# On Discrete Fractional Complex Gaussian Map: Fractal Analysis, Julia Sets Control, and Encryption Application

Authors: Amr Elsonbaty; A. Elsadany; Fatma Kamal
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This work is devoted to present a generalized complex discrete fractional Gaussian map. Analytical and numerical analyses of the proposed map are conducted. The dynamical behaviors and stability of fixed points of the map are explored. The existence of fractal Mandelbrot and Julia sets is examined along with the corresponding fractal characteristics. The influences of the key parameters of the map and fractional order are examined. Moreover, nonlinear controllers are designed in the complex domain to control Julia sets generated by the map or to achieve synchronization between two Julia sets in master/slave configuration. Numerical simulations are provided to attain a deep understanding of nonlinear behaviors of the proposed map. Then, a suggested efficient chaos-based encryption technique is introduced by integrating the complicated dynamical behavior and fractal sets of the proposed map with the pseudo-chaos generated from the modified lemniscate hyperchaotic map.


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## Body

## 1. Introduction

Mathematical models are used to describe and understand the interesting behaviors of nonlinear systems, which arise in different disciplines of science. There are a plethora of mathematical tools, which have proved their efficacy in mathematical modeling of biological, physical, engineering, economic, and natural systems. Among these tools, the differential equations, difference equations, and statistical methods have attracted a considerable interest [1–5].However, when dealing with systems with memory, that is, the associated rate of changes depends on the past values of state variables in addition to the present values, the conventional continuous-time differential equation and discrete-time maps cannot describe these systems properly. To address this issue, mathematicians and engineers employ fractional calculus to formulate nonlocal differential operators, which are necessary to study systems with memory. Firstly, they focused on the fractional-order differential equations (FDEs) for the past two decades. The electric circuits, fluid mechanics, electromagnetics, immune systems, nanofluids, epidemics, and biological and financial systems are only examples of the fields, where FDEs are of great importance [6–13]. There are a few definitions for fractional-order derivatives and integrals, which have been developed so far such as Riemann–Lioville, Caputo Grunwald–Letinkov, and Wyl–Riesez fractional operators. More details are provided in references [14–18]. In reference [19], a fractional-order model based on Atangana–Baleanu–Caputo fractional derivative was proposed to understand the dynamics of differentiation of stem cells. The state-of-the-art developments in special functions and mathematical analysis tools associated with fractional-order differential equations are provided in reference [20].The numerical solutions of FDEs are usually carried out with high computational cost and induce several types of numerical errors.Therefore, while searching for an efficient and reasonable alternative, it is recognized that the fractional difference operators can be applied in a straightforward way to the mathematical modeling of different nonlinear systems. More recently, attention has been turned to the discrete fractional difference equations [21–25], where they have been successfully applied in different fields.On the other side, complex maps are found to exhibit very interesting and fascinating geometrical structures known as Julia and Mandelbrot fractal sets [26–28]. These sets are known to have fractal dimensions and have many interesting applications. The nonlinear dynamics and chaotic behavior of discrete fractional Gauss maps are investigated in the literature. It has been observed that the fractional Gauss map is more stable compared with the associated integer map. The width of period-3 windows is found to increase with the decrement in the value of fractional order [29]. Also, the synchronization for standard integer-order Gauss maps and discrete fractional Gauss maps has been studied using a parameter estimation scheme [30]. The emerging nonlinear dynamics and synchronization in coupled integer-order and fractional-order Gauss maps with different topologies have been explored in reference [31, 32]. The motivation of this study is based on the observation that the nonlinear characteristics and dynamics of the fractional complex maps are still almost an unexplored point in literature. Indeed, there are very few works that begin to investigate only the case of fractional-order complex differential equations [33, 34]. The present work extends the aforementioned works to the more general and unexplored case, where the state variable of the map has complex values, and it also investigates the emerging Julia and Mandelbrot fractal sets along with synchronization methodology of discrete fractional Gaussian map in complex domain for the first time, to the best of authors’ knowledge. Moreover, the present work combines the induced fractal sets into a proposed efficient chaos-based encryption technique.The very complicated behaviors of chaotic systems along with noise-like dynamics, very broadband spectrum, and ability to attain synchronization between distant systems have been utilized efficiently in a plethora of schemes for chaos-based communications [35–52]. In the last two decades, the chaos-based cryptography has become a focus research point of great interest. The critical evaluation of chaos-based encryption systems reveals that it is essential to keep high complexity and dimensionality of chaotic dynamics in encryption schemes along with effectively preventing any information leakage by possible eavesdroppers attacks [40–42]. The chaotic maps, in particular, are easily implementable on digital hardware, which can be straightforwardly integrated with modern communication systems. However, several works have highlighted the problem of degradation and suppression of chaotic behavior in simple structure and low-dimensional chaotic maps. These problems result from hardware finite precision of floating numbers [43–45]. Also, the small key space in these chaotic maps is another drawback. The employment of multiple chaos systems and switching between their outputs is offered along with sufficient long finite precision computations to improve the performance of chaotic maps [46]. The pseudo-chaotic orbits can be employed as another solution to the aforementioned chaos degradation issue [47, 48]. More specifically, pseudo-chaotic time series can be attained by subtracting the output sequences of two mathematically equivalent chaotic maps, which are nonequivalent in computations when machine finite precision is considered [47, 48]. The application of discrete fractional complex maps in the field of chaos-based encryption systems is also an unexplored research point, to the best of our knowledge. So, this article aims also at investigating this challenging task and providing a reliable encryption machine based on the complicated dynamics of a proposed fractional complex map.This study is organized as follows: the mathematical model of the proposed discrete fractional complex Gaussian map is presented in Section2. The control and synchronization of Julia sets generated by the proposed map are examined in Section 3. The proposed hybrid chaos-fractal encryption scheme is presented in Section 4, while the associated security analysis is addressed in Section 5. Section 6 contains conclusion and final discussion.


## 2. Discrete Fractional Complex Gaussian Map

The discrete fractional complex Gaussian map is proposed in the following form:(1)CΔ0αzt=e−az2t+α−1+b,where z, a≠0, and b≠0 take complex values, whereas the fractional order α∈0,1. The complex discrete fractional map (1) has infinite number of fixed points, which can be evaluated from the following equation:(2)e−az∗2=−b,or(3)z∗=lnb+iθ0+2k+1πa1/2,k=0,±1,±2,…,where θ0 denotes the principal argument of complex-valued constant b. This means that the equilibrium points of the proposed map are determined according to the assigned values for a,b, and k.The asymptotic stability analysis of fixed points in the complex fractional Gaussian map (1) is conducted in the following subsection:


