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Dynamic analysis of an improper fractional-order laser chaotic system and its image encryption application
Optics and Lasers in Engineering 129 (2020) 106031
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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Dynamic analysis of an improper fractional-order laser chaotic system and its image encryption application Feifei Yang, Jun Mou∗, Chenguang Ma, Yinghong Cao School of Information Science and Engineering, Dalian Polytechnic University, Dalian, Liaoning 116034, China
a r t i c l e
i n f o
Keywords: Improper fractional-order laser chaotic system DSP Implementation Image encryption algorithm
a b s t r a c t The fractional-order chaotic systems have characteristics of the all chaotic systems, however, the improper fractional-order chaotic systems have more complexity random sequences, which more suitable for chaotic cryptosystems. To investigate the application of improper fraction order chaotic system in chaotic cryptography, in this paper, an improper fractional-order laser chaotic system is constructed and applied in image encryption algorithm. The dynamic performances of the system are studied through phase diagrams, Lyapunov exponents spectrum, bifurcation diagrams and C0 complexity. Meanwhile, the improper fractional-order laser chaotic system is realized based on DSP platform. In addition, the performances of the designed encryption scheme are analyzed by key space, correlation coefficients, information entropy, histogram, differential attacks and robustness analysis. The experimental simulation results indicate that the improper fractional-order laser chaotic system not only has the abundant dynamic characteristics, but also has better security when it is used to image encryption algorithm. Therefore, this research would provide theoretical basis for the improper fractional-order laser chaotic system in security communications application.
1. Introduction Laser chaotic systems not only have various complex phenomena of dissipative systems, but also have more excellent characteristics such as bistable, pulse reproduction, close to ideal model, easy to design, inherent bandwidth chaotic laser signal, noise like and unpredictable, which illustrate that laser chaotic systems have better application potential in chaos cryptography. Since Colet proposed digital communications based on laser in 1994 [1], the study of laser chaotic systems is more and more popular. In recent years, there are many researches of laser chaotic systems [2–10]. For instance, the chaotic masking of synchronous semiconductor laser was researched in [2]. Zhang et al. analyzed brillouin optical correlation domain based on chaotic laser with suppressed time delay signature [3]. One-time pad image encryption based on physical random numbers of chaotic laser was proposed by Zhang et al. in [4]. Guo et al. analyzed photon statistics and bunching of a chaotic semiconductor laser [5]. Secure key distribution of dynamic chaos synchronization with cascaded semiconductor laser system in [6]. However, characteristic of the fractional-order laser chaotic system has not been studied, so the dynamic performance of the improper fractional-order laser chaotic system is investigated in this paper. Due to chaotic system has ergodicity, aperiodic, initial value and parameter sensitivity [11–13], which is widely used in image encryption ∗
algorithm. So far, a variety of image encryption algorithms based on chaotic system have been proposed [14–31]. For instance, Huang et al. proposed a nonlinear optical multi-image encryption algorithm based on Logistic map [14]. An optical image encryption algorithm by using Chen’s hyperchaotic system was studied in [18]. Chai et al. proposed a color image encryption algorithm based four-wing hyperchaotic system and DNA sequences operation [22]. Yang et al. described an image encryption scheme by complex chaotic system [23]. A color pathological image encryption algorithm through coupled hyper-chaotic system was introduced by Liu et al. [24]. A joint color image encryptioncompression algorithm through hyperchaotic syatem was proposed in [28]. Liu et al. introduced a fast image encryption scheme based on 2D chaotc map [32]. A color image encryption scheme through Hopfield chaotic neural network was researched in [33]. The fractionalorder chaotic systems are applied to encrypt image encryption algorithms [34–37]. For example, Yang et al. analyzed dynamic characteristic of fractional-order complex chaotic system, and proposed an image encryption algorithm based on this chaotic system [38]. An image encryption scheme based on fractional-order system and DNA operation was introduced in [39]. In addition, fractional-order systems have more complex dynamic performances, recently, there are some researches on fractional-order chaotic system [40–45]. For instance, Yang et al. analyzed dynamic performance of a fractional-order memristive chaotic system using ADM de-
Corresponding author. E-mail address: [email protected] (J. Mou).
https://doi.org/10.1016/j.optlaseng.2020.106031 Received 28 November 2019; Received in revised form 20 December 2019; Accepted 15 January 2020 0143-8166/© 2020 Elsevier Ltd. All rights reserved.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 1. Attractors phase with different planes: (a) 𝑥1 − 𝑥2 plane, (b) 𝑥1 − 𝑥4 plane, (c) 𝑥2 − 𝑥4 plane.
composition algorithm in [45]. He et al. [46] researched dynamic and synchronization of conformable fractional-order hyperchaotic systems based on the Homotopy analysis method. These fractional-order chaotic systems are true fractional-order system (the order less than 1). However, improper fractional-order chaotic systems (the order more than 1) have more large key space and complex randomness sequences. So far, the research on image encryption by using improper fraction-order chaotic system is very rare, and it is a valuable research topic. Therefore, in this paper, dynamic characteristics of the improper fractional-order laser chaotic system are analyzed, in addition, it is used in image encryption scheme. The rest of this work is arranged as follows. The dynamic performance analysis of the improper fractional-order laser chaotic and DSP digital hardware implementation in Section 2. In Section 3, the encryption algorithm and decryption algorithm are described. The simulation results and security performances are analyzed in Section 4. In Section 5 drawn a brief conclusion.
2. Improper fractional-order laser chaotic system 2.1. Model of the improper fractional-order laser chaotic system
= 𝜎(𝑥2 − 𝑥1 ) = −𝑥2 − 𝛿𝑥3 + (𝛾 − 𝑥4 )𝑥1 = 𝛿𝑥2 − 𝑥3
,
(1)
= −𝑏𝑥4 + 𝑥1 𝑥2
where, x1 , x2 , x3 and x4 represent the state variables, 𝜎, 𝛿, 𝛾 and b mean the parameters of the 4D laser chaotic system (1). According to the definition of the Caputo fractional-order differentiation [48] and system (1), the equations of the fractional-order laser chaotic system is obtained as ⎧𝐷𝑡𝑞 𝑥1 ⎪ 𝑞 ⎪𝐷 𝑡 𝑥 2 ⎨ 𝑞 ⎪𝐷 𝑡 𝑥 3 ⎪ 𝑞 ⎩𝐷 𝑡 𝑥 4
= 𝜎(𝑥2 − 𝑥1 ) = −𝑥2 − 𝛿𝑥3 + (𝛾 − 𝑥4 )𝑥1 = 𝛿𝑥2 − 𝑥3
,
In this section, solutions of the improper fractional-order laser chaotic system are obtained by Adomian decomposition method (ADM). The linear and nonlinear terms of the system (2) are expressed by ⎡𝐿𝑥1 ⎤ ⎡𝜎(𝑥2 − 𝑥1 ) ⎤ ⎢ ⎥ ⎢ ⎥ 𝐿 𝑥 − 𝑥 − 𝛿𝑥 + 𝛾𝑥 3 1⎥ ⎢ 2⎥ = ⎢ 2 , ⎢𝐿𝑥3 ⎥ ⎢𝛿𝑥2 − 𝑥3 ⎥ ⎢𝐿 𝑥 ⎥ ⎢− 𝑏 𝑥 ⎥ 4 ⎣ 4⎦ ⎣ ⎦
(2)
= −𝑏𝑥4 + 𝑥1 𝑥2
where, q is the fractional-order, x1 , x2 , x3 and x4 represent the state variables, 𝜎, 𝛿, 𝛾 and b are the system parameters. If q < 1, the system (2) is the true fractional-order system, in addition, the system is integerorder system when q = 1, when q > 1, the system is called improper fractional-order system.
