Encryption algorithm based on fractional order chaotic system combined with adaptive predefined time synchronization

An image encryption and decryption method of fractional order chaotic systems (FOCS) with predefined time synchronization is proposed in this article. Compared with the existing integer order chaotic systems (IOCS), fractional order chaotic systems has the advantage of increasing the complexity of the ciphertext. At the same time, by using the predefined synchronization time, the key space is expanded, the complexity of the key is increased, and the security of the algorithm is improved. To further improve the security of encryption and decryption process, this article uses a combination of DNA encoding, row/column cyclic shift and XOR diffusion, position scrambling and Arnold scrambling. The simulation tests of image encryption and decryption are carried out, and the effectiveness and advantages of the proposed encryption/decryption method are verified by histogram analysis, correlation analysis, entropy analysis, key sensitivity analysis and plaintext sensitivity analysis.

1 Introduction

Chaos is a kind of irregular and complex movement in nature [1]. Chaotic phenomena are seemingly random, irregular movements that occur in a deterministic system. The development of chaotic systems originated from the discovery of the Lorentz system [2]. Information encryption through chaotic signals or direct communication using chaotic systems is the future direction of chaotic applications [3]. In the past 2 decades, the algorithms combining chaotic systems with image encryption [4]; [5]; [6]; [7]; [8]; [9]; [10]; [11]; [12]; [13]; [14]; [15] have been investigated. Chaotic systems can generate extensive chaotic sequences (chaotic signal) by using different initial conditions and parameter settings [9]. Moreover, the signal is extremely sensitive to the change of system parameters and the disturbance of initial conditions [9]; [10]. Thus, chaotic systems provide a less consumption and higher speed means of implementing encryption tasks for large-scale data such as images [16]; [17]; [18], video and audio [11]; [19]. In order to combine chaotic systems with encryption algorithms, many properties of chaotic systems have been studied and developed [20]; [21]; [22]; [23]; [24]; [25]; [26]. In [20], the synchronization of two chaotic systems is studied, and the chaotic synchronization system is applied to the encryption algorithm. The synchronization technique of generalized chaotic systems, which excellently eliminates the negative effects of channel noise, are studied in [24]. The combination of analog chaotic system and digital chaotic system can effectively weaken the time delay of analog chaotic system and the periodicity of digital chaotic system, then improving the security of chaotic system [25]. To deal with the degradation of the dynamics leading to quantization, digital chaotic systems with higher dimensions than universal chaotic systems are constructed [27]. However, most of the existing encryption algorithms are based on IOCS. The description of many phenomena by IOCS is flawed. FOCS is proposed to overcome the deficiency of description ability of IOCS. Compared with the dynamic system described by integer order calculus, fractional order system contains more complex dynamic characteristics and genetic characteristics, and its memory function is nearly infinite. Classical fractional order systems include FOCS of Duffing Lu [28], FOCS of Rossler [29], FOCS of Chua [30]. Information security has always been a major issue in secure communications and related applications of information science. Achieving information security relies on reliable information encryption techniques. Research [31]; [32] has shown that using fractional order systems for information encryption has better performance. An image cryptosystem using fractional order chaotic systems is scrutinized for security applications [33]. An integral analysis framework based on a comprehensive security analysis with the aim to establish a basis in security analysis for chaos-based image cryptosystems were presented in [34].

The key to secure communication is to realize chaotic synchronization. The encryption process corresponds to the drive system and the decryption process corresponds to the response system. Under different initial conditions, the drive system and the response system often present completely different motion states, which is extremely unfavorable to the encryption algorithm. Therefore, how to design the synchronization method of chaotic systems is very important. The synchronization process of chaotic system includes asynchronous stage and synchronous stage. As mentioned before, once the chaotic signal is used in the asynchronous phase, the decryption of the final information cannot be achieved. Therefore, we need to accurately estimate the synchronization time of chaotic systems. We can predefined the synchronization time (i.e., adjusting the synchronization time) to enhance the effect of confidentiality. The definition of predefined time stability was proposed in [35], and the upper limit of its stability time can be acquired by adjusting system parameters [36]; [37]. In addition, a number synchronization methods of chaotic systems [38,39]; [37] within predefined time are proposed. In [37], an active controller with predefined time sliding mode synchronization was designed. To realize the predefined time sliding mode synchronization, a new formula was proposed in [39] and a series of controllers were designed.

To realize synchronization of the FOCS within a predefined time, the correction function of the hyperchaotic system was studied and a near sliding synchronization controller was designed in [40]. Inspired by the above work, this article proposes adaptive predefined time synchronization for time-delayed FOCS, and tests the performance of this system by applying it to image encryption/decryption. The novelties of the proposed encryption/decryption method in this article are as follows.

(1) The adjustable synchronization time process is used for generating the key, which increases the complexity of the key by expanding the key space. Compared to most previous image encryption algorithms, this one does not require that the sender transmits the key to the receiver over the channel, which may put the risk of the key being intercepted and cracked.

(2) A new encryption algorithm is carefully designed, and the scrambling and diffusion algorithm which is highly correlated with the plaintext is introduced. The encryption algorithm has a strong ability to resist attacks.

(3) To improve the security of the communication process, IOCS is replaced by FOCS in the proposed encryption/decryption method.

2 Preliminary

2.1 Definitions and lemmas

Some definitions and lemmas related to FOCS are introduced [41].

Definition 1. Arslan et al. [42] The fractional integral of h(t) is defined as

$I^{q}h\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(q\right)}{\int }_{t_0}^t{\left(t-s\right)}^{q-1}h\left(s\right)ds,$

where q > 0 is the order of integral; t₀ is the initial time and t > t₀; h(t): (0, + ∞) → R and Γ(⋅) is the generalized Euler’s Gamma function

$\mathrm{\Gamma }\left(q\right)={\int }_0^{\infty }{t}^{q-1}{e}^{-t}dt.$

Definition 2. Petráš [43]; Song et al. [44] The Caputo definition derivative for h(t) is described by

${}_{t_0}{}^{C}D_t^{\beta }h\left(t\right)=\left\{\begin{array}{c}\frac{1}{\mathrm{\Gamma }\left(n-\beta \right)}{\int }_{t_0}^t\frac{h^{\left(n\right)}\left(s\right)}{{\left(t-s\right)}^{\beta -n+1}}ds,\hfill \\ n-1<\beta <n,\hfill \\ \frac{d^{n}h\left(t\right)}{d{t}^n},\beta =n,\hfill \end{array}\right.$

where β > 0 is the order of derivative, t₀ is the initial time and t ≥ t₀; n is integer and n − 1 < β ≤ n.