### 2.1. Stability Analysis of Fixed Points

Theorem 1.
The fixed pointz∗ of the fractional complex Gaussian map (1) is locally asymptotically stable if(4)−2az∗e−az∗2<2cosArg−2az∗e−az∗2−π2−αα,Arg−2az∗e−az∗2>απ2.Proof.
Assume thatεt=zt−z∗, then the next linearized map is derived from equation (1):(5)CΔ0αεt=−2az∗e−az∗2εt+α−1=γεt+α−1.
Expressing the above equation in terms of its real and imaginary parts, it follows that(6)Δ0αεrt+iCΔaαεit=γr+iγiεrt+α−1+iεit+α−1,and therefore the next equivalent 2D discrete fractional system is attained:(7)Δ0αεrt=γrεrt+α−1−γiεit+α−1,Δ0αεit=γiεrt+α−1+γrεit+α−1.
Now, the above two equations can be expressed as follows:(8)Δ0αεrtΔ0αεit=γr−γiγiγrεrt+α−1εit+α−1,where it can be verified that the eigenvalues of the matrix of coefficients are given by γr±iγi.
DefineΛ by(9)Λ=γr−γiγiγr,such that trΛ=2γr. and detΛ=γr2+γi2>0. The zero equilibrium point of equation (8) is asymptotically stable if its associated eigenvalues satisfy(10)γr2+γi2<2costan−1γi/γr−π2−αα,tan−1γiγr>απ2.
Forz∗=lnb+iθ0+2k+1π/a1/2, k=0,±1,±2,…, the stability conditions reduce to(11)−2az∗e−az∗2<2cosArg−2az∗e−az∗2−π2−ααArg−2az∗e−az∗2>απ2.
In this case, the trajectories which start from small initial perturbationsεr0. and εi0. around the origin will algebraically decay to the equilibrium point such thatεn=On−α. as n⟶∞.
For the special case, where principal argument ofb is considered, that is, k=0, we get(12)z1,2∗=1/2lnbr2+bi2+iθ0+πar+iai1/2=1/2arlnbr2+bi2−aiθ0+π+iarθ0+π+1/2ailnbr2+bi2ar2+ai21/2=1ar2+ai2r∗1/2cosϕ02+isinϕ02,1ar2+ai2r∗1/2cosϕ0+π2+isinϕ0+π2,where a=ar+iai and b=br+ibi,(13)r∗=12arlnbr2+bi2−aiθ0+π2+arθ0+π+12ailnbr2+bi22,ϕ0=tan−1arθ0+π+1/2ailnbr2+bi21/2arlnbr2+bi2−aiθ0+π.
The specific forms ofz1,2∗ can be substituted in above-mentioned stability conditions to investigate their stability.
By the aid of numerical simulations, previous results regarding stability conditions of fixed points are validated for different values ofα,k,a, and b (Figure 1). The obtained solution orbits indicate that the stability conditions are satisfied for selected parameter sets employed in Figure 1.Figure 1
Time series solution of fractional Gaussian map depicting stable fixed points at (a, b)a=0.15−0.15i,α=0.9, b=0.71+0.25i and (c, d) a=0.3+0.2i,α=0.85, b=0.3−0.1i.
(a)(b)(c)(d)


### 2.2. Fractal Sets Induced by Discrete Fractional Complex Gaussian Map

The notions of Julia fractal set and Mandelbrot fractal set in integer-order complex-valued maps can be extended to the general case of discrete fractional-order complex maps. Given the next discrete fractional map of orderα(14)Δ0αzt=Ψzt+α−1,μ,where Ψ:ℂ⟶ℂ and μ∈ℂ. The Julia set generated by map (5) is described in the following definition [26–28, 33, 34]:Definition 2.
The filled-in Julia set of complex-valued discrete fractional map (5) is defined as the set Ω of initial points z∈ℂ, whose solution orbits are bounded. The boundary of Ω set is referred to as ∂Ω and it is known as the Julia set ΥΨα of the map (5).
The main characteristics of Julia setΥΨα are summarized as follows [27, 28, 33, 34]:(1)
ΥΨα≠ϕ (Julia set is nonempty).(2)
ΥΨα is invariant with respect to associated map (5) in the forward and backward directions of time.(3)
Assuming that an attractive fixed pointz^ of the discrete fractional map (5) has period p and exists at α¯, then ΥΨα¯ includes the basin of attraction of z^.
The well-known Mandelbrot set has been investigated by Benoit Mandelbrot in 1979 [27, 28]. Its concept can also be generalized to the discrete fractional case. More specifically, fixing the value of fractional order α, the Mandelbrot set χΨα consists of the set of values of parameter μ∈ℂ at which the values of zt, t>0 are bounded for z0=0.
The space-filling dimension can be employed to quantify the fractal properties of Julia and Mandelbrot sets. In particular, the box-counting measure for dimension is one of the most accessible measures in fractal analysis and it is defined as follows:Definition 3.
Consider the nonempty bounded subsetΞ of ℝn and suppose that there are Nρ boxes with side length ρ, which are required to cover the set Ξ. Then, the box-counting dimension (Minkowski–Bouligand dimension) is determined by the following equation:(15)dimΞ=limρ⟶0logNρlog1/ρ,where Nρ is the number of boxes to cover Ξ. In addition, the upper box dimension (entropy dimension) and the lower box dimension (lower Minkowski dimension) of Ξ are also defined, respectively, by the following equations:(16)dim¯Ξ=limρ⟶0¯logNρlog1/ρ,dim¯Ξ=lim¯ρ⟶0logNρlog1/ρ.
The generation of Mandelbrot and Julia sets is explored through numerical simulations at different values of parameters. The following table summarizes the obtained results at different values of fractional orderα, constant q, and exponent p. In addition, the box-counting dimensions for the different cases considered in simulations are also presented in Table 1 The corresponding Mandelbrot and Julia sets are depicted in Figures 2 to 4.Table 1
Summary of fractal sets generated from complex fractional Gaussian map and their fractal dimensions.
GraphFractal setParametersDimensionFigure2(a)Mandelbrot setα=1,a=0.5+0.3i1.544Figure2(b)Julia setα=1,a=0.5+0.3i, b=0.3+0.3i1.838Figure2(c)Julia setα=1,a=0.5+0.3i, b=−0.15−0.4i1.6438Figure2(d)Mandelbrot setα=1,a=0.19−0.5i1.429Figure2(e)Julia setα=1,a=0.19−0.5i, b=0.5−0.5i1.8321Figure3(a)Mandelbrot setα=0.8,a=0.19−0.5i1.753Figure3(b)Mandelbrot setα=0.5,a=0.19−0.5i1.4775Figure3(c)Mandelbrot setα=0.3,a=0.19−0.5i1.512Figure3(d)Julia setα=0.8,a=0.19−0.5i, b=0.5−0.5i1.781Figure3(e)Julia setα=0.5,a=0.19−0.5i, b=0.5−0.5i1.6574Figure3(f)Julia setα=0.3,a=0.19−0.5i, b=0.5−0.5i1.932Figure4(a)Mandelbrot setα=0.9,a=−0.59+0.93i1.5016Figure4(b)Julia setα=0.9,a=−0.59+0.93i, b=−0.75−0.05i1.753Figure4(c)Mandelbrot setα=0.5,a=1.15−0.7i1.483Figure4(d)Julia setα=0.5,a=1.15−0.7i, b=0.5−0.5i1.4485Figure 2
The Mandelbrot and Julia sets of generalized fractional Gaussian map obtained at specified values in Table1.
(a)(b)(c)(d)(e)Figure 3
The Mandelbrot and Julia sets of generalized fractional Gaussian map obtained at specified values in Table1.
(a)(b)(c)(d)(e)(f)Figure 4
The Mandelbrot and Julia sets of generalized fractional Gaussian map obtained at specified values in Table1.
(a)(b)(c)(d)


