1 Introduction

Nowadays, non-integer or fractional-order chaotic systems are becoming very interesting topics for engineers, physicists, and mathematicians, because most real physical systems are inherently nonlinear. Their application fields have become more and more important because of the utilization of fractional-order mathematics in several real-world fields, such as electronic systems, machine systems, mathematics, signal generators [16, 19], biology, robotics, synchronization, random number generators, communication security and image processing.

It is important to note that many papers have been published presenting implementations of integer-order chaotic systems, and systems using analog integrated circuit technology and digital hardware like micro-controllers, and FPGAs. This is not the case for fractional-order chaotic systems that are mathematical models comprising fractional derivatives in nonlinear equations that are difficult to solve, analytically.

However, analog implementations using passive circuit components, multipliers, and amplifiers suffer from the high sensitivity to any variation process, which requires more complex analog circuitry to solve the fractional derivatives. In addition, digital implementations suffer from the problem of degradations due to the reduced number of bits to perform computer arithmetic operations. So, we require more complex digital hardware to implement memory blocks.

The main objective of the recent works is oriented to introducing alternatives for the analog/digital implementations of fractional-order chaotic oscillators, trying to provide details on the synthesis and physical realization using either analog or digital electronics. Moreover, fractional-order chaotic systems are used for cancellable biometric applications to enhance the security of biometric systems. In the past few years, the theoretical design and circuit implementation of many different chaotic systems have attracted more and more attention of engineers. They used them in many real-world applications, such as secure communication [27], random number generators, image processing, encryption [10], and cancellable biometric recognition. After the first family of n-double scroll (multi-scroll) chaotic attractors was introduced by Suykens and Vandewalle, the authors utilized quasi-linear function techniques [22]. These techniques can be used to generate multi-scroll chaotic systems with high complexity in many applications because of their high sensitivity to initial conditions and change of parameters. Badr et al. [4] presented partial encryption for cancellable biometrics, but the proposed system in this paper depends on full encryption.

Recent works have concentrated on fractional-order mathematical models of chaotic systems to evaluate their equilibrium points, and stability, in addition to performing numerical simulations with appropriate methods in the frequency and time domains, such as Caputo, Grünwald-Letnikov, and Adams-Bashforth-Moulton methods. Under certain conditions, analog and digital hardware of integer-order chaotic systems are relatively simple to implement.

The Grünwald-Letnikov method is utilized to calculate the fractional derivative of a function g(t).

$$\begin{aligned} {}_{a} D_{t}^{q} g(t)=\lim _{h \rightarrow 0} \frac{1}{h^{q}} \sum _{i=0}^{\lfloor (t-a) / h\rfloor } w_{i}^{(q)} g(t-i h), \end{aligned}$$
(1)

where g(t) is a continuous function, q is any positive real number, in the range of \(0<q<1\) for the fractional-order chaotic system and h is the step size. \(w_{i}^{(q)}\) denotes the binomial coefficients calculated with the following formula:

$$\begin{aligned} w_{0}^{(q)}=1, \quad w_{i}^{(q)}=\left( 1-\frac{q+1}{i}\right) w_{i-1}^{(q)}, \quad i=1,2,3, \ldots \end{aligned}$$
(2)

Figure 3 shows that the magnitude of binomial coefficient \(w_{i}^{(q)}\) approaches zero, when the index increases. In other words, when fractional order q tends to an integer value, the dependence of the values of the binomial coefficients decreases. A fractional-order system of three differential equations has the following general form:

$$\begin{aligned} \begin{aligned}&D^{q_{1}} x=S(x, y, z, t) \\&D^{q_{2}} y=P(x, y, z, t) \\&D^{q_{3}} z=Q(x, y, z, t) \end{aligned} \end{aligned}$$
(3)