⎡𝑁 𝑥1 ⎤ ⎡0 ⎤ ⎢ ⎥ ⎢ ⎥ 𝑁 𝑥 − 𝑥 𝑥 ⎢ 2 ⎥ = ⎢ 4 1 ⎥. ⎢𝑁 𝑥3 ⎥ ⎢0 ⎥ ⎢𝑁 𝑥 ⎥ ⎢𝑥 𝑥 ⎥ ⎣ 4⎦ ⎣ 1 2 ⎦
(3)
Based on the ADM, the first seven Adomian polynomials for the nonlinear terms of the system (2) are ⎧𝐴 0 ⎪ −𝑥1 𝑥4 ⎪𝐴1−𝑥 𝑥 ⎪ 2 1 4 ⎪𝐴 − 𝑥 1 𝑥 4 ⎪ 3 ⎨𝐴 − 𝑥 1 𝑥 4 ⎪ 4 ⎪𝐴 − 𝑥 1 𝑥 4 ⎪𝐴 5 ⎪ −𝑥1 𝑥4 ⎪𝐴 6 ⎩ −𝑥 𝑥 1 4
A new 4D laser chaotic system is obtained based on Lorenz-Haken Model in [47], its system equations is expressed as 𝑑𝑥 ⎧ 𝑑𝑡1 ⎪ 𝑑 𝑥2 ⎪ 𝑑𝑡 ⎨ 𝑑 𝑥3 ⎪ 𝑑𝑡 ⎪ 𝑑 𝑥4 ⎩ 𝑑𝑡
2.2. Solutions of the improper fractional-order laser chaotic system
⎧𝐴 0 ⎪ 𝑥1 𝑥2 ⎪𝐴1𝑥 𝑥 ⎪ 21 2 ⎪𝐴 𝑥 1 𝑥 2 ⎪ 3 ⎨𝐴 𝑥 1 𝑥 2 ⎪ 4 ⎪𝐴 𝑥 1 𝑥 2 ⎪𝐴 5 ⎪ 𝑥1 𝑥2 ⎪𝐴 6 ⎩ 𝑥1 𝑥2
= −𝑥01 𝑥04 = −𝑥11 𝑥04 − 𝑥01 𝑥14 = −𝑥21 𝑥04 − 𝑥11 𝑥14 − 𝑥01 𝑥24 = −𝑥31 𝑥04 − 𝑥21 𝑥14 − 𝑥11 𝑥24 − 𝑥01 𝑥34
,
(4)
= −𝑥41 𝑥04 − 𝑥31 𝑥14 − 𝑥21 𝑥24 − 𝑥11 𝑥34 − 𝑥01 𝑥44 = −𝑥51 𝑥04 − 𝑥41 𝑥14 − 𝑥31 𝑥24 − 𝑥21 𝑥34 − 𝑥11 𝑥44 − 𝑥01 𝑥54 = −𝑥61 𝑥04 − 𝑥51 𝑥14 − 𝑥41 𝑥24 − 𝑥31 𝑥34 − 𝑥21 𝑥44 − 𝑥11 𝑥54 − 𝑥01 𝑥64
= 𝑥01 𝑥02 = 𝑥11 𝑥02 + 𝑥01 𝑥12 = 𝑥21 𝑥02 + 𝑥11 𝑥12 + 𝑥01 𝑥22 = 𝑥31 𝑥02 + 𝑥21 𝑥12 + 𝑥11 𝑥22 + 𝑥01 𝑥32
(5)
= 𝑥41 𝑥02 + 𝑥31 𝑥12 + 𝑥21 𝑥22 + 𝑥11 𝑥32 + 𝑥01 𝑥42 = 𝑥51 𝑥02 + 𝑥41 𝑥12 + 𝑥31 𝑥22 + 𝑥21 𝑥32 + 𝑥11 𝑥42 + 𝑥01 𝑥52 = 𝑥61 𝑥02 + 𝑥51 𝑥12 + 𝑥41 𝑥22 + 𝑥31 𝑥32 + 𝑥21 𝑥42 + 𝑥11 𝑥52 + 𝑥01 𝑥62
Setting the initial condition x0 = [x1 (0), x2 (0), x3 (0), x4 (0)], then the first term is ⎧𝑥01 ⎪ 0 ⎪𝑥 2 ⎨ 0 ⎪𝑥 3 ⎪ 0 ⎩𝑥 4
= 𝑥1 (𝑡0 ) = 𝑥2 (𝑡0 )
(6)
= 𝑥3 (𝑡0 ) = 𝑥4 (𝑡0 )
Let 𝑐10 = 𝑥01 , 𝑐20 = 𝑥02 , 𝑐30 = 𝑥03 , 𝑐40 = 𝑥04 , and set c0 = [c1 (0), c2 (0), c3 (0), c4 (0)], then other terms are ⎧𝑐11 ⎪ 1 ⎪𝑐 2 ⎨ 1 ⎪𝑐 3 ⎪ 1 ⎩𝑐 4
= 𝜎(𝑐20 − 𝑐10 ) = −𝑐20 − 𝛿𝑐30 + (𝛾 − 𝑐40 )𝑐10 = 𝛿𝑐20 − 𝑐30 = −𝑏𝑐40 + 𝑐10 𝑐20
,
(7)
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 2. Bifurcation diagram with different parameter intervals: (a) 𝜎 ∈ [3, 7], (b) b ∈ [1, 2], (c) 𝛾 ∈ [10, 35], (d) 𝛿 ∈ [0, 2.5], (e) q ∈ [0.7, 1.03].
Fig. 3. LEs with different parameter intervals: (a) 𝜎 ∈ [3, 7], (b) b ∈ [1, 2], (c) 𝛾 ∈ [10, 35], (d) 𝛿 ∈ [0, 2.5], (e) q ∈ [0.7, 1.03].
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 4. Complexity of the time sequences for q ∈ [0.7, 1.03]: (a) 𝜎 ∈ [3, 7], (b) b ∈ [1, 2], (c) 𝛾 ∈ [10, 35], (d) 𝛿 ∈ [0, 2.5].
⎧𝑐12 ⎪ 2 ⎪𝑐 2 ⎨ 2 ⎪𝑐 3 ⎪ 2 ⎩𝑐 4
= 𝜎(𝑐21 − 𝑐11 )
⎧𝑐 1 ⎪ 3 ⎪𝑐 2 ⎨𝑐 3 ⎪ 3 ⎪𝑐 3 ⎩ 4
= 𝜎(𝑐22 − 𝑐12 )
⎧𝑐 1 ⎪ 4 ⎪𝑐 2 ⎨ 4 ⎪𝑐 3 ⎪𝑐 4 ⎩ 4
= 𝜎(𝑐23 − 𝑐13 )
𝑞+1) = −𝑏𝑐43 + 𝑐13 𝑐20 + (𝑐11 𝑐22 + 𝑐12 𝑐21 ) Γ(𝑞Γ(2 + 𝑐10 𝑐23 +1)(2𝑞 +1)
2.3. Dynamic performances of the improper fractional-order laser chaotic system
⎧𝑐15 ⎪ 5 ⎪𝑐2 ⎨𝑐 5 ⎪ 3 ⎪𝑐 5 ⎩ 4
= 𝜎(𝑐24 − 𝑐14 )
The dynamic characteristics of the improper fractional-order laser chaotic system are analyzed by attractor phase, bifurcation diagram and Lyapunov exponents spectrum (LEs). Setting the parameters of the system 𝜎 = 4, 𝛿 = 0.5, 𝛾 = 27, b = 1.8 and q = 1.005, initial condition x1 (0) = 2, x2 (0) = 1, x3 (0) = 1 and x4 (0) = 2, iteration time step h = 0.001, then the system (2) is iterated 7000 times. The attractor
= 𝛿𝑐21 − 𝑐31 =
−𝑏𝑐41
+
⎧𝑐 1 ⎪ 6 ⎪𝑐 2 ⎨𝑐 6 ⎪ 3 ⎪𝑐 6 ⎩ 4 6
= −𝑐21 − 𝛿𝑐31 + 𝛾𝑐11 − 𝑐41 𝑐10 − 𝑐40 𝑐11 𝑐11 𝑐20
+
,
(8)
𝑐10 𝑐21
= 𝜎(𝑐25 − 𝑐15 ) 𝑞+1) 𝑞+1) = −𝑐25 − 𝛿𝑐35 + 𝛾𝑐15 − 𝑐45 𝑐10 − (𝑐41 𝑐14 + 𝑐44 𝑐11 ) Γ(𝑞Γ(5 − (𝑐43 𝑐12 + 𝑐43 𝑐12 ) Γ(2Γ(5 − 𝑐40 𝑐15 +1)(4𝑞 +1) 𝑞 +1)(3𝑞 +1)
= 𝛿𝑐25 − 𝑐35
𝑞+1) 𝑞+1) = −𝑏𝑐45 + 𝑐15 𝑐20 + (𝑐11 𝑐24 + 𝑐14 𝑐21 ) Γ(𝑞Γ(5 + (𝑐13 𝑐22 + 𝑐13 𝑐22 ) Γ(2Γ(5 + 𝑐10 𝑐25 +1)(4𝑞 +1) 𝑞 +1)(3𝑞 +1)
(12) 3
4
= −𝑐22 − 𝛿𝑐32 + 𝛾𝑐12 − 𝑐42 𝑐10 − 𝑐41 𝑐11 Γ(2𝑞+1)2 − 𝑐40 𝑐12 Γ(𝑞+1)
= 𝛿𝑐22 − 𝑐32
,
(9)
𝑥̃ 𝑗 (𝑡) = 𝑐𝑗0 + 𝑐𝑗1
= −𝑏𝑐42 + 𝑐12 𝑐20 + 𝑐11 𝑐21 Γ(2𝑞+1)2 + 𝑐10 𝑐22 Γ(𝑞+1)
+𝑐𝑗4
𝑞+1) = −𝑐23 − 𝛿𝑐33 + 𝛾𝑐13 − 𝑐43 𝑐10 − (𝑐41 𝑐12 + 𝑐42 𝑐11 ) Γ(𝑞Γ(3 − 𝑐40 𝑐13 +1)(2𝑞 +1)
= 𝛿𝑐23 − 𝑐33
,
𝑞+1) 𝑞+1) = −𝑐24 − 𝛿𝑐34 + 𝛾𝑐14 − 𝑐44 𝑐10 − (𝑐41 𝑐13 + 𝑐43 𝑐11 ) Γ(𝑞Γ(4 − 𝑐42 𝑐12 Γ(3 − 𝑐40 𝑐14 +1)(3𝑞 +1) Γ(𝑞+1)3
= 𝛿𝑐24 − 𝑐34 =
−𝑏𝑐44
+
𝑐14 𝑐20
Therefore, the solution of the improper fractional-order laser chaotic system (2) is expressed as
+
(𝑐11 𝑐23
+
𝑞+1) 𝑐13 𝑐21 ) Γ(𝑞Γ(4 +1)(3𝑞 +1)
+
𝑞+1) 𝑐12 𝑐22 Γ(3 Γ(𝑞+1)3
+
(10)
,
𝑐10 𝑐24
(11)
ℎ𝑞 ℎ2𝑞 ℎ3𝑞 + 𝑐𝑗2 + 𝑐𝑗3 Γ(𝑞 + 1) Γ(2𝑞 + 1) Γ(3𝑞 + 1)