## 2.1. Stability Analysis of Fixed Points

Theorem 1.
The fixed pointz∗ of the fractional complex Gaussian map (1) is locally asymptotically stable if(4)−2az∗e−az∗2<2cosArg−2az∗e−az∗2−π2−αα,Arg−2az∗e−az∗2>απ2.Proof.
Assume thatεt=zt−z∗, then the next linearized map is derived from equation (1):(5)CΔ0αεt=−2az∗e−az∗2εt+α−1=γεt+α−1.
Expressing the above equation in terms of its real and imaginary parts, it follows that(6)Δ0αεrt+iCΔaαεit=γr+iγiεrt+α−1+iεit+α−1,and therefore the next equivalent 2D discrete fractional system is attained:(7)Δ0αεrt=γrεrt+α−1−γiεit+α−1,Δ0αεit=γiεrt+α−1+γrεit+α−1.
Now, the above two equations can be expressed as follows:(8)Δ0αεrtΔ0αεit=γr−γiγiγrεrt+α−1εit+α−1,where it can be verified that the eigenvalues of the matrix of coefficients are given by γr±iγi.
DefineΛ by(9)Λ=γr−γiγiγr,such that trΛ=2γr. and detΛ=γr2+γi2>0. The zero equilibrium point of equation (8) is asymptotically stable if its associated eigenvalues satisfy(10)γr2+γi2<2costan−1γi/γr−π2−αα,tan−1γiγr>απ2.
Forz∗=lnb+iθ0+2k+1π/a1/2, k=0,±1,±2,…, the stability conditions reduce to(11)−2az∗e−az∗2<2cosArg−2az∗e−az∗2−π2−ααArg−2az∗e−az∗2>απ2.
In this case, the trajectories which start from small initial perturbationsεr0. and εi0. around the origin will algebraically decay to the equilibrium point such thatεn=On−α. as n⟶∞.
For the special case, where principal argument ofb is considered, that is, k=0, we get(12)z1,2∗=1/2lnbr2+bi2+iθ0+πar+iai1/2=1/2arlnbr2+bi2−aiθ0+π+iarθ0+π+1/2ailnbr2+bi2ar2+ai21/2=1ar2+ai2r∗1/2cosϕ02+isinϕ02,1ar2+ai2r∗1/2cosϕ0+π2+isinϕ0+π2,where a=ar+iai and b=br+ibi,(13)r∗=12arlnbr2+bi2−aiθ0+π2+arθ0+π+12ailnbr2+bi22,ϕ0=tan−1arθ0+π+1/2ailnbr2+bi21/2arlnbr2+bi2−aiθ0+π.
The specific forms ofz1,2∗ can be substituted in above-mentioned stability conditions to investigate their stability.
By the aid of numerical simulations, previous results regarding stability conditions of fixed points are validated for different values ofα,k,a, and b (Figure 1). The obtained solution orbits indicate that the stability conditions are satisfied for selected parameter sets employed in Figure 1.Figure 1
Time series solution of fractional Gaussian map depicting stable fixed points at (a, b)a=0.15−0.15i,α=0.9, b=0.71+0.25i and (c, d) a=0.3+0.2i,α=0.85, b=0.3−0.1i.
(a)(b)(c)(d)


## 2.2. Fractal Sets Induced by Discrete Fractional Complex Gaussian Map

The notions of Julia fractal set and Mandelbrot fractal set in integer-order complex-valued maps can be extended to the general case of discrete fractional-order complex maps. Given the next discrete fractional map of orderα(14)Δ0αzt=Ψzt+α−1,μ,where Ψ:ℂ⟶ℂ and μ∈ℂ. The Julia set generated by map (5) is described in the following definition [26–28, 33, 34]:Definition 2.
The filled-in Julia set of complex-valued discrete fractional map (5) is defined as the set Ω of initial points z∈ℂ, whose solution orbits are bounded. The boundary of Ω set is referred to as ∂Ω and it is known as the Julia set ΥΨα of the map (5).
The main characteristics of Julia setΥΨα are summarized as follows [27, 28, 33, 34]:(1)
ΥΨα≠ϕ (Julia set is nonempty).(2)
ΥΨα is invariant with respect to associated map (5) in the forward and backward directions of time.(3)
Assuming that an attractive fixed pointz^ of the discrete fractional map (5) has period p and exists at α¯, then ΥΨα¯ includes the basin of attraction of z^.
The well-known Mandelbrot set has been investigated by Benoit Mandelbrot in 1979 [27, 28]. Its concept can also be generalized to the discrete fractional case. More specifically, fixing the value of fractional order α, the Mandelbrot set χΨα consists of the set of values of parameter μ∈ℂ at which the values of zt, t>0 are bounded for z0=0.
The space-filling dimension can be employed to quantify the fractal properties of Julia and Mandelbrot sets. In particular, the box-counting measure for dimension is one of the most accessible measures in fractal analysis and it is defined as follows:Definition 3.
Consider the nonempty bounded subsetΞ of ℝn and suppose that there are Nρ boxes with side length ρ, which are required to cover the set Ξ. Then, the box-counting dimension (Minkowski–Bouligand dimension) is determined by the following equation:(15)dimΞ=limρ⟶0logNρlog1/ρ,where Nρ is the number of boxes to cover Ξ. In addition, the upper box dimension (entropy dimension) and the lower box dimension (lower Minkowski dimension) of Ξ are also defined, respectively, by the following equations:(16)dim¯Ξ=limρ⟶0¯logNρlog1/ρ,dim¯Ξ=lim¯ρ⟶0logNρlog1/ρ.
The generation of Mandelbrot and Julia sets is explored through numerical simulations at different values of parameters. The following table summarizes the obtained results at different values of fractional orderα, constant q, and exponent p. In addition, the box-counting dimensions for the different cases considered in simulations are also presented in Table 1 The corresponding Mandelbrot and Julia sets are depicted in Figures 2 to 4.Table 1
Summary of fractal sets generated from complex fractional Gaussian map and their fractal dimensions.
GraphFractal setParametersDimensionFigure2(a)Mandelbrot setα=1,a=0.5+0.3i1.544Figure2(b)Julia setα=1,a=0.5+0.3i, b=0.3+0.3i1.838Figure2(c)Julia setα=1,a=0.5+0.3i, b=−0.15−0.4i1.6438Figure2(d)Mandelbrot setα=1,a=0.19−0.5i1.429Figure2(e)Julia setα=1,a=0.19−0.5i, b=0.5−0.5i1.8321Figure3(a)Mandelbrot setα=0.8,a=0.19−0.5i1.753Figure3(b)Mandelbrot setα=0.5,a=0.19−0.5i1.4775Figure3(c)Mandelbrot setα=0.3,a=0.19−0.5i1.512Figure3(d)Julia setα=0.8,a=0.19−0.5i, b=0.5−0.5i1.781Figure3(e)Julia setα=0.5,a=0.19−0.5i, b=0.5−0.5i1.6574Figure3(f)Julia setα=0.3,a=0.19−0.5i, b=0.5−0.5i1.932Figure4(a)Mandelbrot setα=0.9,a=−0.59+0.93i1.5016Figure4(b)Julia setα=0.9,a=−0.59+0.93i, b=−0.75−0.05i1.753Figure4(c)Mandelbrot setα=0.5,a=1.15−0.7i1.483Figure4(d)Julia setα=0.5,a=1.15−0.7i, b=0.5−0.5i1.4485Figure 2
The Mandelbrot and Julia sets of generalized fractional Gaussian map obtained at specified values in Table1.
(a)(b)(c)(d)(e)Figure 3
The Mandelbrot and Julia sets of generalized fractional Gaussian map obtained at specified values in Table1.
(a)(b)(c)(d)(e)(f)Figure 4
The Mandelbrot and Julia sets of generalized fractional Gaussian map obtained at specified values in Table1.
(a)(b)(c)(d)


## 3. Control and Synchronization of Julia Sets

The problem of achieving control and synchronization of Julia sets generated by the discrete fractional complex Gaussian map is discussed in this section.For two discrete fractional-order complex Gaussian maps, the first map is known as the master map and it produces the outputz1t, while the second map, with the output z2t, will be referred to as the slave one.Definition 4.
The synchronization between the master and slave maps is achieved, ifz2t⟶z1t as t⟶∞. In other words, it can be expressed as follows [33, 34]:(17)limt⟶∞z2t−z1t=0.
When the synchronization is attained between two trajectories, it implies that the corresponding characteristics of convergence and divergence are identical. Assume thatΥ1α and Υ2α denote the Julia sets induced by fractional-order master and fractional-order slave Gaussian maps, respectively, at fractional order α. Therefore, the synchronization between the mentioned two Julia sets can be defined as follows [29,30]:Definition 5.
The asymptotic synchronization of the two Julia setsΥ1α and Υ2α is satisfied if(18)limt⟶∞Υ1α∪Υ2α−Υ1α∩Υ2α=∅.