Lemma 1. Song et al. [45] An equation describes the Caputo fractional derivative and Riemann–Liouville fractional derivative, that is:

${}_{t_0}D_t^{\alpha }\left({}_{t_0}D_t^{m}f\left(t\right)\right)={}_{t_0}D_t^{\alpha +m}f\left(t\right),$

where m ∈ N and n − 1 < α < n ∈ N⁺.

Lemma 2. Arslan et al. [42] If f(t) ∈ C^m[0, ∞), n − 1 < q < n ∈ N⁺, then

${}_{t_0}D_t^q{}_{t_0}D_t^{-q}f\left(t\right)=f\left(t\right).$

Lemma 3. Li et al. [46] For constant γ and δ, the linear characteristic of Caputo fractional order derivatives describes as:

${}_{t_0}D_t^{\alpha }\left[\gamma f\left(t\right)+\delta g\left(t\right)\right]={\gamma }_{t_0}{D}_t^{\alpha }f\left(t\right)+{\delta }_{t_0}{D}_t^{\alpha }g\left(t\right).$

Definition 3. Jiménez-Rodríguez et al. [47] T_c is an important parameter in the process of predefined time stability. The value of the stable time T(x₀) for fixed-time synchronization has a functional relationship with the value of T_c, that is, the stable time T_x can be set by adjusting the parameter T_c,

$T\left(x_0\right)\le T_c,\forall x_0\in {\mathit{R}}^n$

Remark 1. Obviously, when the dynamic error system achieves convergence within predefined time, predefined time synchronization of the drive-response chaotic systems can be realized. Therefore, a controller u_i(t) will be designed to force the error system to converge within the predefined time.

Lemma 4. Lin [48] If there exists a continuous radially unbounded and positive definite function V: Rⁿ → R such that satisfies

${\displaystyle \dot{V}}\le -\frac{T_{\mathit{max}}^2}{T_c}{\left(\alpha V^q+\lambda \right)}^k(1)$

where α, λ, q, k > 0, qk > 1 and

$T_{\mathit{max}}^2=\frac{{\lambda }^{\frac{1}{q}-k}}{{\alpha }^{\frac{1}{q}}q}B\left(\sigma ,\theta \right),(2)$

where B(σ, θ) is the complete beta function. Then, the formula (1) is globally predefined time stable, and the predefined time is T_c.

Lemma 5. Zuo [49] For constants η_j ∈ R⁺(j = 1, 2, …, n) and ϵ ∈ R⁺, the following inequalities is satisfied:

$\left\{\begin{array}{c}{\left({\displaystyle \sum _{j=1}^N}\left|{\eta }_j\right|\right)}^{ϵ}\le {\displaystyle \sum _{j=1}^N}{\left|{\eta }_j\right|}^{ϵ},0<ϵ\le 1\hfill \\ n^{1-ϵ}{\left({\displaystyle \sum _{j=1}^N}\left|{\eta }_j\right|\right)}^{ϵ}\le {\displaystyle \sum _{j=1}^N}{\left|{\eta }_j\right|}^{ϵ},ϵ>1.\hfill \end{array}\right.$

2.2 Secure hash algorithm (SHA-256)

Hash function [50] is a way to convert the information we want to encrypt into a short string. The hash function maps the input information into a unique string digest, which significantly reduces the space occupied by the information and unifies the information storage format. The information mapped by the hash function becomes a string digest that no longer has any characteristics of the original information. This string digest is called a hash value. Secure hash algorithm is a family of cryptographic hash functions. It is designed by the National Security Agency (NSA) and published by the National Institute of standards and Technology (NIST) and is widely used [14]; [51]. Including SHA-0 series, SHA-1 series, SHA-2 series and ShA-3 series. SHA-256 is one of SHA-2 series functions. SHA-256 has two features: 1. No matter how long the input is, it outputs 64 characters, a total of 32 bytes and 256 bits; 2. The output contains only the number 0 through 9 and the letters A through F, which are not case-sensitive.

3 Predefined time synchronization of fractional order systems with different structures

3.1 System introduction

The FOCS has more complex dynamic behaviors and mathematical explanations than the IOCS, so they have higher degrees of nonlinearity. The fractional order time delay drive system is described as [52].

$\left\{\begin{array}{cc}\hfill D^{\alpha }{x}_1& =-0.1x_1\left(t\right)+12\left(x_2\left(t\right)-x_1\left(t\right)\right)\hfill \\ \hfill & +0.1\tanh \left(x_1\left(t-0.8\right)\right)\hfill \\ \hfill D^{\alpha }{x}_2& =-0.1x_2\left(t\right)+20x_1\left(t\right)-7.3x_1\left(t\right)x_3\left(t\right)\hfill \\ \hfill & +{\mathit{x}}_4\left(t\right)+0.1\tanh \left(x_2\left(t-0.8\right)\right)\hfill \\ \hfill D^{\alpha }{x}_3& =-0.1x_3\left(t\right)+h{\left(x_1\left(t\right)\right)}^2-4.5x_3\left(t\right)\hfill \\ \hfill & +{\mathit{x}}_4\left(t\right)+0.1\tanh \left(x_3\left(t-0.8\right)\right)\hfill \\ \hfill D^{\alpha }{x}_4& =-0.1x_4\left(t\right)-4x_2\left(t\right)\hfill \\ \hfill & +0.1\tanh \left(x_4\left(t-0.8\right)\right)\hfill \end{array}\right.(3)$

where x_i(i = 1, 2, 3, 4) is the state variable, 0 < α < 1 is the order of system (3), and $\tanh \left(x_i(t-0.8)\right.$ is the delayed function. The corresponding response system is described as