ℎ4𝑞 ℎ5𝑞 ℎ6𝑞 + 𝑐𝑗5 + 𝑐𝑗6 , Γ(4𝑞 + 1) Γ(5𝑞 + 1) Γ(6𝑞 + 1)
(13)
where, j=1,2,3,4, h is the time step.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 5. Sequences time domain waveform: (a) sequence x1 , (b) sequence x2 , (c) sequence x3 , (d) sequence x4 .
phases of the system (2) with different planes are shown in Fig. 1. As we can see, the attractor trajectories of the improper fractional-order laser chaotic system are distributed over a large area, which means that the improper fractional-order laser chaotic system has good ergodic property. To explore the parameter intervals of the system (2) dynamics characteristics, we fix 𝜎 ∈ [3, 7], b ∈ [1, 2], 𝛾 ∈ [10, 35], 𝛿 ∈ [0, 2.5] and q ∈ [0.7, 1.03], and other conditions as above. The bifurcation diagram and LEs with different parameter intervals as shown in Figs. 2 and 3. The bifurcation diagrams and LEs illustrate that the improper fractionalorder laser chaotic system has complex dynamic performances. In addition, the randomness of time sequences is tested by C0 complexity algorithm. The C0 complexity algorithm is described as follows. Step 1 Discrete fourier transformation of the sequence 𝑥(𝑛), 𝑛 = 0, 1, 2, … 𝑁 − 1 is 𝑋(𝑘) =
𝑁−1 ∑
2𝜋
𝑥(𝑛)−𝑗 𝑁 𝑛𝑘 =
𝑛=0
𝑁−1 ∑ 𝑛=0
𝑥(𝑛)𝑊𝑁𝑛𝑘 ,
(14)
where k = 0, 1, 2, … , 𝑁 − 1. Step 2 The X(k) is processed by ⎧ ⎪ 𝑋(𝑘) ⎪ 𝑋̃ (𝑘) = ⎨ ⎪ 0 ⎪ ⎩
|𝑋(𝑘)|2 > 𝑟 𝑁1 |𝑋(𝑘)| ≤ 2
𝑟 𝑁1
𝑁−1 ∑ 𝑛=0 𝑁−1 ∑ 𝑛=0
|𝑋(𝑘)|2 (15) |𝑋(𝑘)|
2
where k = 0, 1, 2, … , 𝑁 − 1, r is the introduced parameter.
Step 3 Inverse Fourier transform of X(k) is 𝑥̃ (𝑘) =
𝑁−1 𝑁−1 2𝜋 1 ∑ ̃ 1 ∑ ̃ 𝑋 (𝑘)𝑒𝑗 𝑁 𝑛𝑘 = 𝑋 (𝑘)𝑊𝑁−𝑛𝑘 , 𝑁 𝑛=0 𝑁 𝑛=0
where n = 0, 1, 2, … , 𝑁 − 1. Step 4 The C0 complexity is expressed as ∑𝑁−1 |𝑥(𝑛) − 𝑥̃ (𝑛)|2 𝐶0 (𝑟, 𝑁) = 𝑛=0 . ∑𝑁−1 2 𝑛=0 |𝑥(𝑛)|
(16)
(17)
Keeping the above parameters and initial values, then according to the C0 complexity algorithm, the complexity of chaotic sequences is shown in Fig. 4. This shows that sequences generated by the improper fractional-order laser chaotic system have good randomness. The sensitivity of the improper fractional-order laser chaotic system to the initial values are analyzed by time domain waveform. Keeping the other initial values, a initial value is respectively changed 10−5 and 10−7 . Then getting the sequences time domain waveform is Fig. 5. The results indicate that the sequences generated by the improper fractional-order laser chaotic system when initial value slight change are significantly different. Therefore, the improper fractional-order laser chaotic system to the initial values are extremely sensitive. 2.4. DSP Implementation of the improper fractional-order laser chaotic system Due to DSP digital signal processor has the advantages of fast processing speed, strong programmable, high anti-interference and easy to
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 6. Attractors phase with different planes: (a) 𝑥1 − 𝑥2 plane, (b) 𝑥1 − 𝑥4 plane, (c) 𝑥2 − 𝑥4 plane.
Fig. 7. The flowchart of encryption algorithm.
realize, in this paper, DSP platform is used to digital hardware implementation of the improper fractional-order laser chaotic system. DSP platform and the corresponding of results are shown in Fig. 6. The results illustrate that the chaotic signal generated by DSP platform is consistent with the MATLAB simulation results under the same system parameters and initial condition, which realizes the digital hardware implementation of the improper fractional-order laser chaotic system.
3. Image encryption algorithm of the improper fractional-order laser chaotic system 3.1. Encryption algorithm In this section, we designed an image encryption algorithm based on the improper fractional-order laser chaotic system. The flowchart of the proposed image encryption algorithm is given in Fig. 7. The corresponding detailed process is expressed as follows.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 8. The flowchart of decryption algorithm.
