

Table 8 Correlation coefficient analysis in all directions of the plain and the encrypted Lena images

5.5 Evaluating the pixels randomness

It is observed from Fig. 4d-f that the encrypted images appear to be noise and similar to random images, so the proposed cryptosystem sucessfully masked the perceptual semantic of the plain images. we use the information entropy metric and the Chi-Square test to evaluate this randomness.

5.5.1 Information entropy

The information entropy is evaluated to analyze the spreading of the gray scales of the image, in other words to measure the randomness of the image. The ideal entropy score of encrypted message is 8 in higher the value higher will be the uniform distribution. Mathematically, we can represent the entropy H for a data source x is characterized as:

$$ H(x)=-\sum\limits_{i=0}^{K-1}Pr(x_{i})log_{2}Pr(x_{i}) $$

(48)

where Pr(x_i) and k are the probability and the total number of the gray value x_i, respectivly.

We have calculated the information entropy for five plain images and their corresponding encrypted images by using our encryption algorithm and some other [13, 39, 40]. The results are given in Table 9. It’s well shown that the information entropy values obtained by our proposed cryptosystem is far better than [39, 40] and comparable to [13] and we can see that our scores are very close to the theoretical value.

Table 9 The Information Entropy of the encrypted images

5.5.2 The Chi-Square test

The variance of an histogram is the output which represents the variation in the frequency of gray levels and the Chi-square score is a measurement of how expectations are compare with the output. The low scores of Chi-square demonstrate that we have a better randomness in the encrypted image. The Equation for Chi-Square test can be defined as follow,

$$ X^{2} = \sum\limits_{i=0}^{255}\frac{(z_{i}-kk/256)^{2}}{kk/256} $$

(49)

where z_i is the number of pixels at ith gray level and kk/256 is the expected frequency at i^th gray level. The scores of the Chi-square test for three different images are listed in Table 10 and compared with [13, 14, 40]. From the results The are comparable to [13, 14], and better than [40].

Table 10 Chi-square and variance values for encrypted images

5.6 Pixel modification based measurements

The quality of an image depends upon the pixel difference which is calculated by means the mean square error (MSE) and peak signal to noise ratio) values (PSNR). Those metrics are for the comparison of unlike images.

5.6.1 The mean square error

An encrypted image should not be equivalent to the plain image due to the application of the encryption procedure, which surely adds some noise to the actual digital content. We compute the Mean Square Error (MSE) between the plain and encrypted images to analyze the level of enciphering. Mathematically, MSE is defined as:

$$ MSE= \frac{{\sum}_{i=1}^{N} {\sum}_{j=1}^{M} \left( E (i,j)-I (i,j) \right)^{2} }{MN} $$

(50)

where E and I represent the encrypted and the plain images respectively. M and N indicate the width and the height of the test image, respectively. A larger value of the MSE enhances the security. Table 11 provides a comparison of the MSE scores of our proposed cryptosystem with the one in [10]. The table shows that our proposed cryptosystem has clear advantage.

Table 11 MSE Comparison

5.6.2 The Peak Signal Noise Ratio

PSNR metric is a ratio between the plain and the encrypted images [8,50]. It employed as a security evaluation parameter when the plain and the encrypted images are taken as a signal and a noise respectivtly. A higher value of PSNR declare that the encrypted image is close to the plain image which is of course not desirable in any encryption procedure. Mathematically, PSNR can be written as:

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

(51)

The lower PSNR value provide an evidence that a plain image is significantly different from its corresponding encrypted and also become very difficult to retrieve it. The result for PSNR metric is listed in Table 12. As compared with the other algorithms in [13, 39, 40], the effectiveness of the proposed cryptosystem is evident by lower values of PSNR.

Table 12 PSNR Comparison

5.6.3 The gray difference degree (GVD)

The gray difference degree is another measure of pixel modification by comparing the plain and the encrypted image. The GVD score approaches to 1 indicates that the two images are completely different.

$$ GN(x,y)=\frac{\sum \left[I(x-y)-I(x^{\prime},y^{\prime})\right]^{2}}{4} $$

(52)

where the pair $(x^{\prime },y^{\prime })$ comes in fourth cases like $(x^{\prime },y^{\prime })=\left \{ \begin {array}{lr} (x-1,y)\\ (x,y-1)\\ (x+1,y)\\ (x,y+1) \end {array} \right .$. I(x,y) and $I(x^{\prime },y^{\prime })$ represents the gray score at position (x,y) and $(x^{\prime },y^{\prime })$ respectively.

The GVD, can be computed by using Eq. (53) as

$$ GVD(x,y)=\frac{AN^{\prime}[GN(x,y)]-AN[GN(x,y)]}{AN^{\prime}[GN(x,y)]+AN[GN(x,y)]} $$

(53)

where

$$ AN[GN(x,y)]=\frac{{\sum}_{x=2}^{M-1} {\sum}_{y=2}^{N-1}GN(x,y)}{(M-2)(N-2)} $$

(54)

and AN and $AN^{\prime }$ are the Average Neighborhood gray value before and after the encrypting respectively. Table 13 contains the GVD values of images from USC-SIPI database set (Table 14). Further we compute the GVD score of Lena and Baboon images as shown in Table 15. The listed results are comparable to [13, 40].