### 3.1. Control of Julia Sets of Discrete Fractional Complex Gaussian Map

In this section, the appropriate controller is designed in order to change the characteristics and geometry of Julia sets generated by the proposed fractional map via varying the type of stability of one of the fixed points of the present map. More specifically, we consider the feedback controller in the following form:(19)ϱt=−ςzt−z˜−e−az2t−b,where z˜ is the selected unstable fixed point intended to be stabilized under the influence of controller and ς=ςr+iςi represents the complex-valued gain of the controller, which can be evaluated as follows:Theorem 6.
Assume that the gainς of controller ϱt of the controlled fractional-order complex Gaussian map(20)Δ0αzt=e−az2t+α−1+b+ϱt+α−1,fulfills the two inequalities(21)ςr>0,ςr2+ςi2<2α,then the fixed point z˜ become stable, such that the associated Julia set in its neighborhood is changed.Proof.
By applying the control signal (6), we get the following controlled fractional-order complex map:(22)Δ0αzt=−ςzt+α−1−z˜.
Definingδt=zt−z˜∈ℂ, equation (22) takes the following form:(23)Δ0αδt=−ςδt+α−1.
The corresponding two-dimensional real-valued fractional map can be expressed as follows:(24)Δ0αδrt=−ςrδrt+α−1+ςiδit+α−1,Δ0αδit=−ςiδrt+α−1−ςrδit+α−1.
Then, the coefficients of the above system can be put in the following matrix:(25)Λ=−ςrςi−ςi−ςr,and the associated eigenvalues are computed as −ςr±iςi. Hence, the sufficient conditions required for local asymptotic stability of z˜ can be formulated as ςr>0 and ςr2+ςi2<2α.


### 3.2. Synchronization of Julia Sets

The discrete fractional master system is defined in the following form:(26)Δ0αz1t=e−az12t+α−1+b,whereas the corresponding slave system is formulated as follows:(27)Δ0αz2t=e−az22t+α−1+b+ϕz1,z2,t+α−1,where ϕz1,z2,t+α−1 is the adequate controller to be designed. Note that the initial values of two systems are assumed to be different and since the present map has infinite number of fixed points, the solutions z1 and z2 may converge to different fixed points in the way that they induce distinct filled Julia sets. When the synchronization is achieved between the two maps, it is achieved for the associated Julia sets.Theorem 7.
The two fractional maps (8) and (9) are synchronized under the influence of the following controller:(28)ϕz1,z2,t+α−1=e−az12t+α−1−e−az22t+α−1−κz2t+α−1−z1t+α−1,where the gain κ=κr+iκi, satisfying κ<2α and κr>0.Proof.
The discrete fractional error map is obtained by subtracting equation (8) from equation (9) as follows:(29)Δ0αet=e−az22t+α−1−e−az12t+α−1+ϕz1,z2,t+α−1,et=z2t−z1t.
Using the proposed controller (10) into the above fractional error system, it results in(30)Δ0αet=−κet+α−1,or(31)Δ0αert+ieit=−κr−iκiert+α−1+ieit+α−1,which can be expressed in the following two dimensional system:(32)Δ0αert=−κrert+α−1+κieit+α−1,ΔCeit=−κiert+α−1−κreit+α−1.
It is obvious that the eigenvalues of error system are−κr±iκi, so that the asymptotic stability to zero fixed point of error system is attained provided that κ<2α and κr>0.
Numerical simulations are now employed to validate the theoretical results acquired in this section. The synchronization between orbits of two fractional-order complex Gaussian maps initiated from different initial conditions is shown in Figure5.Figure 5
Synchronization errors between master and slave systems ata=0.15−0.15i,α=0.9, b=0.71+0.25i, where initial value for master system is −1.43−3i and that for slave system is −1.4−2.8i, whereas κ=1+i.
(a)(b)


## 3.1. Control of Julia Sets of Discrete Fractional Complex Gaussian Map

In this section, the appropriate controller is designed in order to change the characteristics and geometry of Julia sets generated by the proposed fractional map via varying the type of stability of one of the fixed points of the present map. More specifically, we consider the feedback controller in the following form:(19)ϱt=−ςzt−z˜−e−az2t−b,where z˜ is the selected unstable fixed point intended to be stabilized under the influence of controller and ς=ςr+iςi represents the complex-valued gain of the controller, which can be evaluated as follows:Theorem 6.
Assume that the gainς of controller ϱt of the controlled fractional-order complex Gaussian map(20)Δ0αzt=e−az2t+α−1+b+ϱt+α−1,fulfills the two inequalities(21)ςr>0,ςr2+ςi2<2α,then the fixed point z˜ become stable, such that the associated Julia set in its neighborhood is changed.Proof.
By applying the control signal (6), we get the following controlled fractional-order complex map:(22)Δ0αzt=−ςzt+α−1−z˜.
Definingδt=zt−z˜∈ℂ, equation (22) takes the following form:(23)Δ0αδt=−ςδt+α−1.
The corresponding two-dimensional real-valued fractional map can be expressed as follows:(24)Δ0αδrt=−ςrδrt+α−1+ςiδit+α−1,Δ0αδit=−ςiδrt+α−1−ςrδit+α−1.
Then, the coefficients of the above system can be put in the following matrix:(25)Λ=−ςrςi−ςi−ςr,and the associated eigenvalues are computed as −ςr±iςi. Hence, the sufficient conditions required for local asymptotic stability of z˜ can be formulated as ςr>0 and ςr2+ςi2<2α.


## 3.2. Synchronization of Julia Sets

The discrete fractional master system is defined in the following form:(26)Δ0αz1t=e−az12t+α−1+b,whereas the corresponding slave system is formulated as follows:(27)Δ0αz2t=e−az22t+α−1+b+ϕz1,z2,t+α−1,where ϕz1,z2,t+α−1 is the adequate controller to be designed. Note that the initial values of two systems are assumed to be different and since the present map has infinite number of fixed points, the solutions z1 and z2 may converge to different fixed points in the way that they induce distinct filled Julia sets. When the synchronization is achieved between the two maps, it is achieved for the associated Julia sets.Theorem 7.
The two fractional maps (8) and (9) are synchronized under the influence of the following controller:(28)ϕz1,z2,t+α−1=e−az12t+α−1−e−az22t+α−1−κz2t+α−1−z1t+α−1,where the gain κ=κr+iκi, satisfying κ<2α and κr>0.Proof.
The discrete fractional error map is obtained by subtracting equation (8) from equation (9) as follows:(29)Δ0αet=e−az22t+α−1−e−az12t+α−1+ϕz1,z2,t+α−1,et=z2t−z1t.
Using the proposed controller (10) into the above fractional error system, it results in(30)Δ0αet=−κet+α−1,or(31)Δ0αert+ieit=−κr−iκiert+α−1+ieit+α−1,which can be expressed in the following two dimensional system:(32)Δ0αert=−κrert+α−1+κieit+α−1,ΔCeit=−κiert+α−1−κreit+α−1.
It is obvious that the eigenvalues of error system are−κr±iκi, so that the asymptotic stability to zero fixed point of error system is attained provided that κ<2α and κr>0.
Numerical simulations are now employed to validate the theoretical results acquired in this section. The synchronization between orbits of two fractional-order complex Gaussian maps initiated from different initial conditions is shown in Figure5.Figure 5
Synchronization errors between master and slave systems ata=0.15−0.15i,α=0.9, b=0.71+0.25i, where initial value for master system is −1.43−3i and that for slave system is −1.4−2.8i, whereas κ=1+i.
(a)(b)