Step 1: Inputting a plain image I and it’s size is defined to M × N. Step 2: Setting the parameter and initial values of the improper fractional-order laser chaotic system. The system Eq. (2) is iterated (𝐿 = 𝑀 × 𝑁) times. Then improper fractional-order laser chaotic sequences {x1 (i), x2 (i), x3 (i), x4 (i)} are obtained, here they both have length L. Step 3: The sequences x1 (i) and x2 (i) are processed by ⎧ 16 ⎪𝑏𝑟 = mod (𝑓 𝑙𝑜𝑜𝑟(abs(𝑥1 (𝑖)) × 10 , 𝑀∕2) ⎪𝑏𝑐 = mod (𝑓 𝑙𝑜𝑜𝑟(abs(𝑥2 (𝑖)) × 1016 , 𝑁∕2) , ⎨𝑋 = 𝑥 (1 ∶ 𝑀) 1 ⎪ ⎪𝑌 = 𝑥2 (1 ∶ 𝑁) ⎩
(18)
where br, bc, X and Y are used to scramble the plane image I. Step 4: According to the br and X, the scrambled row image T is generated through ⎧𝑋(𝑖) > 0 ⎪ ⎪𝑇 (𝑖, end − 𝑏𝑟(𝑖) + 1 ∶ end) = 𝐼(𝑖, 1 ∶ 𝑏𝑟(𝑖)) ⎪𝑇 (𝑖, 1 ∶ end − 𝑏𝑟(𝑖)) = 𝐼(𝑖, 𝑏𝑟(𝑖) + 1 ∶ end) , ⎨𝑋(𝑖) < 0 ⎪ ⎪𝑇 (𝑖, 1 ∶ 𝑏𝑟(𝑖)) = 𝐼(𝑖, end − 𝑏𝑟(𝑖) + 1 ∶ end) ⎪𝑇 (𝑖, 𝑏𝑟(𝑖) + 1 ∶ end) = 𝐼(𝑖, 1 ∶ end − 𝑏𝑟(𝑖)) ⎩ where 𝑖 = 1, 2, … , 𝑀.
Step 5: Based on bc and Y, the scrambled column image T1 is obtained by
⎧𝑌 ( 𝑖 ) > 0 ⎪ ⎪𝑇 1(end − 𝑏𝑐(𝑗) + 1 ∶ end, 𝑖) = 𝑇 (1 ∶ 𝑏𝑟(𝑗 ), 𝑗 ) ⎪𝑇 1(1 ∶ end − 𝑏𝑐(𝑗 ), 𝑗 ) = 𝑇 (𝑏𝑐(𝑗) + 1 ∶ end, 𝑗) , ⎨𝑌 ( 𝑖 ) < 0 ⎪ ⎪𝑇 1(1 ∶ 𝑏𝑐(𝑗 ), 𝑗 ) = 𝑇 (end − 𝑏𝑐(𝑗) + 1 ∶ end, 𝑗) ⎪𝑇 1(𝑏𝑐(𝑗) + 1 ∶ end, 𝑗) = 𝑇 (1 ∶ end − 𝑏𝑟(𝑗), 𝑗) ⎩
where 𝑗 = 1, 2, … , 𝑁. Step 6: Two matrixes Z and W based on two chaotic sequences x3 (i) and x4 (i) are given as {
(19)
(20)
𝑍(𝑖, 𝑗) = mod (𝑓 𝑙𝑜𝑜𝑟( mod (𝑥3 ((𝑖 − 1) × 𝑁 + 𝑗 + 500, 1) × 1016 )), 256) , 𝑊 (𝑖, 𝑗) = mod (𝑓 𝑙𝑜𝑜𝑟( mod (𝑥4 ((𝑖 − 1) × 𝑁 + 𝑗 + 500, 1) × 1016 )), 256) (21)
where 𝑖 = 1, 2, … , 𝑀. 𝑗 = 1, 2, … , 𝑁.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 9. Encryption and decryption test results: (a) “Couple”, “Camera”, “Lake” and “Boat” of original image, (b) “Couple”, “Camera”, “Lake” and “Boat” of encrypted image, (c) “Couple”, “Camera”, “Lake” and “Boat” of decrypted image.
Step 7: The diffused image T2 is obtained by using matrix Z and Eq. (22). ⎧ ⎪𝑇 2(𝑀 , 𝑁 ) = 𝑇 1(𝑀 , 𝑁 ) + 𝑍(𝑀 , 𝑁 ) mod 256 ⎪𝑇 2(𝑀, 𝑗) = 𝑇 1(𝑀, 𝑗) + 𝑇 2(𝑀, 𝑗 + 1) + 𝑍(𝑀, 𝑗) mod 256 , ⎨𝑇 2(𝑖, 𝑁) = 𝑇 1(𝑖, 𝑁) + 𝑇 2(𝑖 + 1, 𝑁) + 𝑍(𝑖, 𝑁) mod 256 ⎪ ⎪𝑇 2(𝑖, 𝑗) = 𝑇 1(𝑖, 𝑗) + 𝑇 2(𝑖 + 1, 𝑗) + 𝑇 2(𝑖, 𝑗 + 1) + 𝑍(𝑖, 𝑗) mod 256 ⎩
(22)
where 𝑖 = 𝑀 − 1, 𝑀 − 2, … , 2, 1. 𝑗 = 𝑁 − 1, 𝑁 − 2, … , 2, 1. Step 8: The diffused image T2 is scrambled by Zigzag algorithm. Then we get a confused image B. Step 9: The image B is changed into C through matrix W in step 6. The specific operation is expressed as follows: If 𝑖 = 𝑀, 𝑗 = 𝑁. 𝐶(𝑖, 𝑗) = 𝐵(𝑖, 𝑗) + 𝑊 (𝑖, 𝑗) mod 256.
(26)
When 𝑗 = 𝑗 − 1, j > 1. 𝐶1(𝑖, 𝑗) = 𝐵(𝑖, 𝑗) − 𝐶1(𝑖, 𝑗 + 1) − 𝑊 (𝑖, 𝑗) mod 256.
(27)
Other, 𝑗 = 𝑁, 𝑖 = 𝑖 − 1 and i ≥ 1. 𝐶1(𝑖, 𝑗) = 𝐵(𝑖, 𝑗) − 𝑠𝑢𝑚(𝐶1(𝑖 + 1, 1 ∶ 𝑁)) − 𝑊 (𝑖, 𝑗) mod 256.
(24)
Step 5: The scrambled image C1 is recovered to T1 by inverse zigzag algorithm. Step 6: The image T1 is restored by
Other, 𝑗 = 𝑁, 𝑖 = 𝑖 − 1 and i ≥ 1. 𝐶(𝑖, 𝑗) = 𝐵(𝑖, 𝑗) + 𝑠𝑢𝑚(𝐶(𝑖 + 1, 1 ∶ 𝑁)) + 𝑊 (𝑖, 𝑗) mod 256.
𝐶1(𝑖, 𝑗) = 𝐶(𝑖, 𝑗) − 𝑊 (𝑖, 𝑗) mod 256.
(23)
When 𝑗 = 𝑗 − 1, j > 1. 𝐶(𝑖, 𝑗) = 𝐵(𝑖, 𝑗) + 𝐶(𝑖, 𝑗 + 1) + 𝑊 (𝑖, 𝑗) mod 256.
Step 1: Inputting the cipher image C(M × N). Step 2: As in step 2 of encryption algorithm, four chaotic sequences {x1 (i), x2 (i), x3 (i), x4 (i)} are obtained. Step 3: According to step 6 of encryption algorithm, two matrixes Z and W are generated. Step 4: The restored diffusion image C1 is obtained based on the following principles. If 𝑖 = 𝑀, 𝑗 = 𝑁.
(25)
Step 10: The matrix C is cipher image. 3.2. Decryption algorithm Fig. 8 is a flowchart of decryption algorithm. Decryption algorithm refer to restor the cipher image process. In this paper, the detailed steps of the decryption algorithm is expressed as follows.