$\left\{\begin{array}{cc}\hfill D^{\alpha }{y}_1& =-0.1y_1\left(t\right)+15\left(y_2\left(t\right)-y_1\left(t\right)\right)+y_4\left(t\right)\hfill \\ \hfill & +0.1\tanh \left(y_1\left(t-0.8\right)\right)+u_1\left(t\right)\hfill \\ \hfill D^{\alpha }{y}_2& =-0.1y_2\left(t\right)+40y_1\left(t\right)-y_2\left(t\right)-y_1\left(t\right)y_3\left(t\right)\hfill \\ \hfill & +0.1\tanh \left(y_2\left(t-0.8\right)\right)+u_2\left(t\right)\hfill \\ \hfill D^{\alpha }{y}_3& =-0.1y_3\left(t\right)+y_1\left(t\right)y_2\left(t\right)-2y_3\left(t\right)\hfill \\ \hfill & +0.1\tanh \left(y_3\left(t-0.8\right)\right)+u_3\left(t\right)\hfill \\ \hfill D^{\alpha }{y}_4& =-0.1y_4\left(t\right)-y_2\left(t\right)y_3\left(t\right)-y_4\left(t\right)\hfill \\ \hfill & +0.1\tanh \left(y_4\left(t-0.8\right)\right)+u_4\left(t\right)\hfill \end{array}\right.(4)$

where y_i(i = 1, 2, 3, 4) is the state variable and $\tanh \left(y_i(t-0.8)\right.$ is the delayed function. The chaotic behaviours of (3) and (4) are shown in Figures 1A, B. Since predefined time synchronization of two FOCSs can improve communication security, we designed a scheme for RGB image encryption/decryption.

FIGURE 1

FIGURE 1. Chaotic images of driving and response system: (A) driving system (3); (B) response system (4) without u_i(t).

Theorem 1. The response fractional order delay system (4) can achieve predefined time synchronization with the drive fractional order delay system (3) via the following control law:

$\left\{\begin{array}{cc}\hfill u_1\left(t\right)& =0.1e_1\left(t\right)-15\left(y_2\left(t\right)-y_1\left(t\right)\right)-y_4\left(t\right)+12\left(x_2\left(t\right)\right.\hfill \\ \hfill & \left.-{\mathit{x}}_1\left(t\right)\right)-D^{\alpha -1}\left[\frac{2^{k-1}}{N^{1-qk}}{\alpha }_1\mathrm{sign}\left(e_1\left(t\right)\right)\cdot \right.\hfill \\ \hfill & \left.{\left|e_1\left(t\right)\right|}^{qk}+2^{k-1}{\lambda }_1\mathrm{sign}\left(e_1\left(t\right)\right)\right]-\mathrm{sign}\left(e_1\left(t\right)\right)L\hfill \\ \hfill u_2\left(t\right)& =0.1e_2\left(t\right)-40y_1\left(t\right)+y_2\left(t\right)+y_1\left(t\right)y_3\left(t\right)+20x_1\left(t\right)\hfill \\ \hfill & -7.3x_1\left(t\right)x_3\left(t\right)+x_4\left(t\right)-D^{\alpha -1}\left[\frac{2^{k-1}}{N^{1-qk}}{\alpha }_1\cdot \right.\hfill \\ \hfill & \left.\mathrm{sign}\left(e_2\left(t\right)\right){\left|e_2\left(t\right)\right|}^{qk}+2^{k-1}{\lambda }_1\mathrm{sign}\left(e_2\left(t\right)\right)\right]\hfill \\ \hfill & -\mathrm{sign}\left(e_2\left(t\right)\right)L\hfill \\ \hfill u_3\left(t\right)& =0.1e_3\left(t\right)-y_1\left(t\right)y_2\left(t\right)+2y_3\left(t\right)+2{\left(x_1\left(t\right)\right)}^2\hfill \\ \hfill & -4.5x_3\left(t\right)+x_4\left(t\right)-D^{\alpha -1}\left[\frac{2^{k-1}}{N^{1-qk}}{\alpha }_1\mathrm{sign}\left(e_3\left(t\right)\right)\cdot \right.\hfill \\ \hfill & \left.{\left|e_3\left(t\right)\right|}^{qk}+2^{k-1}{\lambda }_1\mathrm{sign}\left(e_3\left(t\right)\right)\right]-\mathrm{sign}\left(e_3\left(t\right)\right)L\hfill \\ \hfill u_4\left(t\right)& =0.1e_4\left(t\right)+y_2\left(t\right)y_3\left(t\right)+y_4\left(t\right)-4x_2\left(t\right)\hfill \\ \hfill & -{\mathit{D}}^{\alpha -1}\left[\frac{2^{k-1}}{N^{1-qk}}{\alpha }_1\mathrm{sign}\left(e_4\left(t\right)\right){\left|e_4\left(t\right)\right|}^{qk}+2^{k-1}\cdot \right.\hfill \\ \hfill & \left.{\lambda }_1\mathrm{sign}\left(e_4\left(t\right)\right)\right]-\mathrm{sign}\left(e_4\left(t\right)\right)L\hfill \end{array}\right.(5)$

where α₁, λ₁, q and k meet the requirements of Lemma 4 and L ≥ 0.2.

Proof. According to the definition of errors, e_i(t) is described as

$e_i\left(t\right)=y_i\left(t\right)-x_i\left(t\right),i=\mathrm{1,2},\dots ,n.(6)$

Supposing that $E(t)={\left(e_1(t),e_2(t),\dots ,e_n(t)\right)}^{\mathrm{T}}\in R^n$ is the state vector of error system. Obviously, E(0) = Y(0) − X(0) is the initial error value. System (6) is described as

${}_{t_0}D_t^{\alpha }{e}_i\left(t\right)={}_{t}D_t^{\alpha }{y}_i\left(t\right)-{}_{t}D_t^{\alpha }{x}_i\left(t\right),(7)$

Substituted formula (3) and (4) into (6), one can obtain that:

$D^{\alpha }{e}_i\left(t\right)=-{\mathit{\sigma }}_i{e}_i\left(t\right)+h_i\left(e_i\left(t\right)\right)+H_i\left(e_i\left(t-\tau \right)\right)+u_i\left(t\right)(8)$

Continuous function ‖e(t)‖₁ is chosen as Lyapunov function

$V_2\left(t\right)=\Vert e\left(t\right){\Vert }_1={\displaystyle \sum _{i=1}^n}\left|e_i\right|$