## 4. Proposed Encryption Algorithm

The objective of this section is to introduce an efficient chaos-based encryption technique, which utilizes the idea of pseudo-chaotic dynamics along with complicated fractal patterns to boost its security performance.Consider the following two modified chaotic lemniscate maps [47]:(33)x1n+1=cos23/2+rcos2rx1nsin2rx1n/1+sin22rx1n1+sin223/2+rcos2rx1nsin2rx1n/1+sin22rx1n,y1n+1=22cos2rcos2ry1n/1+sin22ry1nsin2rcos2ry1n/1+sin22ry1n1+sin22rcos2ry1n/1+sin22ry1n,(34)x2n=cos23/2cos2rx2n/1+sin22rx2n×2rsin2rx2n/1+sin22rx2n1+sin223/2cos2rx2n/1+sin22rx2n×2rsin2rx2n/1+sin22rx2n,y2n=2cos2r/1+sin22ry2n×cos2ry2n/1+sin22ry2n1+sin22rcos2ry2n/1+sin22ry2n×sin2rsin2rx2n/1+sin22rx2n1/2.It is obvious that these two maps are mathematically equivalent, yet the finite floating-point representation renders the corresponding orbits diverge exponentially from each other even in the case where identical initial conditions are used. Now, a set ofq random perturbation values, b1,b2,…,pq, is chosen and used to update the generated sequences from the above two systems as follows:Forn=1:1000(35)Xin=xin+b1,Yin=yin+b1,i=1,2.Forn=1001:2000(36)Xin=xin+b2,Yin=yin+b2,i=1,2.….Forn=q−11000+1:q×1000,(37)Xin=xin+bq,Yin=yin+bq,i=1,2.The modular one operations are employed to get(38)X^in=modXin,1,Y˜in=modYin,1,and hence, the associated lower bound errors can be obtained by setting(39)eXn=X^1n−X^2n2,eXn=Y^1n−Y^2n2.Fractal images are used in the proposed encryption technique to boost the security performance of the technique via incorporating additional layers of encryption. More specifically, the color components of each pixel in randomly selected two fractal images are used to confuse the values of each color component in the way that the first fractal image is used with the plain image and the second one is concerned with the shuffled plain image. In order to control and reduce the computation cost, a catalog of secretly pregenerated fractal images can be saved and then employed as one of the secret keys in the scheme. The advantages of using discrete fractional complex maps in the generation of fractal images are that they significantly increase key space. In particular, the two complex-valued parametersa and b in addition to the real-valued parameters α, r, x0, and y0 are the key parameters in the system in addition to the random perturbing values for pseudo-chaotic signals. This implies that using IEEE 754 double-precision floating-point format, the established key space is approximately 23922 for 256×256 plain images and increases considerably for larger plain images. The pseudo-chaotic time series represented by the obtained lower bound errors are utilized in the encryption process as illustrated in the next section.Figure 6
Example of fractal images that are generated by the proposed fractional complex map (1).
(a)(b)


### 4.1. Steps of the Proposed Algorithm

Step 1. The original color image is separated into R-channelPr, G-channel Pg, and B-channel Pb, which are arranged into three matrices of size M×N.Step 2. Establish three time-varying and plain-image dependent perturbation valuesξr,g,b by evaluating(40)ξr,g,b=ντt+13M×N2∑i=1M∑j=1NPr,g,bi,j,where the value ofτt refers to a scaled value of time difference between the moment when the plain image was supplied to encryption machine and another secretly specified moment in the past, for example, 10 : 45 : 12 : 73 Jan 1, 2000. The difference can be taken in units of milliseconds. Also, the scaling factor ν is used to render ντtspans the required range of time range. Moreover, i and j are pixel positions of the R-channel, G-channel, and B-channel matrices of plain images, that is, Pr,Pg,Pb, respectively. We use ξr,g,b. as perturbation values for chaotic map parameter r, such that(41)r1,2,3=r0+ξr,g,b,wherer0 is a base-value for r. Therefore, three pseudo-chaotic sequences are generated and utilized in permutation and diffusion processes of the aforementioned three plain image channels.Step 3. The chaotic lemniscate map is used to generate two pseudo-chaotic sequencesexi,eyi and used in creating the following sequences:(42)rowColi=modfloorexi×1015,450+1,ksi=modflooreyi×1015,256.We usemod operation between variables xi and M=N to get a sequence to build a new position for pixels value image matrices IR,IG,IB as shuffling process. Also, we use mod operation between the variable yi and 256 to get a random sequence that we used it in encryption process as a secret key.Step 4. We getrowj and columnj as a new position of image pixels, where j=1,2,…,M, from rowColi sequence.Step 5. Rearrange the pixel position as shuffle process as follows:(43)IRshi,j=IRrowi,columnj,IGshi,j=IGrowi,columnj,IBshi,j=IBrowi,columnj,whereIRsh and IR are the matrix for shuffled and plain images, respectively, where i=1,2,…,M and j=1,2,…,N are the image matrix dimensions.Step 6. We use two randomly selected fractal images from previously constructed catalog, for example, Figures6(a) and 6(b), as secret keys Keyf1 and Keyf2 for each red, green, and blue color images by separating each color image from each fractal image and using it as secret keys with corresponding color in the plain image. Therefore, we have six secret keys based on the two fractal images.In addition, to enhance the confusion of the secret key, we do a shuffle process as in step 4 to R-channel, G-channel, and B-channel of fractal image (Figure6(a)) before using them as a secret key.Step 7. We divide the sequenceks to three sequences ksr,ksg,ksb for each color in the plain image. To set the secret keys in matrix form, the reshape function is used as follows:(44)ksR=reshapeksr,M,N,ksG=reshapeksg,M,N,ksB=reshapeksb,M,N,to be used as secret keysksR,ksG,ksB for red, green, and blue channels in the plain images, respectively.Step 8. Apply two bitwise XOR operation betweenKeyf,ks,Ish to establish the encrypted image Ien as follows:(45)IReni,j=IRshi,j⊕KeyfR1i,j⊕KeyfR2i,j⊕ksRi,j,IGeni,j=IGshi,j⊕KeyfG1i,j⊕KeyfG2i,j⊕ksGi,j,IBeni,j=IBshi,j⊕KeyfB1i,j⊕KeyfB2i,j⊕ksBi,j,whereIRen,IGen, and IBen are the encrypted images for each color component in plain image.The process of decryption is carried out using the reverse approach. The proposed encryption scheme is applied to three colored images. The three perturbation constants that are used in the proposed scheme are3.9724×10−4, 3.7782×10−4, and 4.0288×10−4 for baboon, pepper, and house images, respectively. The values of r0,x0,y0 are taken as 35,0.5,0.5, respectively. Figure 7 depicts the original, shuffled, and encrypted images for the three images with size M=N=450 after applying the presented algorithm.Figure 7
The plain, shuffled, and encrypted images in (a), (b), and (c), respectively, for baboon, pepper, and house images.
(a)(b)(c)