(28)
⎧ ⎪𝑇 2(𝑀 , 𝑁 ) = 𝑇 1(𝑀 , 𝑁 ) + 256 × 2 − 𝑍(𝑀 , 𝑁 ) mod 256 ⎪𝑇 2(𝑀, 𝑗) = 𝑇 1(𝑀, 𝑗) + 256 × 2 − 𝑇 2(𝑀, 𝑗 + 1) − 𝑍(𝑀, 𝑗) mod 256 , ⎨𝑇 2(𝑖, 𝑁) = 𝑇 1(𝑖, 𝑁) + 256 × 2 − 𝑇 2(𝑖 + 1, 𝑁) − 𝑍(𝑖, 𝑁) mod 256 ⎪ ⎪𝑇 2(𝑖, 𝑗) = 𝑇 1(𝑖, 𝑗) + 256 × 3 − 𝑇 2(𝑖 + 1, 𝑗) − 𝑇 2(𝑖, 𝑗 + 1) − 𝑍(𝑖, 𝑗) mod 256 ⎩ (29) where 𝑖 = 𝑀 − 1, 𝑀 − 2, … , 2, 1. 𝑗 = 𝑁 − 1, 𝑁 − 2, … , 2, 1. Step 7: br, bc, X and Y are obtained based on step 3 of encryption algorithm.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 10. Histograms: (a) “Couple”, “Camera”, “Lake” and “Boat” of original image,(b) “Couple”, “Camera”, “Lake” and “Boat” of encrypted image, (c) “Couple”, “Camera”, “Lake” and “Boat” of decrypted image.
Step 8: The scrambled image T2 of rows and columns is restored through ⎧𝑌 ( 𝑖 ) < 0 ⎪ ⎪𝑇 1(end − 𝑏𝑐(𝑗) + 1 ∶ end, 𝑖) = 𝑇 2(1 ∶ 𝑏𝑟(𝑗 ), 𝑗 ) ⎪𝑇 1(1 ∶ end − 𝑏𝑐(𝑗 ), 𝑗 ) = 𝑇 2(𝑏𝑐(𝑗) + 1 ∶ end, 𝑗) , ⎨𝑌 ( 𝑖 ) > 0 ⎪ ⎪𝑇 1(1 ∶ 𝑏𝑐(𝑗 ), 𝑗 ) = 𝑇 2(end − 𝑏𝑐(𝑗) + 1 ∶ end, 𝑗) ⎪𝑇 1(𝑏𝑐(𝑗) + 1 ∶ end, 𝑗) = 𝑇 2(1 ∶ end − 𝑏𝑟(𝑗), 𝑗) ⎩
4. Experiment test results and security performance analysis
(30)
where 𝑗 = 1, 2, … , 𝑁. ⎧𝑋(𝑖) < 0 ⎪ ⎪𝑇 (𝑖, end − 𝑏𝑟(𝑖) + 1 ∶ end) = 𝑇 1(𝑖, 1 ∶ 𝑏𝑟(𝑖)) ⎪𝑇 (𝑖, 1 ∶ end − 𝑏𝑟(𝑖)) = 𝑇 1(𝑖, 𝑏𝑟(𝑖) + 1 ∶ end) , ⎨𝑋(𝑖) > 0 ⎪ ⎪𝑇 (𝑖, 1 ∶ 𝑏𝑟(𝑖)) = 𝑇 1(𝑖, end − 𝑏𝑟(𝑖) + 1 ∶ end) ⎪𝑇 (𝑖, 𝑏𝑟(𝑖) + 1 ∶ end) = 𝑇 1(𝑖, 1 ∶ end − 𝑏𝑟(𝑖)) ⎩ where 𝑖 = 1, 2, … , 𝑀.
Step 9: The ciphertext image C is restored to plane image T.
(31)
To prove the effectiveness and the security of the image algorithm based on improper fractional-order laser chaotic system, the corresponding of simulation experiments are realized on MATLAB (version R2018a). In this experiment, the “Couple”, “Camera”, “Lake” and “Boat” of size 256 × 256 gray images are tested images of the proposed image encryption algorithm. The parameters of the improper fractional-order laser chaotic are selected 𝜎 = 4, 𝛿 = 0.5, 𝛾 = 27, 𝑏 = 1.8 and 𝑞 = 1.005, initial values are 𝑥1 (0) = 2, 𝑥2 (0) = 1, 𝑥3 (0) = 1 and 𝑥4 (0) = 2. In decryption algorithm, the parameter and initial values are the same as the encryption algorithm. The original grayscale images “Couple”, “Camera”, “Lake” and “Boat” are Fig. 9(a). The ciphertext images are shown in Fig. 9(b). The corresponding of restor ciphertext images as shown in Fig. 9(c). As we can see from Fig. 9, the visual information of the plain-
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 11. The correlation distribution of images in horizontal, vertical and diagonal directions: (a) Plaintext “Couple” image, (b) Ciphertext “Couple” image, (c) Plaintext “Camera” image, (d) Ciphertext “Camera” image, (e) Plaintext “Lake” image, (f) Ciphertext “Lake” image, (g) Plaintext “Boat” image, (h) Ciphertext “Boat” image.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
text image cannot be seen in the ciphertext image, in addition, there is no significant difference between the restored ciphertext images and the plaintext images. Therefore, the proposed encryption algorithm could effectively hide image information. 4.1. Histogram analysis Histogram is an important tool to analyze statistical attack. Fig. 10(a) are the histograms of plaintext images “Couple”, “Camera”, “Lake” and “Boat”. The corresponding of the histograms of ciphertext images are Fig. 10(b). The histograms of restored ciphertext images as shown in Fig. 10(c). The results indicate that the pixel values distribution of the plaintext images are concentrated in some certain ranges, while the pixel values distribution of encrypted images are uniform, in addition, pixel values distribution of the recovered ciphertext images are the same as original images. So the ciphertext images can resist the histogram attack. 4.2. Chi-square test To further verify uniform distribution of the histograms with encrypted image, the Chi-square test is adopted. The computational formula of 𝜒 2 test is expressed by 255 ∑ (𝐶𝑛 − 𝐺)2 𝜒2 = , 𝐺 𝑛
Table 1 Chi-square test results of images. Images
Couple
Plaitext Ciphertext Decision
where Cn represents actual amount of each grayscale level, G means the desired number of each grayscale level. If the histogram distribution of ciphertext image more uniformly is, the Chi-square value is smaller. It shows that it can pass the Chi-square test when the confidence level is
5.4617 × 10 236.3203 pass
Lake
1.1097 × 10 289.5234 pass
5
4.8806 × 10 249.3359 pass
Boat 4
1.0067 × 105 242.0234 pass
0.05 and the Chi-square value not exceed 293.2478. In this experimental, the Chi-square test results of the different images as Table 1. The results indicate that histogram distribution of the ciphertext image are uniformly. 4.3. Correlation of adjacent pixels The correlation coefficient of two pixels refer to the degree of correlation for the between different pixels. The correlation coefficient of the adjacent pixels for the image is calculated by 𝑛 ∑
𝑟= √
𝑖=1 𝑛 ∑ 𝑖=1
(32)
Camera 4
(𝑥𝑖 −
(𝑥𝑖 −
1 𝑛
1 𝑛
∑𝑛
𝑖=1
∑𝑛
𝑥𝑖 )(𝑦𝑖 − 𝑛 2 ∑
𝑖=1 𝑥𝑖 )
𝑖=1
1 𝑛
∑𝑛
(𝑦𝑖 −
𝑖=1 𝑦𝑖 )
1 𝑛
∑𝑛
,
(33)
2
𝑖=1 𝑦𝑖 )
where xi and yi represent the gray value of the two adjacent pixels. n is the number of the different pixels pairs. To measure the pixel correlation of encrypted image, in this test, we randomly select 2000 pixels in plaintext image and ciphertext image to calculate the correlation coefficients in horizontal, vertical and diagonal directions. The correlation distribution of images is Fig. 11. The Fig. 12. Key sensitivity analysis results: (a1) 𝜎 + 10−15 , (a2) 𝛿 + 10−15 , (a3) 𝛾 + 10−15 , (a4) 𝑞 + 10−15 , (a5) 𝑏 + 10−15 , (a6) 𝑥1 (0) + 10−15 , (a7) 𝑥2 (0) + 10−15 , (a8) 𝑥3 (0) + 10−15 , (a9) 𝑥4 (0) + 10−15 .