Taking the derivative of V₂(t), one can obtain that

$\begin{array}{c}{{\displaystyle \dot{V}}}_2\left(t\right)={\displaystyle \sum _{i=1}^N}{{\displaystyle \dot{e}}}_i\left(t\right)\mathrm{sign}\left(e_i\left(t\right)\right)\hfill \\ ={\displaystyle \sum _{i=1}^N}\left(D^{1-\alpha }\left(D^{\alpha }{e}_i\left(t\right)\right)\right)\mathrm{sign}\left(e_i\left(t\right)\right)\hfill \\ ={\displaystyle \sum _{i=1}^N}\mathrm{sign}\left(e_i\left(t\right)\right)\left(D^{1-\alpha }\left(-\sigma e_i\left(t\right)+h_i\left(e_i\left(t\right)\right)\right.\right.\hfill \\ \left.\left.+H_i\left(e_i\left(t-\tau \right)\right)+{\mathit{u}}_i\left(t\right)\right)\right)\hfill \\ ={\displaystyle \sum _{i=1}^N}\mathrm{sign}\left(e_i\left(t\right)\right)\left(D^{1-\alpha }\left(H_i\left(e_i\left(t-\tau \right)\right)-L\mathrm{sign}\left(e_i\left(t\right)\right)\right.\right.\hfill \\ \left.\left.-\left(\frac{T_{\mathit{max}}^2}{T_c}\cdot \left({\alpha }_2\mathrm{sign}\left(e_i\left(t\right)\right){\left|e_i\left(t\right)\right|}^{qk}+{\lambda }_2\mathrm{sign}\left(e_i\left(t\right)\right)\right)\right)\right)\right)\hfill \\ \le -\frac{T_{\mathit{max}}^2}{T_c}{\displaystyle \sum _{i=1}^N}\mathrm{sign}\left(e_i\left(t\right)\right)\left(D^{1-\alpha }\left(D^{\alpha -1}\left({\alpha }_2\mathrm{sign}\left(e_i\left(t\right)\right){\left|e_i\left(t\right)\right|}^{qk}\right.\right.\right.\hfill \\ \left.\left.\left.+{\mathit{\lambda }}_2\mathrm{sign}\left(e_i\left(t\right)\right)\right)\right)\right)\hfill \\ =-\frac{T_{\mathit{max}}^2}{T_c}{\displaystyle \sum _{i=1}^N}\mathrm{sign}\left(e_i\left(t\right)\right)\left({\alpha }_2\mathrm{sign}\left(e_i\left(t\right)\right){\left|e_i\left(t\right)\right|}^{qk}+{\mathit{\lambda }}_2\mathrm{sign}\left(e_i\left(t\right)\right)\right)\hfill \\ =-\frac{T_{\mathit{max}}^2}{T_c}{\displaystyle \sum _{i=1}^N}\left({\alpha }_2{\left|e_i\left(t\right)\right|}^{qk}+{\lambda }_2\right)\hfill \\ =-\frac{T_{\mathit{max}}^2}{T_c}\left(N{\lambda }_2+{\displaystyle \sum _{i=1}^N}\left({\alpha }_2{\left|e_i\left(t\right)\right|}^{qk}\right)\right).\hfill \end{array}$

According to Lemma 5, the following inequality holds

${\displaystyle \sum _{i=1}^N}\left({\alpha }_2{\left|e_i\left(t\right)\right|}^{qk}\right)\ge N^{1-qk}{\left({\displaystyle \sum _{i=1}^N}{\alpha }_2\left|e_i\left(t\right)\right|\right)}^{qk}.$

Substitute the above formula into ${{\displaystyle \dot{V}}}_2(t)$, one can obtain that

$\begin{array}{c}{{\displaystyle \dot{V}}}_2\left(t\right)\le -\frac{T_{\mathit{max}}^2}{T_c}\left(N^{1-qk}{\alpha }_2{\left({\displaystyle \sum _{i=1}^N}\left|e_i\left(t\right)\right|\right)}^{qk}+N{\lambda }_2\right)\hfill \\ =-\frac{T_{\mathit{max}}^2}{T_c}\left(N^{1-qk}{\alpha }_2{V}_2^{qk}+N{\lambda }_2\right).\hfill \end{array}(9)$

Since N^1−qk > 1 and N > 1. formula (9) has

${{\displaystyle \dot{V}}}_2\left(t\right)\le -\frac{T_{\mathit{max}}^2}{T_c}\left({\beta }_2{V}_2^{qk}+{\lambda }_2\right)$

According to Lemma 5, the following inequality holds

${\left({\alpha }_2^{\frac{1}{k}}{V}_2^q\right)}^k+{\left({\lambda }_2^{\frac{1}{k}}\right)}^k\ge {\left({\alpha }_2^{\frac{1}{k}}{V}_2^q+{\lambda }_2^{\frac{1}{k}}\right)}^k(10)$

Then substitute (10) into ${{\displaystyle \dot{V}}}_2(t)$, one can obtain that

${{\displaystyle \dot{V}}}_2\left(t\right)\le -\frac{T_{\mathit{max}}^2}{T_c}{\left({\alpha }_2^{\frac{1}{k}}{V}_2^q+{\lambda }_2^{\frac{1}{k}}\right)}^k(11)$

where α₂ and λ₂ are positive parameters. If ${\displaystyle \tilde{\alpha }}={\alpha }_2^{\frac{1}{k}}$ and ${\displaystyle \tilde{b}}={\lambda }_2^{\frac{1}{k}}$, the formula (11) can be simplified as

${{\displaystyle \dot{V}}}_2\left(t\right)\le -\frac{T_{\mathit{max}}^2}{T_c}{\left({\displaystyle \tilde{\alpha }}{V}_2^q+{\displaystyle \tilde{b}}\right)}^k.$

It can be obtained by combining Lemma 4 that

$T_{\mathit{max}}^2=\frac{{{\displaystyle \tilde{b}}}^{\frac{1}{q}-k}}{{{\displaystyle \tilde{\alpha }}}^{\frac{1}{q}}q}B\left(\sigma ,\theta \right).$

Based on the above discussion, for any t ≥ T_c, there are V₂(t) = 0. That is, the system can achieve synchronization within a predefined time T_c.

Remark 2. The complexity of the FOCS can be fully utilized to increase the complexity of the encrypted text and improve the security of the encryption/decryption algorithm. The predefined time synchronization of the FOCSs is used to generate the decrypted text. By using the predefined time as the key, the key space is expanded and the complexity of the key is increased.