This system can be simulated based on GL definition with the following set of equations:

$$\begin{aligned} \begin{aligned}&x_{t_{k}}=S\left( x\left( t_{k-1}\right) , y\left( t_{k-1}\right) , z\left( t_{k-1}\right) \right) h^{q_{1}}-\sum _{i=1}^{n} w_{i}^{\left( q_{1}\right) } x\left( t_{k-i}\right) \\&y_{t_{k}}=P\left( x\left( t_{k-1}\right) , y\left( t_{k-1}\right) , z\left( t_{k-1}\right) \right) h^{q_{2}}-\sum _{i=1}^{n} w_{i}^{\left( q_{2}\right) } y\left( t_{k-i}\right) \\&z_{t_{k}}=Q\left( x\left( t_{k-1}\right) , y\left( t_{k-1}\right) , z\left( t_{k-1}\right) \right) h^{q_{3}}-\sum _{i=1}^{n} w_{i}^{\left( q_{3}\right) } z\left( t_{k-i}\right) \end{aligned} \end{aligned}$$
(4)

where \(n=L\) denotes the approximated window variation of the GL operator, and if \(n = k\), this means that all state memory is used in the calculations. The proposed system is able to overcome the drawbacks of the previous methods. It allows complete hiding of biometric patterns instead of partial hiding in [4]. It helps to enhance user privacy, while maintaining a high degree of security. Moreover, a fractional-order multi-scroll chaotic system is exploited in the generation of cancellable templates. The utilization of multiple keys enhances the degree of security.

In this paper, we introduce a proposed chaotic system that generates multi-scroll chaotic attractors. The paper is organized as follows. Section 2 presents the literature survey. Section 3 present a construction of the proposed multi-scroll chaotic system and its properties. Section 4 presents the description of the fractional-order multi-scroll chaotic system. In Section 5, circuit realization is presented for the proposed chaotic system. In Sections 6, FPGA realizations are studied to implement the multi-scroll chaotic system and present some experimental results. Section 7 presents the simulation results. Section 8 presents the cancellable face recognition scheme based on the proposed fractional-order multi-scroll chaotic system. Finally, the conclusions are introduced in the last section.

2 Literature survey

Multiple chaotic attractors can be generated by chaotic systems. Several valuable results have been introduced in [9, 25]. For example, Cui et al [9] presented a multi-scroll chaotic system, which generates multi-scroll chaotic attractors based on sinusoidal nonlinear terms. A new color image encryption scheme was produced based on the fractional-order multi-scroll Chen chaotic system and the DNA mutation principle [11].

Liang et al. [12] presented a multi-scroll chaotic system, which generates multi-scroll chaotic attractors based on a nonlinear exponential function and Chua system. San-Um et al. [18] designed the multi-scroll chaotic system by using a delay differential equation, which employs a piecewise-linear nonlinear function with a tiny delay time. An X- and Heart-shape multi-scroll chaotic system was produced based on the Lorenz circuit and adding a staircase nonlinear function to the system. Soliman et al. [20] designed a multi-scroll chaotic system. Chang et al. [5] presented a multi-scroll chaotic system based on piecewise nonlinear functions to generate a multi-scroll chaotic attractor that is used in secure audio communication.

In addition, Ozoguz et al. [15] presented a multi-scroll chaotic attractor using smooth hyperbolic tangent functions. With different system parameters, Xiong et al. [25] proposed single, double, three, and four chaotic attractors and designed a circuit schematic diagram for realization.

The methods of implementation of chaotic systems represent a vital topic to be studied in real engineering applications. The methods that generate multi-scroll chaotic attractors comprise analog circuits of simple passive components (resistors and capacitors) and an operational amplifier [7, 26], and digital FPGA circuits (embedded system) [5, 24]. For example, in [7], Chen et al. designed an electronic circuit to generate multi-scroll chaotic attractors based on piecewise-linear functions. In addition, based on a sgn function, Cui et al. presented n-scroll chaotic attractors and their circuit realizations to implement the multi-scroll chaotic system [8]. Additionally, in [26], Yan et al. introduced a circuit realization to implement the new 5D fractional-order mutli-scroll chaotic system, whose equations contain cross products and power functions.