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Table 2 Correlation coefficients of different images. Images
Horizontal
Plain image Vertical
Diagonal
Horizontal
Cipher image Vertical
Diagonal
Couple Camera Lake Boat
0.9337 0.9597 0.9568 0.9451
0.9255 0.9342 0.9584 0.9268
0.8773 0.9062 0.9295 0.8835
−0.0095 −0.0003 0.0044 −0.0037
−0.0044 −0.0052 −0.0047 −0.0009
0.0070 −0.0027 −0.0021 −0.0027
Table 3 Correlation coefficients of different algorithms. Algorithm
Images
Horizontal
Vertical
Diagonal
Ours Ref [14] Ref [15] Ref [16] Ref [17] Ours Ref [18] Ref [19] Ref [20] Ref [21] Ours Ref [18] Ref [19] Ref [20] Ref [21] Ours Ref [14] Ref [15] Ref [16] Ref [17]
Encrypted “Couple”
−0.0095 0.0086 −0.0382 0.1539 −0.0290 −0.0003 0.0079 −0.0264 0.0341 0.0685 0.0044 −0.0012 −0.0306 0.0600 0.2071 −0.0037 0.0086 0.0250 0.1830 −0.0037
−0.0044 0.0060 0.0083 0.1342 −0.0166 −0.0052 −0.0005 0.26375 −0.0509 0.0821 −0.0047 0.0095 0.3188 0.0196 0.2128 −0.0009 0.0060 0.0186 0.1316 −0.0177
0.0070 0.0017 0.0277 −0.0029 0.0281 −0.0027 −0.0044 0.0272 −0.0216 0.0821 −0.0021 −0.0093 −0.0137 −0.0231 0.2102 −0.0027 0.0017 −0.0051 0.0599 −0.0131
Encryption “Camera”
Encrypted “Lake”
Encrypted “Boat”
Table 5 Information entropy of ciphertext images with different algorithms.
Images
Couple
Camera
Lake
Boat
Plaitext Ciphertext
7.4000 7.9974
7.0097 7.9968
7.4584 7.9973
7.1587 7.9973
The information entropy reflect the uncertainty of image information. It is calculated by 𝑝(𝑦𝑗 )log2 (𝑝(𝑦𝑗 )),
Ref [16]
Ref [17]
7.9953 7.9948 7.9948 7.9956
7.4101 7.4740 7.7877 7.4232
7.9937 7.9938 7.9939 7.9944
4.6. Key sensitivity analysis
4.4. Information entropy
𝑗=1
Ref [15]
7.9974 7.9968 7.9973 7.9973
The size of the key space shows the ability of the resist violent attacks for encryption algorithm. Theoretically, for a good encryption algorithm, its key space should be larger than 2100 . In this paper, the key space of the proposed encryption algorithm is composed of 𝜎, 𝛿, 𝛾, q, b, x1 (0), x2 (0), x3 (0) and x4 (0). The experimental results find that the key space for 𝜎, 𝛿, 𝛾, q, b, x1 (0), x2 (0), x3 (0) and x4 (0) is about 1015 . So all key space of the proposed encryption algorithm is 2449 much large than 2100 , which shows that the key space of the proposed encryption algorithm is large enough to resist violent attacks.
correlation coefficients are shown in Table 2. The results show that the correlation adjacent pixels of original image at horizontal, vertical and diagonal directions are almost close to 1, distribution is linear 𝑥 = 𝑦. However, the correlation coefficients adjacent pixels of ciphertext image are almost 0, distribution in the whole plane. Therefore, the proposed algorithm can effectively destroy correlation of the between adjacent pixels. In addition, comparison results of correlation coefficients with existing algorithms [14–21]are listed in Table 3.
𝑀 ∑
Ours
Couple Camera Lake Boat
4.5. Key space analysis
Table 4 Information entropy test results of images.
ℎ=−
Images
(34)
where yj represent the jth grayscale value of the M-level grayscale image, p(yj ) is the probability of the yj . The information entropy value is 8 for an ideal encrypted image, which indicates that the ciphertext image is entirely random. The Table 4 listed information entropy values of the different images. The Table 4 illustrate that information entropy values of encrypted images are very close to 8, therefore, the proposed encryption algorithm can resist entropy value attacks. In addition, comparison results of information entropy with existing algorithms [15–17]are listed in Table 5.
For the chaotic cryptosystem, the sensitivity of chaotic system to initial value and parameter value determines the sensitivity of encryption system to the key. In this experimental, the decryption key 𝜎, 𝛿, 𝛾, q, b, x1 (0), x2 (0), x3 (0) and x4 (0) are respectively changed 10−15 , then decryption “Boat” images as Fig. 12. It shows that the decryption image has not any valuable information of original image, which indicate that the proposed image algorithm is extremely sensitivity for the key. 4.7. Differential attacks The Number of Pixels Change Rate (NPCR) and the Unified Average Changing Intensity (UACI) are two methods to test the ability of the differential attack. The corresponding of calculation equation is { 0, 𝐶1 (𝑖, 𝑗) = 𝐶2 (𝑖, 𝑗) 𝐷(𝑖, 𝑗) = , (35) 1, 𝐶1 (𝑖, 𝑗) ≠ 𝐶2 (𝑖, 𝑗) ∑ D(i,j) ⎧ ⎪UPCR = i,j M×N × 100% , ⎨ ∑ |𝐶1 (𝑖,𝑗 )−𝐶2 (𝑖,𝑗 )| ⎪UACI = 𝑖,𝑗 255×𝑀×𝑁 × 100% ⎩
(36)
where C1 (i, j) represents the pixel value of the normal ciphertext image. C2 (i, j) is the pixel value of the encrypted image for the changed pixel value of plaintext image. Recently, Wu et al. [49] proposed the criterion values of NPCR and UACI. If a significance level is 𝛼, its critical NPCR value 𝑁𝛼∗ is calculated by √ 𝐹 − Φ−1 (𝛼) 𝐹 ∕𝐿 𝑁𝛼∗ = . (37) 𝐹 +1 For an image encryption algorithm, if the NPCR value is more than 𝑁𝛼∗ , which shows that it has a high ability to resist differential attack.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 13. The Anti-shear attack analysis results: (a) Ciphertext “Couple” image of data loss, (b) Restored “Couple” image, (c) Ciphertext “Camera” image of data loss, (d) Restored “Camera” image, (e) Ciphertext “Lake” image of data loss, (f) Restored “Lake” image, (g) Ciphertext “Boat” image of data loss, (h) Restored “Boat” image.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Table 6 The theoretical values of critical NPCR and UACI for different size of images. Images size
NPCR (%) 𝑁0∗.05
𝑁0∗.01
𝑁0∗.001
UACI(%) (𝑈0∗− , 𝑈0∗+ ) .05 .05
(𝑈0∗− , 𝑈0∗+ ) .01 .01
(𝑈0∗− , 𝑈0∗+ ) .001 .001
256 × 256 512 × 512 1024 × 1024
99.5693 99.5893 99.5994
99.5527 99.5810 99.5952
99.5341 99.5717 99.5906
(33.2824, 33.6447) (33.3730, 33.5541) (33.4183, 33.5088)
(33.2255, 33.7016) (33.3445, 33.5826) (33.4040, 33.5231)
(33.1594, 33.7677) (33.3115, 33.6156) (33.3875, 33.5396)
The critical UACI values (𝑈𝛼∗ −, 𝑈𝛼∗ +) for the significance level 𝛼 are obtained as { ∗− 𝑈𝛼 = 𝜇𝑢 − Φ−1 (𝛼∕2)𝜎𝑢 , (38) 𝑈𝛼∗+ = 𝜇𝑢 + Φ−1 (𝛼∕2)𝜎𝑢
NPCR and UACI are shown in Tables 7 and 8. What’s more, Table 9 listed the comparison results of NPCRs and UACIs with existing algorithms [15,16,18,22]. The analysis results illustrate that the proposed encryption scheme has ability to resist differential attacks.
where 𝐹 +2 𝜇𝑢 = , 3𝐹 + 3
4.8. Robustness analysis (39)
in addition 𝜎𝑢 =
(𝐹 + 2)(𝐹 2 + 2𝐹 + 3) 18(𝐹 + 1)2 𝐹 𝐿
.