Remark 3. Here, the delay function is designed as $\tanh \left(x_i(t-0.8)\right)$ or $\tanh \left(y_i(t-0.8)\right)$, where $|\tanh \left(x_i(t-0.8)\right)|\le 1$ and $|\tanh \left(y_i(t-0.8)\right)|\le 1$. It can be offset by ${\displaystyle \tilde{b}}$. Therefore, the controller input (5) can be a non-delayed system.

3.2 Synchronization performance analysis

The variation of state variables for systems (3) and (4) with and without controller input (5) are shown in Figures 2A, B; Figure 2A shows the time response curves of the state variables of the drive system and the response system without controller input (5). At this time, the change amplitude of the state variable y_i is obviously greater than that of x_i, and there is no relationship between them. Figure 2B shows the time response of x_i and y_i under the controller input (5). Among the four state variables, the state variable y_i of the response system (4) quickly approaches the state variable x_i of the drive system (3) from a larger fluctuation amplitude, thereby showing good tracking performance of y_i on x_i. The parameters of systems (3) and (4) are completely the same, but due to the change of the initial values of the system, the state variables in Figure 2A are significantly different from those in Figure 1, especially in response to the change amplitude of y_i in the system. This also reflects the sensitivity of the proposed FOCSs (3) and (4) to the initial values. To illustrate the validity of the predefined time synchronization method, the controller u_i(t) (i = 1,2,3,4) is designed with setting parameter T_c = 1 and T_c = 1.5, α = 0.82, k = 5.2, λ₂ = 11, q = 12.9, α₂ = 0.5, L = 0.2. The simulation performances are shown in Figure 2C with predefined time T_c = 1s; Figure 2B with predefined time T_c = 1.5s. When T_c = 1 (Figure 2C), the errors of drive system (3) and response system (4) gradually tends to 0 in the synchronization time period 0–1s. Whereas, when T_c = 1.5 (Figure 2D) the errors converge to 0 in the synchronization time period 0–1.5s. The results show that, by setting the controller and variables x_i and y_i, the realized predefined time synchronization system is effective under different T_c conditions.

FIGURE 2

FIGURE 2. State behaviors and synchronization error: (A) without controller input; (B) with controller input; (C) with q = 0.5, T_c = 1; (D) with q = 0.5, T_c = 1.5.

3.3 Randomness performance analysis

To further illustrate the proposed chaotic system is suitable for image encryption algorithm, the National Institute of Standards and Technology (NIST) SP800-22 is adopted to test the randomness of the output sequences. The NIST SP800-22 has 15 sub-tests and a p-value may be generated by every sub-test. According to the suggestion of [53], 100 binary streams with 1000000 bits are used as input data and the corresponding p-value is expected to fall into the range of 0.01 and 1 to pass the test. Table 1 lists the test results. It is clear that the binary streams of the four-wing FOCS can pass all the sub-tests, which indicates that FOCS is very suitable for image encryption.

TABLE 1

TABLE 1. NIST SP800-22 test results.

4 Encryption/decryption algorithm and process

The encryption algorithm designed in this article includes four steps: chaotic sequence generation, image DNA coding, image scrambling, diffusion encryption and Arnold scrambling. In order to achieve the effect of diffusion, at least two rounds of encryption are required.

4.1 Generating initial value of chaotic system and chaotic sequence

In the key generating step, SHA-256 hash function is used for generating the initial key. The image information represented as matrix in plaintext form is input into SHA-256 function to form a string of 64-bit hexadecimal summaries with a length of 256 bits. The 64-bit hexadecimal digest is firstly converted to decimal then divided into 8 sequences: $H=\left\{H_1,H_2,H_3,H_4,H_5,H_6,H_7,H_8\right\}$. The numbers in H are XORed to get 4 numbers: $x=\left\{x_1,x_2,x_3,x_4\right\}$. The four numbers in x are used as the initial values of drive system to generate chaotic sequences X.

4.2 DNA coding encryption rules

The aim of DNA coding is to scramble and diffuse the information of the origin image and generate a new image. DNA coding follows the principle of base complementary pairing, which is composed of Adenine (A), Thymine (T), Cytosine (C) and Guanine (G). The principle is similar to the complementary principle of binary numbers. For binary numbers, 00 and 11 are complementary, 01 and 10 are complementary. While, A and T are complementary and G and C are complementary. Thus, there are 24 coding combinations, but only eight meet the principle of base complementary pairing, by using four bases A, T, C and G to encode 00, 01, 10 and 11, which are listed in Table 2. According to DNA coding rules, image pixels value can be encoded. For example, if the binary number of a pixel value is [01100100] (decimal nuber 100), it can be encoded as [CGCA] according to DNA coding rule 1 from Table 2. Whereas, DNA decoding is the opposite process of DNA coding, which can recover the pixel value from the DNA sequence. If the DNA sequence is [CGCA], it can be decoded as binary number [01100100] (decimal number 100) according to rule 1 or as binary number [10011000] (decimal number 152) according to rule 2. In view of the above example, pixel value of original image can be changed by firstly encoding its binary sequence with rule R₁ and then decoding the output DNA sequence with rule R₂ to convert the original image pixel value and hide the original image information, where the rule R₂ and R₂ are two different coding rules in Table 2. The details of the above process are as follows:

TABLE 2

TABLE 2. DNA coding rules.

Step 1. get the pixel value from the original image, assuming the pixel value is 100;

Step 2. convert pixel values into binary sequences and use the coding rule R₁ to encode the pixel value. For example, the binary expression of pixel value 100 is [01100100], if R₁ = 3, then the output DNA code is [ATAC];

Step 3. using coding rule R₂ to decode the encoded pixel value to obtain a new pixel value. For example, it is encoded as [ATAC], when R₂ = 5, the decoded binary sequence is [10011000], which is 152 in decimal nubler. That is, the original pixel value 100 is converted to a new value 152.Determination of Rule R₁,R₂: randomly take M × N × 3 × 2 groups of data in chaotic sequence X to obtain chaotic sequence: $S_1=\left\{X_1,Y_1,Z_1\right\},S_2=\left\{X_2,Y_2,Z_2\right\}$

$\left\{\begin{array}{c}{\mathrm{R}}_1=\mathrm{mod}\left(\left[S_1\left(1:M\right)\times 10^8,8\right]\right)\hfill \\ {\mathrm{R}}_2=\mathrm{mod}\left(\left[S_2\left(1:M\right)\times 10^8,8\right]\right)\hfill \end{array}\right.(12)$

Thus, a new image hiding the original image information is obtained, and the new image is scrambled and diffused.