## 4.1. Steps of the Proposed Algorithm

Step 1. The original color image is separated into R-channelPr, G-channel Pg, and B-channel Pb, which are arranged into three matrices of size M×N.Step 2. Establish three time-varying and plain-image dependent perturbation valuesξr,g,b by evaluating(40)ξr,g,b=ντt+13M×N2∑i=1M∑j=1NPr,g,bi,j,where the value ofτt refers to a scaled value of time difference between the moment when the plain image was supplied to encryption machine and another secretly specified moment in the past, for example, 10 : 45 : 12 : 73 Jan 1, 2000. The difference can be taken in units of milliseconds. Also, the scaling factor ν is used to render ντtspans the required range of time range. Moreover, i and j are pixel positions of the R-channel, G-channel, and B-channel matrices of plain images, that is, Pr,Pg,Pb, respectively. We use ξr,g,b. as perturbation values for chaotic map parameter r, such that(41)r1,2,3=r0+ξr,g,b,wherer0 is a base-value for r. Therefore, three pseudo-chaotic sequences are generated and utilized in permutation and diffusion processes of the aforementioned three plain image channels.Step 3. The chaotic lemniscate map is used to generate two pseudo-chaotic sequencesexi,eyi and used in creating the following sequences:(42)rowColi=modfloorexi×1015,450+1,ksi=modflooreyi×1015,256.We usemod operation between variables xi and M=N to get a sequence to build a new position for pixels value image matrices IR,IG,IB as shuffling process. Also, we use mod operation between the variable yi and 256 to get a random sequence that we used it in encryption process as a secret key.Step 4. We getrowj and columnj as a new position of image pixels, where j=1,2,…,M, from rowColi sequence.Step 5. Rearrange the pixel position as shuffle process as follows:(43)IRshi,j=IRrowi,columnj,IGshi,j=IGrowi,columnj,IBshi,j=IBrowi,columnj,whereIRsh and IR are the matrix for shuffled and plain images, respectively, where i=1,2,…,M and j=1,2,…,N are the image matrix dimensions.Step 6. We use two randomly selected fractal images from previously constructed catalog, for example, Figures6(a) and 6(b), as secret keys Keyf1 and Keyf2 for each red, green, and blue color images by separating each color image from each fractal image and using it as secret keys with corresponding color in the plain image. Therefore, we have six secret keys based on the two fractal images.In addition, to enhance the confusion of the secret key, we do a shuffle process as in step 4 to R-channel, G-channel, and B-channel of fractal image (Figure6(a)) before using them as a secret key.Step 7. We divide the sequenceks to three sequences ksr,ksg,ksb for each color in the plain image. To set the secret keys in matrix form, the reshape function is used as follows:(44)ksR=reshapeksr,M,N,ksG=reshapeksg,M,N,ksB=reshapeksb,M,N,to be used as secret keysksR,ksG,ksB for red, green, and blue channels in the plain images, respectively.Step 8. Apply two bitwise XOR operation betweenKeyf,ks,Ish to establish the encrypted image Ien as follows:(45)IReni,j=IRshi,j⊕KeyfR1i,j⊕KeyfR2i,j⊕ksRi,j,IGeni,j=IGshi,j⊕KeyfG1i,j⊕KeyfG2i,j⊕ksGi,j,IBeni,j=IBshi,j⊕KeyfB1i,j⊕KeyfB2i,j⊕ksBi,j,whereIRen,IGen, and IBen are the encrypted images for each color component in plain image.The process of decryption is carried out using the reverse approach. The proposed encryption scheme is applied to three colored images. The three perturbation constants that are used in the proposed scheme are3.9724×10−4, 3.7782×10−4, and 4.0288×10−4 for baboon, pepper, and house images, respectively. The values of r0,x0,y0 are taken as 35,0.5,0.5, respectively. Figure 7 depicts the original, shuffled, and encrypted images for the three images with size M=N=450 after applying the presented algorithm.Figure 7
The plain, shuffled, and encrypted images in (a), (b), and (c), respectively, for baboon, pepper, and house images.
(a)(b)(c)


## 5. Security Analysis

The proper encryption scheme must be evaluated to investigate his efficacy in resisting several types of attacks. These involve brute force, statistical, differential, known-plaintext, chosen-plaintext, and chosen-ciphertext attacks. In this section, a thorough security analysis is carried out considering these types of attacks.


### 5.1. Histogram

The histogram analysis is used to visualize the distribution of pixels in an image before and after the encryption process. Uniformity of pixels distribution in encrypted data implies that statistical features of input data are efficiently hidden by encryption operation. Histograms for red, green, and blue plain, shuffled, and encrypted images for baboon image are shown in Figure8 whereas histograms for red, green, and blue plain, shuffled, and encrypted images for pepper image are shown in Figure 9. Finally, histograms for red, green, and blue plain, shuffled, and encrypted images for house image are shown in Figure 10.Figure 8
Histograms for (a) red, (b) green, and (c) blue baboon image for plain, shuffled, and encrypted image, respectively.
(a)(b)(c)Figure 9
Histograms for (a) red, (b) green, and (c) blue pepper image for plain, shuffled, and encrypted image, respectively.
(a)(b)(c)Figure 10
Histograms for (a) red, (b) green, and (c) blue house image for plain, shuffled, and encrypted image, respectively.
(a)(b)(c)In order to quantify the uniformity of histograms, the variance of histogram is utilized as a useful measure. The variance of histogram is calculated as follows [51]:(46)Varh=12562∑i=1256∑j=125612hi−hj2,where h represents the histogram values arranged in vector form and hi and hj denote the numbers of pixels having values of i and j, respectively. The variance of histogram for original and ciphered images is depicted in Table 2 with the percentage of reduction between the plain and encrypted images. Noting that the percentage of reduction is greater than 99.6% in the red, green, blue baboon images and greater than 99.8% in three separated colors for pepper and house images. These results confirm the efficiency of the proposed technique.Table 2
The histogram variance and its reduction for the original and cipher images for baboon, pepper, and house images.
VariancePlainEncryptedReduction (%)BaboonRed176920701.442999.6035Green348200755.011599.7832Blue188610650.03999.6553PepperRed520530818.901799.8427Green695920672.101799.9034Blue1122000694.697899.9381HouseRed440620710.893999.8387Green756780764.344999.899Blue577050800.117499.8613


### 5.2. Key Space Analysis

Evaluating the size of secret key space in a specific encryption technique is a crucial step to evaluate its performance against brute force attacks. When the capabilities and characteristics of the state-of-the-art computer are taken into account, it is found that a threshold value for a minimum sufficient key space is a size of2100 to ensure that the brute-force attacks are unfeasible [47, 51]. In our suggested scheme, the two complex-valued parameters a and b in addition to the real-valued parameters α, r, x0, and y0 are the key parameters in the system in addition to the random perturbing values for pseudo-chaotic signals. This implies that using IEEE 754 double-precision floating-point format, the attained key space is approximately 23922 for 256×256 plain images and increases considerably for larger plain images. Accordingly, the presented scheme has key space that is much greater than the minimum value of 2100.


### 5.3. Correlation Analysis

The correlation analysis utilized to measure and quantify the similarity among adjacent pixels throughout the image under consideration, which can be the plain image or the encrypted image. The efficient encryption scheme should make the correlation coefficient as small as possible to boost the security against conventional statistical attacks. The correlation coefficient can be defined as follows:(47)r=covx,yσxσy,where σϕ=varϕ,σψ=varψ.(48)varϕ=1N∑i=1Nϕi−Eϕ2,covϕ,ψ=1N∑i=1Nϕi−Eϕψi−Eψ,where the values of pixels of plain and encrypted images are denoted by ϕ and ψ, respectively. The correlation values between adjacent pixels in horizontal, vertical, and diagonal directions are acquired for baboon, pepper, and house images and listed in Table 3. It is obvious that the proposed algorithm is immune to statistical attacks because it is successfully minimized the values of correlation coefficients in the encrypted images to about zero.Table 3
The correlation values between adjacent pixels, in all directions, were obtained for red, green, and blue color components in baboon, pepper, and house images, respectively.
Correlation coefficientsHorizontalVerticalDiagonalBaboonRedPlain0.91930.8640.8403Cipher−0.0005−0.00390.001GreenPlain0.87950.79970.7628Cipher0.0032−0.001−0.0028BluePlain0.92850.88270.8597Cipher−0.0021−0.00130.0027PepperRedPlain0.96810.97030.9519Cipher0.0001−0.0000−0.0007GreenPlain0.97860.9790.9616Cipher0.0000−0.0036−0.0003BluePlain0.96540.96430.9414Cipher−0.0048−0.0044−0.0029HouseRedPlain0.94840.94670.9087Cipher−0.00010.00240.0005GreenPlain0.92860.94810.8893Cipher−0.0005−0.0003−0.0004BluePlain0.97040.97180.9472Cipher−0.00050.0013−0.0008


### 5.4. Information Entropy

The information entropy is another powerful analysis tool used to find the unpredictability and randomness in the proposed scheme. It is reported that the optimum value is 8. The information entropy of a given image is outlined as follows:(49)Hm=∑i=12N−1pilog21pi,where Hm denotes the entropy in bits, m is an input parameter, and finally the value of probability for parameter m is referred to as pi.The entropy values for red, green, and blue images have been evaluated for baboon, pepper, and house encrypted images and summarized in Table4. It is cleared that the entropy values for the three images are very close to 8; therefore, the proposed scheme is less feasible to expose information of the plain image.Table 4
The entropy for encrypted image for red, green, and blue images for baboon, pepper, and house image, respectively.
PlainRed (%)Green (%)Blue (%)Baboon7.99927.99917.9993Pepper7.99917.99927.9992House7.99927.99917.9991