(40)
If the UACI value is more than 𝑈𝛼∗− and it is less than 𝑈𝛼∗+ , which shows that encryption algorithm can pass test and resist differential attack. Theoretical values of critical NPCR and UACI for different size of images are calculated in [49], the corresponding of results are listed in Table 6. In this experiment, the pixel of original image is randomly selected and changed its pixel value. For the “Couple”, “Camera”, “Lake” and “Boat” of different images, we test 100 times, calculate the results of
In this section, the anti-shear attack and anti-noise pollution attack are used to analyze the robustness of the proposed algorithm. To measure the Anti-shear attack of the algorithm, the ciphertext images lost with varying degrees of data, then ciphertext images of loss data are restored. The corresponding of results are shown in Fig. 13. The results illustrate that although the restored ciphertext images have noise, however, it does not affect the effect of overall image. So the proposed scheme can resist cropping attack of different degree. To test anti-noise pollution attack ability of the proposed algorithm, the ciphertext images are added by the Gaussian noise with intensity of 0.05, 0.06, 0.07 and 0.08, then to decrypt the ciphertext image of the Gaussian noise with pollution, the results are Fig. 14. The results illustrate that the proposed algorithm can resist the noise pollution. Fig. 14. The anti-noise pollution attack analysis results: (a) Restored “Couple” image, (b) Restored “Camera” image, (c) Restored “Lake” image, (d) Restored “Boat” image.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Table 7 NPCR of the ciphertext images. Critical NPCR(%) Images
Minimum(%)
Maximum(%)
Mean(%)
Couple Camera Lake Boat
98.84 99.54 99.49 99.54
99.95 99.98 99.90 99.68
99.58 99.64 99.64 99.61
𝑁0∗.05 = 99.5693%
𝑁0∗.01 = 99.5527%
𝑁0∗.001 = 99.5341%
pass pass pass pass
pass pass pass pass
pass pass pass pass
Table 8 UACI of the ciphertext images. Critical UACI(%) Images
Minimum(%)
Maximum(%)
Mean(%)
= 33.2824% U∗− 0.05 = 33.6447% U∗+ 0.05
Couple Camera Lake Boat
34.18 32.19 32.46 33.13
33.21 34.88 34.65 33.77
33.55 33.27 33.54 33.43
pass pass pass pass
U∗− = 33.2255% 0.01 U∗+ = 33.7016% 0.01
U∗− = 33.1594% 0.001 U∗+ = 33.7677% 0.001
pass pass pass pass
pass pass pass pass
Table 9 NPCRs and UACIs of the ciphertext images with different encryption methods. Images
Ref[15] NPCRs(%), UACIs(%)
Ref[16] NPCRs(%), UACIs(%)
Ref[18] NPCRs(%), UACIs(%)
Ref[22] NPCRs(%), UACIs(%)
Ours NPCRs(%), UACIs(%)
Couple Camera Lake Boat
99.49, 99.47, 99.50, 99.47,
98.31, 98.43, 98.49, 98.26,
No 99.61, 33.47 99.61, 33.45 No
99.62, 99.59, 99.58, 99.61,
99.58, 99.64, 99.64, 99.61,
33.44 33.31 33.68 33.45
16.78 16.77 18.27 16.17
33.40 33.56 33.57 33.28
33.55 33.27 33.54 33.43
5. Conclusion
Acknowledgements
Dynamic characteristics of an improper fractional-order laser chaotic system and its image encryption application are investigated in this paper. The dynamic analysis illustrate that chaotic state of the improper fractional-order laser chaotic system distribution is in a large parameters range when parameters and order are changed, what’s more, the chaotic sequences are generated by the improper fractional-order laser system have good pseudo-randomness. Therefore, the improper fractional-order laser systems are more suitable for chaotic security communication. The chaotic system is hardware implementation based on DSP platform, which provide theoretical direction for chaotic system in secure communication applications. The encryption algorithm performances results indicate that the proposed encryption algorithm not only has the superior image encryption effect, but also improves the anti-cracking ability, which illustrates that the proposed algorithm more suit for image encryption, further illustrate that the improper fractional-order laser chaotic system is suitable for image encryption and other security communication.
This work is supported by the Basic Scientific Research Projects of Colleges and Universities of Liaoning Province (Grant Nos. 2017J045); Provincial Natural Science Foundation of Liaoning (Grant Nos. 20170540060).
Author Statement Feifei yang designed and carried out experiments, data analyzed and manuscript wrote. Jun Mou made the theoretical guidance for this paper. Chenguang Ma did the DSP implementation of the laser chaotic system. Yinghong Cao improved the algorithm. All authors reviewed the manuscript.
Declaration of Competing Interest No conflict of interest about the publication by all authors.
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Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Dynamic analysis of an improper fractional-order laser chaotic system and its image encryption application Feifei Yang, Jun Mou∗, Chenguang Ma, Yinghong Cao School of Information Science and Engineering, Dalian Polytechnic University, Dalian, Liaoning 116034, China
a r t i c l e
i n f o
Keywords: Improper fractional-order laser chaotic system DSP Implementation Image encryption algorithm
a b s t r a c t The fractional-order chaotic systems have characteristics of the all chaotic systems, however, the improper fractional-order chaotic systems have more complexity random sequences, which more suitable for chaotic cryptosystems. To investigate the application of improper fraction order chaotic system in chaotic cryptography, in this paper, an improper fractional-order laser chaotic system is constructed and applied in image encryption algorithm. The dynamic performances of the system are studied through phase diagrams, Lyapunov exponents spectrum, bifurcation diagrams and C0 complexity. Meanwhile, the improper fractional-order laser chaotic system is realized based on DSP platform. In addition, the performances of the designed encryption scheme are analyzed by key space, correlation coefficients, information entropy, histogram, differential attacks and robustness analysis. The experimental simulation results indicate that the improper fractional-order laser chaotic system not only has the abundant dynamic characteristics, but also has better security when it is used to image encryption algorithm. Therefore, this research would provide theoretical basis for the improper fractional-order laser chaotic system in security communications application.
1. Introduction Laser chaotic systems not only have various complex phenomena of dissipative systems, but also have more excellent characteristics such as bistable, pulse reproduction, close to ideal model, easy to design, inherent bandwidth chaotic laser signal, noise like and unpredictable, which illustrate that laser chaotic systems have better application potential in chaos cryptography. Since Colet proposed digital communications based on laser in 1994 [1], the study of laser chaotic systems is more and more popular. In recent years, there are many researches of laser chaotic systems [2–10]. For instance, the chaotic masking of synchronous semiconductor laser was researched in [2]. Zhang et al. analyzed brillouin optical correlation domain based on chaotic laser with suppressed time delay signature [3]. One-time pad image encryption based on physical random numbers of chaotic laser was proposed by Zhang et al. in [4]. Guo et al. analyzed photon statistics and bunching of a chaotic semiconductor laser [5]. Secure key distribution of dynamic chaos synchronization with cascaded semiconductor laser system in [6]. However, characteristic of the fractional-order laser chaotic system has not been studied, so the dynamic performance of the improper fractional-order laser chaotic system is investigated in this paper. Due to chaotic system has ergodicity, aperiodic, initial value and parameter sensitivity [11–13], which is widely used in image encryption ∗
algorithm. So far, a variety of image encryption algorithms based on chaotic system have been proposed [14–31]. For instance, Huang et al. proposed a nonlinear optical multi-image encryption algorithm based on Logistic map [14]. An optical image encryption algorithm by using Chen’s hyperchaotic system was studied in [18]. Chai et al. proposed a color image encryption algorithm based four-wing hyperchaotic system and DNA sequences operation [22]. Yang et al. described an image encryption scheme by complex chaotic system [23]. A color pathological image encryption algorithm through coupled hyper-chaotic system was introduced by Liu et al. [24]. A joint color image encryptioncompression algorithm through hyperchaotic syatem was proposed in [28]. Liu et al. introduced a fast image encryption scheme based on 2D chaotc map [32]. A color image encryption scheme through Hopfield chaotic neural network was researched in [33]. The fractionalorder chaotic systems are applied to encrypt image encryption algorithms [34–37]. For example, Yang et al. analyzed dynamic characteristic of fractional-order complex chaotic system, and proposed an image encryption algorithm based on this chaotic system [38]. An image encryption scheme based on fractional-order system and DNA operation was introduced in [39]. In addition, fractional-order systems have more complex dynamic performances, recently, there are some researches on fractional-order chaotic system [40–45]. For instance, Yang et al. analyzed dynamic performance of a fractional-order memristive chaotic system using ADM de-
Corresponding author. E-mail address: [email protected] (J. Mou).
https://doi.org/10.1016/j.optlaseng.2020.106031 Received 28 November 2019; Received in revised form 20 December 2019; Accepted 15 January 2020 0143-8166/© 2020 Elsevier Ltd. All rights reserved.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 1. Attractors phase with different planes: (a) 𝑥1 − 𝑥2 plane, (b) 𝑥1 − 𝑥4 plane, (c) 𝑥2 − 𝑥4 plane.
composition algorithm in [45]. He et al. [46] researched dynamic and synchronization of conformable fractional-order hyperchaotic systems based on the Homotopy analysis method. These fractional-order chaotic systems are true fractional-order system (the order less than 1). However, improper fractional-order chaotic systems (the order more than 1) have more large key space and complex randomness sequences. So far, the research on image encryption by using improper fraction-order chaotic system is very rare, and it is a valuable research topic. Therefore, in this paper, dynamic characteristics of the improper fractional-order laser chaotic system are analyzed, in addition, it is used in image encryption scheme. The rest of this work is arranged as follows. The dynamic performance analysis of the improper fractional-order laser chaotic and DSP digital hardware implementation in Section 2. In Section 3, the encryption algorithm and decryption algorithm are described. The simulation results and security performances are analyzed in Section 4. In Section 5 drawn a brief conclusion.