4.3 Scrambling based on circular shifting

Scrambling based on circular shifting is aim to disrupt the order of the DNA-encoding image information and change it into a cluttered image. Scrambling based on circular shifting only operates on the orders of columns (M) and rows(N) of the image matrix. As the output new image in subsection 4.2 consists of three channels: R,G,B, so M + 3N groups of data in chaotic sequence x are arbitrarily taken to obtain chaotic sequence: $S_3=\left\{X_3,Y_3,Z_3\right\}$. Perform the following operations for each sequence in S₁:

$\left\{\begin{array}{c}T=\mathrm{mod}\left(\left[a\left(1:M\right)\times 10^7,3N\right]\right)\hfill \\ W=\mathrm{mod}\left(\left[a\left(M+1:M+3N\right)\times 10^7,M\right]\right)\hfill \end{array}\right.(13)$

where a is from the sequence in S₁. The sequence $T=\left\{T_x,T_y,T_z\right\}$, $W=\left\{W_x,W_y,W_z\right\}$ is obtained after calculation. The sequence for scrambling is obtained according to the following formula:

$\left\{\begin{array}{c}{S}_r=T_x\oplus T_y\oplus T_z\hfill \\ S_c=W_x\oplus W_y\oplus W_z\hfill \end{array}\right.(14)$

where S_r is the sequence required for row cyclic shift and S_c is the sequence required for column cyclic shift. Then, the image matrix is scrambled with the results S_r, S_c as follows:

$\left\{\begin{array}{c}{I}_p\left(i,:\right)=\mathrm{C}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}\left(I_m\left(i,:\right),S_r\left(i\right)\right)\hfill \\ I_p\left(:,j\right)=\mathrm{C}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}\left(I_m\left(:,j\right),S_c\left(j\right)\right)\hfill \end{array}\right.(15)$

where i = 1, 2, …, m; j = 1, 2, …, 3N, the function circshift (a, b) means to circularly shift matrix a by b bits. Finally, the scrambled matrix I_p is obtained.

4.4 Diffusion algorithm

The pixels in each channel of color need to be diffused. The aim is to hide image information. Randomly take M × N groups of data from chaotic sequence X and get the sequence: $S_4=\left\{X_4,Y_4,Z_4\right\}$. Then merge the obtained sequence S₄ into M × 3N matrix, and process S_d according to the following

$S_d=\mathrm{mod}\left(\left.⌊S_d\times 10^7\right],256\right)(16)$

Then conduct diffusion operation according to the following

$\left\{\begin{array}{c}{I}_c\left(1,:\right)=I_p\left(1,:\right)\oplus I_p\left(M,:\right)\oplus S_d\left(1,:\right)\hfill \\ {\mathrm{I}}_c\left(i,:\right)=I_p\left(i,:\right)\oplus I_c\left(i-1,:\right)\oplus S_d\left(i,:\right)\hfill \\ i=\mathrm{2,3},\dots ,M\hfill \end{array}\right.(17)$

$\left\{\begin{array}{c}{I}_c\left(:,1\right)=I_p\left(:,1\right)\oplus I_p\left(:,3N\right)\oplus S_d\left(1,:\right)\hfill \\ {\mathrm{I}}_c\left(:,j\right)=I_p\left(:,j\right)\oplus I_c\left(:,j-1\right)\oplus S_d\left(:,j\right)\hfill \\ j=\mathrm{2,3},\dots ,3N\hfill \end{array}\right.(18)$

Matrix I_c is the output diffused matrix. After Arnold replacement of I_c, $I_c^{\prime }$ is obtained. Then $I_c^{\prime }$ is reconstituted into matrix, I_e of M × N × 3. After two rounds of encryption, the image converted by Matrix, I_e is the encrypted output image.

The proposed encryption algorithm is a symmetric password, that is, the inverse of the encryption process is the decryption process. The encrypted image can be obtained by reverse Arnold scrambling, reverse diffusion, reverse cyclic scrambling and reverse DNA decoding. Because the chaotic system proposed in this article is completely synchronization, the chaotic sequence used in the decryption system is completely consistent with that used in the encryption system. The decryption performance of the algorithm will be given in the numerical simulation in Section 5.

4.5 Encryption and decryption transmission system

The proposed encryption/decryption system consists of three components. They are the transmitter, the receiver and the transmission channel, where the transmitter consists of N encryptors and the receiver consists of N decryptors. Chaotic sequences can be generated by N predefined time chaotic driving systems in the encryptor, and then chaotic sequences can be generated by the synchronization system in the decryptor. Let p(t) represent the plaintext encrypted by the hash function to be transmitted. The block diagram of the system is shown in Figure 3.

FIGURE 3

FIGURE 3. Block diagram of encryption and decryption transmission system.

5 Simulation and performance analysis

5.1 Performance analysis of encryption and decryption

To evaluate the performance of the proposed encryption/decryption algorithm, simulation experiments are conducted. Firstly, the encryption algorithm given in Section 4 is applied to encrypt the original images. Figure 4 shows the original plain-images. The original images are Lena (256 × 256) (from Figure 5A in [54]), Panda (256 × 256) (from Figure 5C in [54]), All white (512 × 512) and All black (512 × 512). After the plaintext images are encrypted, the ciphertext images as shown in Figure 5 is formed. The ciphertext images present a noise state and cannot recognize the original image information. Then, the decryption algorithm given in Section 4 is applied to the encrypted image. The decrypted images are shown in Figure 6. Additionally, the proposed encryption has no limit for image size, and it can be applied for color images of various sizes. To accurately evaluate the consistency of the decrypted images with the original images, an image-subtraction is performed with the decrypted image and the original image. As shown in Figure 7, some images with all pixel values of 0 (a black image) are obtained, which proves that the decrypted images are completely consistent with the original images, so the proposed encryption and decryption algorithm is valid and can achieve a lossless encryption and decryption result.