Table 13 GVD of colored images from USC-SIPI database

Table 14 GVD of Lena and Baboon images

Table 15 Comparison of Gray Difference Degree

5.7 Robustness analysis

In the real issue, the errors often occur in the data while being transmitted by a physical communication system. A unimportant change in the encrypted image may cause a strong distortion in the decryption procedure which results in failure to recover the plain image such that one can loses the plain image completely. A decent cryptosystem should be designed in a way that it does not have domino effect in the decryption procedure [40]. To illustrat the performance of our cryptosystem in those situations, we applied it to the following scenarios:

• 1. Attack in the transmission channel, where we applied occlusion attack to the encrypted images.

• 2. Noise in the transmission channel, in this scenario we add noise to the encrypted images.

5.7.1 Occlusion attack scenario

In the communications channels, lossing some parts of the transmitted (i. e. the encrypted) image can occur, so in this section we test our proposed cryptosystem to show it reaction under an occlusion attack. To show the strength of proposed cryptosystem we remove 1/16, 1/8, 1/4 and 1/2 part respectively of the encrypted image Fig. 4d is removed as shown in Fig. 7a-d and resultant the decrypted images shown in Fig. 7e-h.

Fig. 7

Occlusion attack analysis by removing a part of the encrypted image. a 1 = 16 of encrypted Lena. b 1 = 8 of encrypted Lena. c 1 = 4 of encrypted Lena. d 1 = 2 of encrypted Lena. e Decrypted image of (a). f Decrypted image of (b). g Decrypted image of (c). h Decrypted image of (d)

The PSNR, MSE, NPCR and UCI scores are listed for the lossy decrypted images in Table 16. The scores are far better than [5, 39, 40].

Table 16 PSNR, MSE, NPCR and UACI between plain and decrypted image under different clipping size

5.7.2 Noise addition scenario

In order to show the robustness of our proposed cryptosystem on a noisy envirement, we take for example Fig. 4f and we contaminaed it by the Salt & Pepper noise where the noise density is 0.005, 0.05 and 0.5 which resulting the images in Fig. 8a-c, with their corresponding decrypted images are displayed in Fig. 8d-f. With a similar way, the Gaussian noise with a zero mean and a noise variances of 0.002, 0.05 and 0.3 is add to Fig. 4f resultant the images shown in Fig. 9a-c and the corresponding decrypted images are displayed in Fig. 9a-f.

Fig. 8

Test of Salt & Pepper noise. a Encrypted with density 0.005. b Encrypted with density 0.05. c Encrypted with density 0.5. d Decrypted Panda image from (a). e Decrypted Panda image from (b). f Decrypted Panda image from (c)

Fig. 9

Test of Gaussian noise. a Encrypted with density 0.002. b Encrypted with density 0.05. c Encrypted with density 0.3. d Decrypted Panda image from (a). e Decrypted Panda image from (b). f Decrypted Panda image from (c)

The noisy encrypted images are decrypted and scores of NPCR, UACI and PSNR are computed and compared with [39, 40, 45] shown in Tables 17 and 18. It clear that the PSNR, NPCR and UACI scores obtain by using our propsed cryptosystem are far better than the others cryptosystems.

Table 17 Comparison of resistance to Salt & Pepper noise with Fig. 4f as test image

Table 18 Comparison of Gaussian noise robustness with Fig. 4f as test image

To further investigate the robustness to resist the transmission noise, In a similarty, we perform on Fig. 4e the same scenario as for Fig. 4f, the only difference in this case we change the noise parameters, where the densities of Salt & Pepper noise is choosen as 0.01, 0.05, 0.1 and 0.25, and the Gaussian noise the variances is set as 0.0001, 0.0003 and 0.0005. The results are shown in Tables 19 and 20 which clearly shows that the proposed cryptosystem has better results than [5, 26, 40].

Table 19 Comparison of resistance to Salt & Pepper noise with Fig. 4e as test image

Table 20 Comparison of Gaussian noise robustness with Fig. 4e as test image

6 Conclusions

In this paper, a master-slave synchronization scheme using a static error feedback for fractional-order hyperchaotic systems have been studied for a known time delay existing in the master-slave configuration. The delay-dependent criterion been given based upon a Lyapunov function. The criterion have been applied to fractional-order hyperchaotic Liu systems. The simulation results show that the proposed synchronization scheme gives good performance in the presence of time delay in the outputs of the systems. Further, we implement the synchronized systems into a new cryptosystem for image encryption and decryption. Several performance tests are done such as key space and key sensitivity analysis, pixels randomness valuation and pixel modification based measurement and a comparison with other cryptosystems show that the proposed cryptosystem is stronger than the usual cryptosystem due to the hardness of an additional securities derived from the fractional order derivativess and has a good security performence. In addition, one can observe the hight performance of the proposed cryptosystem on occlusion attack and noise addition applied to the encrypted images. The obtained results show good resistance in those scenarios.