Abdelaty et al. [1] studied an FPGA implementation to realize n-scroll chaotic systems based on Product Integration (PI) rules. Additionally, in [24], FPGA circuits were used to implement the multi-scroll chaotic system based on sgn function series. All of these digital and analog circuits are important to generate multi-scroll chaotic attractors. Generally, FPGA circuits are the most popular ones due to lower cost and higher capacity compared to other methods.

3 Construction of the proposed multi-scroll chaotic system and its properties

Motivated by the structure of the model in [13], a new 3D autonomous chaotic system using sgn function series is expressed as follows:

$$\begin{aligned} \left\{ \begin{array}{l} \dot{x}=-u z+u f(z) \\ \dot{y}=z+x-y \\ \dot{z}=- b y \end{array}\right. \end{aligned}$$
(5)

The nonlinear sgn function is defined as:

$$\begin{aligned} f(z)= \sum _{k=-m}^{m} {\text {sgn}}(z+2 m) \end{aligned}$$
(6)

Here, x, y and z are three dynamic variables, while u and b are positive control parameters. Also, m is a natural number, and the sgn function is denoted by the following equation:

$$\begin{aligned} {\text {sgn}}(z)=\left\{ \begin{array}{ll} 1, &{} z>0 \\ 0, &{} z=0 \\ -1, &{}z<0 \end{array}\right. \end{aligned}$$
(7)

The equilibrium points of the system of (5) can be obtained by setting the right-hand side of the system, represented by (5), equal to zero. This yields the following equations:

$$\begin{aligned} \left\{ \begin{array}{l} -u z+u f(z)=0 \\ z+x-y=0 \\ - b y=0 \end{array}\right. \end{aligned}$$
(8)
Table 1 Eigenvalues of each equilibrium point
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Fig. 1
figure 1

Bifurcation diagram of the system in Equation (5) on variable x (a) at the parameter b, (b) at the parameter u

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where \(b=15\), \(u=3\) and \(m=1\). The Jacobian matrix can be obtained by linearizing the system in Equation (5) at all equilibrium points and at different values of b and u, as shown in the following equation:

$$\begin{aligned} J_{\left( z_{l}\right) }=\left( \begin{array}{cccc} \frac{\partial f_{1}}{\partial x} &{} \frac{\partial f_{1}}{\partial y} &{} \frac{\partial f_{1}}{\partial z} \\ \frac{\partial f_{2}}{\partial x} &{} \frac{\partial f_{2}}{\partial y} &{} \frac{\partial f_{2}}{\partial z} \\ \frac{\partial f_{3}}{\partial x} &{} \frac{\partial f_{3}}{\partial y} &{} \frac{\partial f_{3}}{\partial z} \end{array}\right) = \left[ \begin{array}{ccc} 0 &{} 0 &{} -u(1-2 \delta \left( z_{l}+2m\right) ) \\ 1 &{} -1 &{} 1 \\ 0 &{} -b &{} 0 \end{array}\right] \end{aligned}$$
(9)

From the Jacobian matrix, we can get the characteristic equation of the form \(|\lambda I-J_{\left( z_{i}\right) }|\), leading to the system represented by Equation (10). Therefore, the eigenvalues are evaluated for each equilibrium point. The resulting eigenvalues can be obtained as shown in Table 1.

$$\begin{aligned} \left[ \begin{array}{lll} \lambda &{} 0 &{} 0 \\ 0 &{} \lambda &{} 0 \\ 0 &{} 0 &{} \lambda \end{array}\right] -\left[ \begin{array}{ccc} 0 &{} 0 &{} -u(1-2 \delta \left( z_{i}+2m\right) ) \\ 1 &{} -1 &{} 1 \\ 0 &{} -b &{} 0 \end{array}\right] =\left[ \begin{array}{ccc} \lambda &{} 0 &{} +u(1-2 \delta \left( z_{i}+2m\right) ) \\ -1 &{} \lambda +1 &{} -1 \\ 0 &{} b &{} 0 \end{array}\right] \end{aligned}$$
(10)