FIGURE 4

FIGURE 4. The original images: (A) Lena plain-image; (B) Panda plain-image; (C) All black plain-image; (D) All white plain-image. Reproduced from “A Color Image Encryption Algorithm Based on One Time Key, Chaos Theory, and Concept of Rotor Machine” by Aqeel Ur Rehman, Amnah Firdous, Salman Iqbal, Zahid Abbas, Malik M. Ali Shahid, Huiwei Wang, and Farman Ullah, licensed under [https://creativecommons.org/licenses/by/4.0/], [54]”.

FIGURE 5

FIGURE 5. The cipher-images: (A) Lena cipher-image; (B) Panda cipher-image; (C) All black cipher-image; (D) All white cipher-image.

FIGURE 6

FIGURE 6. The decrypted images: (A) Decrypted image of Lena; (B) Decrypted image of Panda; (C) Decrypted image of All black; (D) Decrypted image of All white.

FIGURE 7

FIGURE 7. The difference images: (A) Difference image of Lena; (B) Difference image of Panda; (C) Difference image of all black; (D) Difference image of All white.

5.2 Key space analysis

The key space is an important indicator to measure the security performance of an encryption system. The larger the key space, the more difficult it is to brute force. In this algorithm, the key space is composed of the proposed chaotic system. The eight initial values $x=\left\{x_1,x_2,x_3,x_4\right\},y=\left\{y_1,y_2,y_3,y_4\right\}$ and predefined time T_c can constitute the system keys. The calculation accuracy of the proposed algorithm is 15, even if only one iteration is performed, the key space can reach 10¹⁵ × 10¹⁵ × 10¹⁵ × 10¹⁵ × 10¹⁵ × 10¹⁵ × 10¹⁵ × 10¹⁵ × 10¹⁵ = 10¹³⁵. According to [55], when the key space is larger than 2²⁰⁰, the cryptosystem can theoretically resist the brute-force attack. Therefore, it can well resist brute force attacks.

5.3 Histogram analysis

The image histogram describes the statistical characteristics of each gray level of the image and counts the number or frequency of each gray level in the image, so intruders can attack encrypted images through histogram analysis even the original image information is hidden in a encrypted image. For a reliable encryption strategy, the histogram of the encrypted image should be smooth and uniform thus no characteristics of original gray-level distribution is reflected. Figures 8, 9 indicates the histogram of the original image and the encrypted image respectively. It can be seen intuitively that the histogram of the plaintext image fluctuates obviously, while the histogram of the ciphertext is evenly and stably distributed. That is to say, through applying the proposed encryption method, the gray value in each channel is close to the uniform distribution, which effectively hides the original plaintext information, and makes it difficult for the attackers to crack the ciphertext image through statistical analysis. Figure 10 indicates the histogram of the decrypted image. It has exactly the same statistical features as the original image histogram, which indicates that the proposed algorithm has excellent decryption.

FIGURE 8

FIGURE 8. Original image histogram.

FIGURE 9

FIGURE 9. Encrypted image histogram.

FIGURE 10

FIGURE 10. Decrypted image histogram.

5.4 Correlation analysis

Data correlation can reflect the characteristics of the original data in terms of organization structure, and an attacker can analyze the data correlation to parse out the original data and cause data leakage. And a reliable encryption algorithm should hide the correlation reflected by the original data. Therefore, data correlation can be used as an indicator to evaluate encryption algorithms. The correlation coefficients of two data sets X and Y with length n are calculated according to equation:

$r_{xy}=\frac{\mathrm{E}\left(\left(\mathrm{X}-\mathrm{E}\left(\mathrm{X}\right)\right)\left(\mathrm{Y}-\mathrm{E}\left(\mathrm{Y}\right)\right)\right)}{\sqrt{\mathrm{D}\left(\mathrm{X}\right)\mathrm{D}\left(\mathrm{Y}\right)}}(19)$

$\mathrm{E}\left(\mathrm{X}\right)=\frac{1}{n}{\displaystyle \sum _{i=1}^n}{x}_i(20)$

$\mathrm{D}\left(\mathrm{X}\right)=\frac{1}{n}{\displaystyle \sum _{i=1}^n}{\left(x_i-\mathrm{E}\left(\mathrm{X}\right)\right)}^2(21)$

In the simulation, 8,000 couples of adjacent pixels in the horizontal, vertical and diagonal directions of plaintext image and ciphertext image are randomly selected, and the correlation coefficient of the adjacent pixel values is calculated, as shown in Tables 3, 4. For all three directions, the ((absolute value of)) adjacent pixels correlation of the original image is higher (which are greater than 0.9) than that of the encrypted image (which approximate to 0). For a more intuitive description, the adjacent pixels correlation of original image and encrypted image in horizontal direction is shown in Figures 11A, B. The scattered points of original image are concentrated on a straight line, which is close to linear correlation, while the scattered points of encrypted image are evenly distributed without obvious aggregation. The above results indicates that the correlation of adjacent pixels in ciphertext image is reduced to a lower level by applying the proposed encryption algorithm, which proofs the encryption effectiveness of the propose method. Figure 11C shows the correlation of the decrypted image, and its statistical features are consistent with the statistical features of the original image correlation, which indicates that the proposed algorithm has excellent decryption results.

TABLE 3

TABLE 3. Correlation coefficient of adjacent pixels of the original image.

TABLE 4

TABLE 4. Correlation coefficient of adjacent pixels of the encrypted image.

FIGURE 11

FIGURE 11. Correlation scatter of adjacent pixels: (A) the original image; (B) the encrypted image; (C) the decrypted image.

Table 5 compares the correlation between the original Lena image and its encrypted version generated using different encryption algorithms (Alawida et al. [51], Gao et al. [56], Patel et al. [57], Yaghouti Niyat and Moattar [58]). The last row of Table 5 shows the correlation coefficient values for this paper. It is clear that the correlation values of the Lena image cipher implemented using the algorithm of this paper are close to the advanced level in the field in the horizontal, vertical and diagonal directions.

TABLE 5

TABLE 5. Lena image correlation coefficient comparison.

5.5 Information entropy analysis

Information entropy indicates the degree of chaos of a chaotic system. The higher the information entropy, the more chaotic the system. The information entropy H of the image with gray level L is:

$H\left(\mathrm{s}\right)=-{\displaystyle \sum _{i=0}^{L-1}}P\left({\mathrm{s}}_i\right){\log }_{2}P\left({\mathrm{s}}_i\right)(22)$

where P(s_i) is the probability that the pixel belongs to gray level s_i.