### 5.5. Differential Attack Analysis

To evaluate the immunity of the proposed cryptosystem against the powerful differential, two useful quantities reevaluated, namely, the number of pixels changing rate (NPCR) and unified average changing intensity (UACI). These measures identify the sensitivity of the encryption scheme to change a single-pixel value of supplied plain image or sensitivity to small changes in the secret key. The equations to evaluate NPCR and UACI are expressed as follows [47]:(50)NPCR%=1M×N∑i=1M∑j=1NsignC1i,j−C2i,j×100,UACI%=1M×N∑i=1M∑j=1NC1i,j−C2i,j255×100,where the well-known sign function is referred to as sign(), while Cis refer to the cipher image. In Table 5, the evaluated values of UACI and NPCR are given for the three submitted plain images. It is observed that the values of NPCR are generally greater than 99.5, while those of UACI are greater than 33.4, which indicates the sensitivity to a pixel change in the proposed encryption algorithm.Table 5
NPCR and UACI results for red, green, and blue images for baboon, pepper, and house images, respectively.
ImageNPCR (%)UACI (%)BaboonRed99.60133.559Green99.601533.4034Blue99.613333.534PepperRed99.628133.5021Green99.59733.4069Blue99.590133.5242HouseRed99.59833.4591Green99.581733.4822Blue99.593633.542


### 5.6. Cropping Attack

In order to detect the robustness of the proposed technique, some blocks of size450×100 of a cipher house image are converted into black. The restored image after is depicted in Figure 11. Although there is a loss of significant information, the encrypted image after the decryption process is still recognizable.Figure 11
The encrypted house image after converting the left, middle, and right blocks, respectively, of the house image into black color (a) and the corresponding recognizable decrypted images (b).
(a)(b)Finally, the aforementioned results are summarized. The proposed encryption technique combines the pseudo-chaos of modified chaotic lemniscate map [47], which has a distinct complicated dynamics and large value of positive Lyapunov exponent with the fractal images generated by complex discrete fractional Gauss map. When compared with different state-of-the-art chaos-based encryption techniques, the main advantages of the present encryption technique are as follows: (a) it deploys superior positive values of maximum Lyapunov exponents. For example, the maximum value of Lyapunov exponent of chaos employed in the image encryption system [48] and bit-level permutation spatial system [48] is less than three, while it is greater than 30 in the present scheme. (b) The pseudo-chaotic time series tame the possible degradation of statistical features of chaos signals in the cases, where they are applied immediately [46]. (c) The assigned keys for the suggested encryption technique are set in a way that renders them controlled by plain data features as well as the time moment of their processing. This means that if identical plain images are encrypted at different instants, different secret keys will be used for the encryption process inducing different cipher images. Moreover, the pseudo-chaos or lower bound errors between the outputs of two interval extensions are employed in the presented scheme instead of applying chaotic signals directly in permutation and diffusion stages. This adds another layer of security and hides the internal characteristics of chaos generators maps. More details about the lower bound errors and analysis of interval extensions can be found in references [49, 50]. Now, the critical scenario of known-plaintext attack (KPA) is considered, where the opponent successfully attains the specific plain image and corresponding cipher image, and then he cannot proceed further to obtain any extra useful information about secret keys’ values, which will be used for upcoming plain images as the scheme utilizes time-varying secret keys. The proposed encryption technique can resist KPA even in special cases when uniform plain images with zero values of pixels are deployed, which may lead to a degenerate performance in other encryption techniques [52–54]. The adoption of fractal images in the scheme boosts complexity, key space range, and security performance. Moreover, if the opponent employs chosen-ciphertext attack (CCA) to supply some specially selected cipher images to decryption part of the scheme, he would not fulfill his target too.The running time of the proposed encryption scheme on personal computer with 16 GB RAM and Intel Core i7-8550U CPU 1.8 GHz is approximately 0.582 s for450×450 colored images. The comparison aspects with some recent chaos-based encryption techniques are summarized in Table 6. The MCC and AVR abbreviations are used to denote the maximum correlation coefficients attained in all directions of encrypted color baboon/pepper images and average running time, respectively.Table 6
Some comparisons with recent chaos-based encryption techniques.
WorkUACINPCRMLEEntropyKey spaceMCCAVR (sec)Proposed work33.53299.814Up to 607.999239220.00310.582Reference [55] (2 rounds)33.48499.80927.90323180.01910.385Reference [56]33.42199.61127.99723120.01311.860Reference [57]33.45299.6070.827.99121870.00820.478Reference [58]33.41199.6106.7567.99823990.01430.8342


## 5.1. Histogram

The histogram analysis is used to visualize the distribution of pixels in an image before and after the encryption process. Uniformity of pixels distribution in encrypted data implies that statistical features of input data are efficiently hidden by encryption operation. Histograms for red, green, and blue plain, shuffled, and encrypted images for baboon image are shown in Figure8 whereas histograms for red, green, and blue plain, shuffled, and encrypted images for pepper image are shown in Figure 9. Finally, histograms for red, green, and blue plain, shuffled, and encrypted images for house image are shown in Figure 10.Figure 8
Histograms for (a) red, (b) green, and (c) blue baboon image for plain, shuffled, and encrypted image, respectively.
(a)(b)(c)Figure 9
Histograms for (a) red, (b) green, and (c) blue pepper image for plain, shuffled, and encrypted image, respectively.
(a)(b)(c)Figure 10
Histograms for (a) red, (b) green, and (c) blue house image for plain, shuffled, and encrypted image, respectively.
(a)(b)(c)In order to quantify the uniformity of histograms, the variance of histogram is utilized as a useful measure. The variance of histogram is calculated as follows [51]:(46)Varh=12562∑i=1256∑j=125612hi−hj2,where h represents the histogram values arranged in vector form and hi and hj denote the numbers of pixels having values of i and j, respectively. The variance of histogram for original and ciphered images is depicted in Table 2 with the percentage of reduction between the plain and encrypted images. Noting that the percentage of reduction is greater than 99.6% in the red, green, blue baboon images and greater than 99.8% in three separated colors for pepper and house images. These results confirm the efficiency of the proposed technique.Table 2
The histogram variance and its reduction for the original and cipher images for baboon, pepper, and house images.
VariancePlainEncryptedReduction (%)BaboonRed176920701.442999.6035Green348200755.011599.7832Blue188610650.03999.6553PepperRed520530818.901799.8427Green695920672.101799.9034Blue1122000694.697899.9381HouseRed440620710.893999.8387Green756780764.344999.899Blue577050800.117499.8613


## 5.2. Key Space Analysis

Evaluating the size of secret key space in a specific encryption technique is a crucial step to evaluate its performance against brute force attacks. When the capabilities and characteristics of the state-of-the-art computer are taken into account, it is found that a threshold value for a minimum sufficient key space is a size of2100 to ensure that the brute-force attacks are unfeasible [47, 51]. In our suggested scheme, the two complex-valued parameters a and b in addition to the real-valued parameters α, r, x0, and y0 are the key parameters in the system in addition to the random perturbing values for pseudo-chaotic signals. This implies that using IEEE 754 double-precision floating-point format, the attained key space is approximately 23922 for 256×256 plain images and increases considerably for larger plain images. Accordingly, the presented scheme has key space that is much greater than the minimum value of 2100.