By fixing the parameters \(u=3\) and \(h=0.009\) and varying the parameter b, the bifurcation diagram is shown in Fig. 1a. In the range of \(b \in [0,50]\), the system is chaotic. Likewise, by fixing the parameters \(b=15\) and \(h=0.009\) and varying the parameter u, the bifurcation diagram is shown in Fig. 1b. In the range of \(u \in [0.02,6]\), the system is chaotic, but in the range of \(u \in [0.001,0.02]\), the system is not chaotic.

Fig. 2
figure 2

At \(u=3, b = 15, \text {and}~ h=0.009\) (a) x, y, z phase portraits of the system represented by Equation (5), (b), (c) and (d) time series of state variables x, y, and z, respectively, using the initial conditions (x(0), y(0), z(0))=(-1, 0.1, 1)

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Fig. 3
figure 3

Graphical representation of the response of binomial coefficient \(w_{i}^{(q)}\) versus index i for (a) \(q=0.6\), (b) \(q=0.7\), (c) \(q=0.8\), (d) \(q=0.9\), (e) \(q=0.91\), (f) \(q=0.92\), (g) \(q=0.93\), (h) \(q=0.95\)

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Figure 2 shows the chaotic attractor of the proposed chaotic system and the chaotic time series in x, y, and z.

4 Description of the fractional-order multi-scroll chaotic system

A proposed 3D autonomous fractional-order multi-scroll chaotic system based on the nonlinear sgn function series is represented as follows:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{d^{q} x}{d t^{q}}=-u z+u (\sum _{k=-n}^{n} {\text {sgn}}(z+2 n)) \\ \frac{d^{q} y}{d t^{q}}=z+x-y \\ \frac{d^{q} z}{d t^{q}}=- b y \end{array}\right. \end{aligned}$$
(11)

Here, x, y and z are three state variables, while u, and b are non-negative parameters. Also, n is a natural number and sgn function is defined by the following equation:

$$\begin{aligned} {\text {sgn}}(z)=\left\{ \begin{array}{ll} 1, &{} z>0 \\ 0, &{} z=0 \\ -1, &{}z<0 \end{array}\right. \end{aligned}$$
(12)

where \(u=3\), \(b=15\), \(h=0.009\), \(q=0.95\) and the initial values of the three state variables are (x(0), y(0), z(0)) = \((-1, 0.1, 1)\). From MATLAB simulation, we obtained phase diagram of the proposed fractional-order chaotic attractor between state variables \(x-y\) and \(x-z\), \(y-z\), and \(x-y-z\) as shown in Fig. 4. Table 2 shows the continuous responses of the proposed fractional-order chaotic system versus the parameter h.

Fig. 4
figure 4

Simulation results of the proposed fractional-order chaotic system projected onto (a) \(x-y\) plane, (b) \(x-z\) plane, and (c) \(y-z\) plane, where and \(x, y, and ~ z\) are state-space portraits of the proposed fractional-order chaotic system based on GL method

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Table 2 Chaotic attractors and their nonlinear behavior
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5 Circuit design of the proposed fractional-order chaotic system

5.1 Fractional integrator approximation

According to Matsuda approximation [14], the linear approximated transfer function F(s) of a fractional integrator of order 0.95 is

$$\begin{aligned} F(s)=\frac{1}{s^{0.95}} \approx \frac{s^3+1.486\times 10^5 s^{2}+1.214\times 10^8 s+5.022\times 10^9}{9.763\times 10^4s^{3}+9.321\times 10^7 s^{2}+4.502\times 10^9 s+1.196\times 10^9} \end{aligned}$$
(13)