The maximum value of information entropy of a 256 gray level image is 8. Tables 6, 7 respectively shows the information entropy in each channel of original image and encrypted image. It can be seen that the information entropy of original image is lower than that of encrypted image, which shows that the proposed encryption algorithm can improve the complexity of image and increase the difficulty of cracking.

TABLE 6

TABLE 6. Information entropy of original image.

TABLE 7

TABLE 7. Information entropy of encrypted image.

5.6 Plaintext sensitivity and key sensitivity analysis

Differential attack is a means for the attacker to decipher the encrypted messages, which obtains the original information (image) by comparing and analyzing the propagation of plaintext with specific differences after encryption. A secure image encryption algorithm should be highly sensitive to changes in pixels of ordinary images, which will change the encrypted image. Plaintext sensitivity and key sensitivity indicate the performance of an encryption algorithm against differential attack, whereas these two sensitivities can be evaluated with two indexes which are pixel number change rate (NPCR) and unified average change intensity (UACI). For each channel of a color image, NPCR and UACI values can be calculated by the following formula

$\mathrm{N}\mathrm{P}\mathrm{C}\mathrm{R}\left(P_1,P_2\right)=\frac{1}{MN}{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^N}\left|\mathrm{S}\mathrm{i}\mathrm{g}\mathrm{n}\left(P_1\left(i,j\right)-P_2\left(i,j\right)\right)\right|(23)$

$\mathrm{U}\mathrm{A}\mathrm{C}\mathrm{I}\left(P_1,P_2\right)=\frac{1}{MN}{\displaystyle \sum _i^M}{\displaystyle \sum _j^N}\frac{\left|P_1\left(i,j\right)-P_2\left(i,j\right)\right|}{255-0}(24)$

As for plaintext sensitivity analysis, it refers to encrypt two plaintext images with slight difference by image encryption system with the same key, then compare the differences between the two encrypted images. If the two ciphertext images are very different, the image encryption system has strong plaintext sensitivity. On the contrary, the image encryption system has weak plaintext sensitivity, which can not resist selective plaintext attack or known plaintext attack.

In the simulation, SHA-256 hash function is used to generate the initial key sequence associated with plaintext, which is highly sensitive to the input image data. The original plaintext image and a changed plaintext image is used as inputs for encryption system. The changed plaintext image is generated by randomly changing one pixel value of the original plaintext image. Through the calculation of Equations 19, 20, the NPCR and UACI (%) of two encrypted images are listed in Table 8.

TABLE 8

TABLE 8. NPCR and UACI between two encrypted images after original image change (%).

The theoretical value of NPCR is 99.6094% and the theoretical value of UACI is 33.4635%. The result of the proposed encryption algorithm approximates to the theoretical value. Table 9 provides the comparison of the NPCR and UACI results on the Lena image using our proposed algorithm and some other schemes. As can be clearly seen, [54] and our algorithm have the closest results to the ideal values of NPCR and UACI. The results indicate that the algorithm has excellent performance against differential attack.

TABLE 9

TABLE 9. Comparison of the average NPCR and UACI (%) on the Lena image by different Encryption algorithm.

As for Key sensitivity analysis, it refers to compute NPCR and UACI of the two ciphertext images after the key is slightly changed. As for plaintext sensitivity analysis, it refers to compute NPCR and UACI of the two ciphertext images after the plaintext image is slightly changed. When NPCR and UACI are close to the theoretical value, it indicates that the encryption method has a strong key sensitivity. Through the calculation of Eqs. 19, 20, the NPCR and UACI of two encrypted images are listed in Table 10.

TABLE 10

TABLE 10. NPCR and UACI between two encrypted images after key change (%).

It can be seen that the experimental values of NPCR and UACI are close to the theoretical values, which proves that the algorithm has strong key sensitivity.

5.7 Application of the algorithm

QR code as a technology to store and identify information has been widely used in many fields. However, QR code encoding algorithm is public and does not implement information encryption, so there are information security problems in some fields. In order to evaluate the performance of the proposed encryption and decryption algorithm, this section applies the encryption algorithm to the QR code image encryption scheme. Firstly, a QR code containing information about the title of this paper (Encryption algorithm based on fractional order chaotic system combined with adaptive predefined time synchronization) is generated as shown in Figure 12A. Then, the QR code is encrypted to get a noisy image that does not contain any information of the original QR code as shown in Figure 12B. Finally, the encrypted QR code is decrypted, and the decrypted image is obtained as shown in Figure 12C. After scanning and recognition, the identical text information is obtained. Therefore, the encryption algorithm can be applied to the process of QR code encryption, and at the same time, it can enhance the security of the process of using QR code.

FIGURE 12

FIGURE 12. QR code: (A) original; (B) encrypted; (C) decrypted.

6 Conclusion

In this article, a new encryption/decription method based on fractional order time-delay chaotic systems is proposed. Compared with the existing method based on IOCS, the proposed encryption method expands the key space and, furthermore, adopts cyclic shift, position disorder and row column XOR diffusion thus obtains higher security. Simulation results show that the proposed algorithm can achieve lossless encryption/decryption for image information. In addition, the proposed method possesses the advantages such as, low histogram distribution, low correlation of adjacent pixels, strong plaintext sensitivity and strong key sensitivity. These characteristics make the proposed method resistant to various attacks such as statistical attack, differential attack. In future research, we will make efforts in the following aspects.

(1) More attention will be paid to the characteristics of fractional order chaotic systems, including bifurcation diagrams and the Lyapunov spectrum [53].

(2) Different attack factors are added, such as noise, cropping, occlusion and so on, and the influence of these factors on the image encryption/decryption process is analyzed in depth.

(3) We will implement the proposed systems and predefined time controller in digital hardware such as FPGAs, DSP, ARMs.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author contributions

The authors declare that the study was realized in collaboration with the same responsibility. Writing—review and editing, LL, YZ, ZX, DY, and DW. All authors contributed to the article and approved the submitted version.

Funding

This research was funded by the Startup Research Project of Jimei University (No. ZQ2022002), the Natural Science Foundation of Fujian Province, China (No. 2019J05024), Xiamen Ocean and Fishery Development Special Fund Project (No. 21CZB013HJ15), Xiamen Key Laboratory of Marine Intelligent Terminal R&D and Application (No. B18208), Jimei University (No. ZP2022042).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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