## 5.3. Correlation Analysis

The correlation analysis utilized to measure and quantify the similarity among adjacent pixels throughout the image under consideration, which can be the plain image or the encrypted image. The efficient encryption scheme should make the correlation coefficient as small as possible to boost the security against conventional statistical attacks. The correlation coefficient can be defined as follows:(47)r=covx,yσxσy,where σϕ=varϕ,σψ=varψ.(48)varϕ=1N∑i=1Nϕi−Eϕ2,covϕ,ψ=1N∑i=1Nϕi−Eϕψi−Eψ,where the values of pixels of plain and encrypted images are denoted by ϕ and ψ, respectively. The correlation values between adjacent pixels in horizontal, vertical, and diagonal directions are acquired for baboon, pepper, and house images and listed in Table 3. It is obvious that the proposed algorithm is immune to statistical attacks because it is successfully minimized the values of correlation coefficients in the encrypted images to about zero.Table 3
The correlation values between adjacent pixels, in all directions, were obtained for red, green, and blue color components in baboon, pepper, and house images, respectively.
Correlation coefficientsHorizontalVerticalDiagonalBaboonRedPlain0.91930.8640.8403Cipher−0.0005−0.00390.001GreenPlain0.87950.79970.7628Cipher0.0032−0.001−0.0028BluePlain0.92850.88270.8597Cipher−0.0021−0.00130.0027PepperRedPlain0.96810.97030.9519Cipher0.0001−0.0000−0.0007GreenPlain0.97860.9790.9616Cipher0.0000−0.0036−0.0003BluePlain0.96540.96430.9414Cipher−0.0048−0.0044−0.0029HouseRedPlain0.94840.94670.9087Cipher−0.00010.00240.0005GreenPlain0.92860.94810.8893Cipher−0.0005−0.0003−0.0004BluePlain0.97040.97180.9472Cipher−0.00050.0013−0.0008


## 5.4. Information Entropy

The information entropy is another powerful analysis tool used to find the unpredictability and randomness in the proposed scheme. It is reported that the optimum value is 8. The information entropy of a given image is outlined as follows:(49)Hm=∑i=12N−1pilog21pi,where Hm denotes the entropy in bits, m is an input parameter, and finally the value of probability for parameter m is referred to as pi.The entropy values for red, green, and blue images have been evaluated for baboon, pepper, and house encrypted images and summarized in Table4. It is cleared that the entropy values for the three images are very close to 8; therefore, the proposed scheme is less feasible to expose information of the plain image.Table 4
The entropy for encrypted image for red, green, and blue images for baboon, pepper, and house image, respectively.
PlainRed (%)Green (%)Blue (%)Baboon7.99927.99917.9993Pepper7.99917.99927.9992House7.99927.99917.9991


## 5.5. Differential Attack Analysis

To evaluate the immunity of the proposed cryptosystem against the powerful differential, two useful quantities reevaluated, namely, the number of pixels changing rate (NPCR) and unified average changing intensity (UACI). These measures identify the sensitivity of the encryption scheme to change a single-pixel value of supplied plain image or sensitivity to small changes in the secret key. The equations to evaluate NPCR and UACI are expressed as follows [47]:(50)NPCR%=1M×N∑i=1M∑j=1NsignC1i,j−C2i,j×100,UACI%=1M×N∑i=1M∑j=1NC1i,j−C2i,j255×100,where the well-known sign function is referred to as sign(), while Cis refer to the cipher image. In Table 5, the evaluated values of UACI and NPCR are given for the three submitted plain images. It is observed that the values of NPCR are generally greater than 99.5, while those of UACI are greater than 33.4, which indicates the sensitivity to a pixel change in the proposed encryption algorithm.Table 5
NPCR and UACI results for red, green, and blue images for baboon, pepper, and house images, respectively.
ImageNPCR (%)UACI (%)BaboonRed99.60133.559Green99.601533.4034Blue99.613333.534PepperRed99.628133.5021Green99.59733.4069Blue99.590133.5242HouseRed99.59833.4591Green99.581733.4822Blue99.593633.542


## 5.6. Cropping Attack

In order to detect the robustness of the proposed technique, some blocks of size450×100 of a cipher house image are converted into black. The restored image after is depicted in Figure 11. Although there is a loss of significant information, the encrypted image after the decryption process is still recognizable.Figure 11
The encrypted house image after converting the left, middle, and right blocks, respectively, of the house image into black color (a) and the corresponding recognizable decrypted images (b).
(a)(b)Finally, the aforementioned results are summarized. The proposed encryption technique combines the pseudo-chaos of modified chaotic lemniscate map [47], which has a distinct complicated dynamics and large value of positive Lyapunov exponent with the fractal images generated by complex discrete fractional Gauss map. When compared with different state-of-the-art chaos-based encryption techniques, the main advantages of the present encryption technique are as follows: (a) it deploys superior positive values of maximum Lyapunov exponents. For example, the maximum value of Lyapunov exponent of chaos employed in the image encryption system [48] and bit-level permutation spatial system [48] is less than three, while it is greater than 30 in the present scheme. (b) The pseudo-chaotic time series tame the possible degradation of statistical features of chaos signals in the cases, where they are applied immediately [46]. (c) The assigned keys for the suggested encryption technique are set in a way that renders them controlled by plain data features as well as the time moment of their processing. This means that if identical plain images are encrypted at different instants, different secret keys will be used for the encryption process inducing different cipher images. Moreover, the pseudo-chaos or lower bound errors between the outputs of two interval extensions are employed in the presented scheme instead of applying chaotic signals directly in permutation and diffusion stages. This adds another layer of security and hides the internal characteristics of chaos generators maps. More details about the lower bound errors and analysis of interval extensions can be found in references [49, 50]. Now, the critical scenario of known-plaintext attack (KPA) is considered, where the opponent successfully attains the specific plain image and corresponding cipher image, and then he cannot proceed further to obtain any extra useful information about secret keys’ values, which will be used for upcoming plain images as the scheme utilizes time-varying secret keys. The proposed encryption technique can resist KPA even in special cases when uniform plain images with zero values of pixels are deployed, which may lead to a degenerate performance in other encryption techniques [52–54]. The adoption of fractal images in the scheme boosts complexity, key space range, and security performance. Moreover, if the opponent employs chosen-ciphertext attack (CCA) to supply some specially selected cipher images to decryption part of the scheme, he would not fulfill his target too.The running time of the proposed encryption scheme on personal computer with 16 GB RAM and Intel Core i7-8550U CPU 1.8 GHz is approximately 0.582 s for450×450 colored images. The comparison aspects with some recent chaos-based encryption techniques are summarized in Table 6. The MCC and AVR abbreviations are used to denote the maximum correlation coefficients attained in all directions of encrypted color baboon/pepper images and average running time, respectively.Table 6
Some comparisons with recent chaos-based encryption techniques.
WorkUACINPCRMLEEntropyKey spaceMCCAVR (sec)Proposed work33.53299.814Up to 607.999239220.00310.582Reference [55] (2 rounds)33.48499.80927.90323180.01910.385Reference [56]33.42199.61127.99723120.01311.860Reference [57]33.45299.6070.827.99121870.00820.478Reference [58]33.41199.6106.7567.99823990.01430.8342


## 6. Conclusion

This study establishes a framework to study dynamical and fractal characteristics, in addition to potential applications, of generalized complex-valued discrete fractional Gaussian map. The occurrence of Mandelbrot and Julia sets of the proposed map is scrutinized at different scenarios for values of parameters. The control and synchronization problems of Julia sets in the complex domain are addressed. A combined pseudo-chaos-fractal image encryption technique is introduced as an efficient tool to resist several kinds of attacks. A thorough security analysis is carried out to validate its robustness and efficiency against statistical, differential, and cropping attacks. Indeed, there is a trade-off between increasing chaoticity and security strength from one side and computational speed from the other side. The present application in this work is the first step and subsequent work will focus on realization aspects on a suitable digital hardware platform, that is, DSP or FPGA, further reduce its running time, and discuss all possible issues that need separate work and cannot be treated here. Future work can also involve extending this study to the case of higher dimensional complex fractional maps [31, 32].



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*Source: 8148831-2022-04-19.xml*