The implementation of Equation (13) in Laplace domain is seen in Fig. 5a. Box "F" represents the fractional-order differential unit between a and b. The transfer function H(s) of this box, which equals \(\frac{1}{C_0 F(s)}\), is given by:

$$\begin{aligned} H(s)= \frac{1}{C_0s^{0.95}}=\frac{1 / C_{1}}{s+1 / R_{1} C_{1}}+\frac{1 / C_{2}}{s+1 / R_{2} C_{2}}+\frac{1 / C_{3}}{s+1 / R_{3} C_{3}}+R_0 \end{aligned}$$
(14)

where \(C_0\) is a unit parameter. Thus, if we consider the capacitor \(C_0= 10~ nF\), by comparing the two equations (13) and (14), we obtain the values of circuit elements: resistors and capacitors as \(R_0=1024~ \Omega \), \(R_1=20~ k\Omega \), \(R_2=360~ k\Omega \), \(R_3=419410~ k\Omega \), \(C_1=47.36~ nF\), \(C_2=55.41~ nF\), and \(C_3=8.92~ nF\). According to Charef approximation [6], at \(q=0.95\), the approximated transfer function F(s) is:

$$\begin{aligned} \frac{1}{s^{0.95}} \approx \frac{1.2834 s^{2}+18.6004 s+2.0833}{s^{3}+18.4738 s^{2}+2.6574 s+0.003} \end{aligned}$$
(15)
Fig. 5
figure 5

The fractional-order circuit unit with order \(q= 0.95\) (a) according to Matsuda approximation, (b) according to Charef approximation

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Fig. 6
figure 6

Electronic circuit schematic diagram to implement the proposed chaotic system of Equation (11) showing the state variables x, y, z at (a), (b) circuit realization of the sgn function \(\sum _{k=-2}^{2} {\text {sgn}}(z+2 m)\)

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The transfer function \(H(s)= \frac{1}{C_0s^{0.95}}\) is given by:

$$\begin{aligned} H(s)=\frac{R_{1}}{s R_{1} C_{1}+1}+\frac{R_{2}}{s R_{2} C_{2}+1}+\frac{R_{3}}{s R_{3} C_{3}+1} \end{aligned}$$
(16)

The realization of Equation (16) is given in Fig. 5b. If \(C_0\) is equal to 10  nF, we obtain the values of resistances and capacitances as \(R_1=15.266~ k\Omega \), \(R_2=151100~ k\Omega \), \(R_3=69.291~ G\Omega \), \(C_1=35.739~ nF\), \(C_2=46.06~ nF\), and \(C_3=12.67~ nF\).

5.2 Description of the circuit

In this section, the circuit is designed in order to realize the proposed system of Equation (11). An electronic circuit diagram of a proposed chaotic system is shown in Fig. 6. The circuit shown in Fig. 6b is for the summation of the nonlinear sgn function signal. The implementation of the proposed fractional-order chaotic system depends on electronic components like resistors, capacitors, and TL082 operational amplifiers. The voltage power supply is equal to \(\pm 15\). By analyzing the the circuit implementation of the proposed system shown in Fig. 6, the circuit equations are given by:

$$\begin{aligned} \left\{ \begin{array}{l} D^{q}x=\frac{R_4}{C_{0}R_7}\left[ \left( \frac{1}{R_{5}}\right) z-\left( \frac{1}{R_{6}}\right) f(z) \right] \\ D^{q}y=\frac{R_8}{C_{0}R_{12}}\left[ \left( \frac{1}{R_{9}}\right) x+\left( \frac{1}{R_{10}}\right) y-\left( \frac{1}{R_{11}}\right) z\right] \\ D^{q} z=\frac{R_{15}}{C_{0} R_{17}}\left[ \left( \frac{1}{R_{16}}\right) y\right] \end{array}\right. \end{aligned}$$
(17)

where the parameter values of the circuit schematic diagram are given by \(R_4=72~ k\Omega \), \(R_5=24~ k\Omega \), \(R_6=24~ k\Omega \), \(R_7=200~ k\Omega \), \(R_8= 72~ k\Omega \), \(R_9=72~ k\Omega \), \(R_{10}=72~ k\Omega \), \(R_{11}=72~ k\Omega \), \(R_{12}=200~ k\Omega \), \(R_{13}=10~ k\Omega \), \(R_{14}=10~ k\Omega \), \(R_{15}=72~ k\Omega \), \(R_{16}=4.8~ k\Omega \), \(R_{17}=200~ k\Omega \), \(R_{18}=10~ k\Omega \), \(R_{19}=10~ k\Omega \), \(R_{20}=13.5~ k\Omega \), \(R_{21}=13.5~ k\Omega \), \(R_{22}=13.5~ k\Omega \), \(R_{23}=13.5~ k\Omega \), \(R_{24}=13.5~ k\Omega \), \(R_{25}=1~ k\Omega \), \(R_{26}=10~ k\Omega \), and \(R_{27}=10~ k\Omega \). Simulation results using OrCAD PSpice software are shown in Figs. 7 and 8. They are compatible with the MATLAB numerical simulation results as shown in Fig. 4.

Fig. 7
figure 7

Pspice simulation results of the fractional-order chaotic system at fractional order \(q=0.95\)

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Fig. 8
figure 8

Pspice simulation results of the fractional-order chaotic system at a fractional order \(q=0.95\)

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Fig. 9
figure 9

Hardware architecture of the proposed fractional-order chaotic system

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6 FPGA implementation

A large capacitor can be used to store the system state, when we use analog integrators, which can be used to create chaotic generators. Chaos generated from analog integration is sensitive to temperature and process variations. Digital implementation does not require capacitors, but it is based on registers to store the system state with a compact size and improved performance. The hardware implementation of the proposed system is shown in Fig. 9, according to the numerical solution given in Equation (4). Three registers are used to store the numerical solution of state variables x, y, and z. To present these state variables in FPGA implementation, we can use 32-bit fixed point numbers that are divided into 8 bits for the integer part and 24 bits for the fractional part.

Fig. 10
figure 10

Hardware design of GL derivatives

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Fig. 11
figure 11

The piecewise function f(z)

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Fig. 12
figure 12

Experimental results of the proposed chaotic system for parameters \(q=0.95\), \(b=14\), \(u=3\) and \(h=0.009\) (a) \(x-y\) plane, (b) \(x-z\) plane, (c) \(y-z\) plane

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Fig. 13
figure 13

Experimental results of the proposed multi-scroll fractional-order chaotic system for parameters \(q=0.95\), \(b=14\), \(u=3\) and \(h=0.05\) (a) \(x-y\) plane, (b) \(x-z\) plane, (c) \(y-z\) plane

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Fig. 14
figure 14

Block diagram of image encryption based on the proposed fractional-order chaotic system

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Fig. 15
figure 15

Encryption stages of a color face with the fractional-order chaotic system (a) R component, (b) encrypted R component, (c) G component, (d) encrypted G component, (e) B component, (f) encrypted B component, (g) generated cancellable template

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Fig. 16
figure 16

Results on FERET dataset (a) encrypted faces with the proposed system (b) encrypted faces with the fractional-order Lorenz chaotic system [4], (c) encrypted faces with RP after homomorphic transform scheme [2], (d) encrypted faces with Gaussian RP after IFL scheme [2], (e) histograms of images in (a), (f) histograms of images in (b), (g) histograms of images in (c), (h) histograms of images in (d), (i) PDFs for the proposed system, (j) PDFs for cancellable face recognition based on fractional-order Lorenz chaotic system, (k) PDFs for cancellable face recognition based on RP after homomorphic transform [2], (l) PDFs for cancellable face recognition based on Gaussian RP after IFL scheme [2], (m) ROC curve for the proposed system, (n) ROC curve for cancellable face recognition based on fractional-order Lorenz chaotic system [4], (o) ROC curve for cancellable face recognition based on RP after homomorphic transform [2], (p) ROC curve for cancellable face recognition based on Gaussian RP after IFL scheme [2]

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Fig. 17
figure 17

Results on LFW dataset (a) encrypted faces with the proposed system (b) encrypted faces with the fractional-order Lorenz chaotic system [4], (c) encrypted faces with RP after homomorphic transform scheme [2], (d) encrypted faces with Gaussian RP after IFL scheme [2], (e) histograms of images in (a), (f) histograms of images in (b), (g) histograms of images in (c), (h) histograms of images in (d), (i) PDFs for the proposed system, (j) PDFs for cancellable face recognition based on fractional-order Lorenz chaotic system, (k) PDFs for cancellable face recognition based on RP after homomorphic transform scheme [2], (l) PDFs for cancellable face recognition based on Gaussian RP after IFL scheme [2], (m) ROC curve for the proposed system, (n) ROC curve for cancellable face recognition based on fractional-order Lorenz chaotic system [4], (o) ROC curve for cancellable face recognition based on RP after homomorphic transform scheme [2], (p) ROC curve for cancellable face recognition based on Gaussian RP after IFL scheme [2]

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Fig. 18
figure 18

Results on ORL dataset (a) encrypted faces with the proposed system (b) encrypted faces with the fractional-order Lorenz chaotic system [4], (c) encrypted faces with RP after homomorphic transform scheme [2], (d) encrypted faces with Gaussian RP after IFL scheme [2], (e) histograms of images in (a), (f) histograms of images in (b), (g) histograms of images in (c), (h) histograms of images in (d), (i) PDFs for the proposed system, (j) PDFs for cancellable face recognition based on fractional-order Lorenz chaotic system, (k) PDFs for cancellable face recognition based on RP after homomorphic transform scheme [2], (l) PDFs for cancellable face recognition based on Gaussian RP after IFL scheme [2], (m) ROC curve for the proposed system, (n) ROC curve for cancellable face recognition based on fractional-order Lorenz chaotic system [4], (o) ROC curve for cancellable face recognition based on RP after homomorphic transform scheme [2], (p) ROC curve for cancellable face recognition based on Gaussian RP after IFL scheme [2]

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The first LUT in Fig. 10 stores the values of coefficients with 22 fixed-point numbers, a 2-bit integer part and a 20-bit fractional part. The proposed design shown in Fig. 9 needs 6 multipliers, a piecewise function f(z), 7 adders, and 3 GL blocks to calculate the numerical solution for state variables x, y, and z. The GL block is implemented in Fig. 10 based on [23] to compute the summation in Equation (4). The piecewise function f(z) is utilized to calculate the non-linear function \(\sum _{k=-n}^{n} {\text {sgn}}(z+2 n)\) as shown in Fig. 11. In the proposed design, the quantity \(h^q\) is constant. It is computed with a step size h, and fractional order q. The carry of each adder is one. To perform the subtraction operations, the two’s complement is used.

7 Measurements

The digital chaotic generator of the proposed system is realized based on Verilog HDL, Xilinx ISE 14.7 and Xilinx FPGA Artix-7 XC7A100T. To validate the results, we can take the data from the RTL simulation and use MATLAB. Experimental results are shown in Figs. 12 and 13 using the same parameters described before on the oscilloscope. To display the chaotic attractor on the oscilloscope, the FPGA generates two serial outputs with 12 bits in digital format. In order to display the output waveform on the digital oscilloscope DPO 4104, the output must be converted from digital to analog using the Pmod DAC 2.

8 Application of fractional-order chaotic system

By using the same datasets shown in Fig. 14, MATLAB is used to separate the three color components of the input color biometric template. This figure illustrates the experimental ciphering outcomes in terms of RGB 227×227×3 images. In the proposed system, the XOR process is accomplished between the RGB color components and the LSBs of x, y,  and z, which are the state variables of the fractional-order multi-scroll chaotic system. So, the R component of face biometric images is XORed with the LSB of x state variable. Likewise, the LSBs of z and y state variables are XORed with the B and G components, respectively, as clarified in Fig. 15.

The evaluation of the suggested system discussed previously is presented in this section. In addition, the performance and efficiency of the suggested cancellable biometric system are compared with those of other different related schemes. Furthermore, we compare it with other different schemes from several perspectives. These comparisons are based on calculating correlation scores, AROC, encryption efficiency, and histograms of face biometrics. All obtained outcomes tested on the ORL, LFW, and FERET datasets confirm the high performance of the suggested cancellable biometric system compared to other related schemes.

The security performance of the suggested cancellable biometric system is verified by comparing its efficacy with those of the cancellable biometric approaches based on Haar wavelet fusion, Gaussian Random Projection (RP) after Intuitionistic Fuzzy Logic (IFL), Gaussian RP after homomorphic transform, and fractional-order Lorenz chaotic system. The encryption outcomes are shown in Figs. 16 (a)-(d), while the histogram outcomes of the ciphered biometrics are demonstrated in Figs. 16 (e)-(h) for the suggested system and other related schemes. It is evident from the attained distributions of the collected histograms that they are very different from those of the original biometric samples.

The probability curves of the correlation coefficients of unlicensed and licensed users are presented in Figs. 16 (i)-(l). The ROC distributions are shown in Figs. 16 (m)-(p). The achieved outcomes for the suggested and related cancellable face recognition systems demonstrate that the attained AROC values reach \(100\%\) for the suggested cancellable biometric system, which guarantees the high robustness and security of the suggested system (Figs.  17 and 18).

The estimated values of all assessment metrics used in the simulation experiments to evaluate the security performance and robustness strength of all examined cancellable biometric systems are summarized in Table 3. It is noticed from the obtained outcomes that the suggested system has high AROC values that are close to 1 and the lowest EER values. In addition, the obtained average correlation values of licensed users for the proposed system are higher than those of the related schemes.

Moreover, Table 4 offers all assessment parameters of the suggested and related cancellable biometric systems for the LFW biometric dataset. The same assessment is performed on the ORL biometric dataset. The primary difference between the three examined LFW, ORL, and FERET biometric datasets is that the ORL biometric dataset includes gray-scale biometric images. Consequently, we carry out the XOR operation on a single image component. Likewise, the findings on the ORL biometric dataset are organized in Table 5. From all presented comparisons between the proposed and related cancellable biometric systems, it is revealed that the proposed system has superior performance in terms of lower EER, lower computational cost, and higher AROC values. Moreover, a comparison of the results with those of other state-of-the-art schemes for cancellable face recognition is shown in Table 6.

Table 3 Results on LFW dataset
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Table 4 Results on FERET dataset
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Table 5 Results on ORL dataset
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Table 6 Comparison between the state-of-the-art (SOTA) methods based on AROC
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9 Conclusions

This paper presented a fractional-order chaotic system and a multi-scroll chaotic system. In our proposed multi-scroll chaotic system, the number of scrolls is controlled by the parameters instead of changing the discontinuous functions and generating a multi-scroll chaotic system. In addition, the circuit realization of this system is introduced and discussed. Moreover, the proposed system is implemented with FPGA. OrCAD-PSpice simulation results, experimental results of FPGA-based implementation of the proposed system and numerical simulation confirm each other. Cancellable face recognition based on the proposed fractional-order chaotic system has been implemented on FERET, LFW, and ORL datasets, and the results are compared with those of other schemes.