2. Improper fractional-order laser chaotic system 2.1. Model of the improper fractional-order laser chaotic system
= 𝜎(𝑥2 − 𝑥1 ) = −𝑥2 − 𝛿𝑥3 + (𝛾 − 𝑥4 )𝑥1 = 𝛿𝑥2 − 𝑥3
,
(1)
= −𝑏𝑥4 + 𝑥1 𝑥2
where, x1 , x2 , x3 and x4 represent the state variables, 𝜎, 𝛿, 𝛾 and b mean the parameters of the 4D laser chaotic system (1). According to the definition of the Caputo fractional-order differentiation [48] and system (1), the equations of the fractional-order laser chaotic system is obtained as ⎧𝐷𝑡𝑞 𝑥1 ⎪ 𝑞 ⎪𝐷 𝑡 𝑥 2 ⎨ 𝑞 ⎪𝐷 𝑡 𝑥 3 ⎪ 𝑞 ⎩𝐷 𝑡 𝑥 4
= 𝜎(𝑥2 − 𝑥1 ) = −𝑥2 − 𝛿𝑥3 + (𝛾 − 𝑥4 )𝑥1 = 𝛿𝑥2 − 𝑥3
,
In this section, solutions of the improper fractional-order laser chaotic system are obtained by Adomian decomposition method (ADM). The linear and nonlinear terms of the system (2) are expressed by ⎡𝐿𝑥1 ⎤ ⎡𝜎(𝑥2 − 𝑥1 ) ⎤ ⎢ ⎥ ⎢ ⎥ 𝐿 𝑥 − 𝑥 − 𝛿𝑥 + 𝛾𝑥 3 1⎥ ⎢ 2⎥ = ⎢ 2 , ⎢𝐿𝑥3 ⎥ ⎢𝛿𝑥2 − 𝑥3 ⎥ ⎢𝐿 𝑥 ⎥ ⎢− 𝑏 𝑥 ⎥ 4 ⎣ 4⎦ ⎣ ⎦
(2)
= −𝑏𝑥4 + 𝑥1 𝑥2
where, q is the fractional-order, x1 , x2 , x3 and x4 represent the state variables, 𝜎, 𝛿, 𝛾 and b are the system parameters. If q < 1, the system (2) is the true fractional-order system, in addition, the system is integerorder system when q = 1, when q > 1, the system is called improper fractional-order system.
⎡𝑁 𝑥1 ⎤ ⎡0 ⎤ ⎢ ⎥ ⎢ ⎥ 𝑁 𝑥 − 𝑥 𝑥 ⎢ 2 ⎥ = ⎢ 4 1 ⎥. ⎢𝑁 𝑥3 ⎥ ⎢0 ⎥ ⎢𝑁 𝑥 ⎥ ⎢𝑥 𝑥 ⎥ ⎣ 4⎦ ⎣ 1 2 ⎦
(3)
Based on the ADM, the first seven Adomian polynomials for the nonlinear terms of the system (2) are ⎧𝐴 0 ⎪ −𝑥1 𝑥4 ⎪𝐴1−𝑥 𝑥 ⎪ 2 1 4 ⎪𝐴 − 𝑥 1 𝑥 4 ⎪ 3 ⎨𝐴 − 𝑥 1 𝑥 4 ⎪ 4 ⎪𝐴 − 𝑥 1 𝑥 4 ⎪𝐴 5 ⎪ −𝑥1 𝑥4 ⎪𝐴 6 ⎩ −𝑥 𝑥 1 4
A new 4D laser chaotic system is obtained based on Lorenz-Haken Model in [47], its system equations is expressed as 𝑑𝑥 ⎧ 𝑑𝑡1 ⎪ 𝑑 𝑥2 ⎪ 𝑑𝑡 ⎨ 𝑑 𝑥3 ⎪ 𝑑𝑡 ⎪ 𝑑 𝑥4 ⎩ 𝑑𝑡
2.2. Solutions of the improper fractional-order laser chaotic system
⎧𝐴 0 ⎪ 𝑥1 𝑥2 ⎪𝐴1𝑥 𝑥 ⎪ 21 2 ⎪𝐴 𝑥 1 𝑥 2 ⎪ 3 ⎨𝐴 𝑥 1 𝑥 2 ⎪ 4 ⎪𝐴 𝑥 1 𝑥 2 ⎪𝐴 5 ⎪ 𝑥1 𝑥2 ⎪𝐴 6 ⎩ 𝑥1 𝑥2
= −𝑥01 𝑥04 = −𝑥11 𝑥04 − 𝑥01 𝑥14 = −𝑥21 𝑥04 − 𝑥11 𝑥14 − 𝑥01 𝑥24 = −𝑥31 𝑥04 − 𝑥21 𝑥14 − 𝑥11 𝑥24 − 𝑥01 𝑥34
,
(4)
= −𝑥41 𝑥04 − 𝑥31 𝑥14 − 𝑥21 𝑥24 − 𝑥11 𝑥34 − 𝑥01 𝑥44 = −𝑥51 𝑥04 − 𝑥41 𝑥14 − 𝑥31 𝑥24 − 𝑥21 𝑥34 − 𝑥11 𝑥44 − 𝑥01 𝑥54 = −𝑥61 𝑥04 − 𝑥51 𝑥14 − 𝑥41 𝑥24 − 𝑥31 𝑥34 − 𝑥21 𝑥44 − 𝑥11 𝑥54 − 𝑥01 𝑥64
= 𝑥01 𝑥02 = 𝑥11 𝑥02 + 𝑥01 𝑥12 = 𝑥21 𝑥02 + 𝑥11 𝑥12 + 𝑥01 𝑥22 = 𝑥31 𝑥02 + 𝑥21 𝑥12 + 𝑥11 𝑥22 + 𝑥01 𝑥32
(5)
= 𝑥41 𝑥02 + 𝑥31 𝑥12 + 𝑥21 𝑥22 + 𝑥11 𝑥32 + 𝑥01 𝑥42 = 𝑥51 𝑥02 + 𝑥41 𝑥12 + 𝑥31 𝑥22 + 𝑥21 𝑥32 + 𝑥11 𝑥42 + 𝑥01 𝑥52 = 𝑥61 𝑥02 + 𝑥51 𝑥12 + 𝑥41 𝑥22 + 𝑥31 𝑥32 + 𝑥21 𝑥42 + 𝑥11 𝑥52 + 𝑥01 𝑥62
Setting the initial condition x0 = [x1 (0), x2 (0), x3 (0), x4 (0)], then the first term is ⎧𝑥01 ⎪ 0 ⎪𝑥 2 ⎨ 0 ⎪𝑥 3 ⎪ 0 ⎩𝑥 4
= 𝑥1 (𝑡0 ) = 𝑥2 (𝑡0 )
(6)
= 𝑥3 (𝑡0 ) = 𝑥4 (𝑡0 )
Let 𝑐10 = 𝑥01 , 𝑐20 = 𝑥02 , 𝑐30 = 𝑥03 , 𝑐40 = 𝑥04 , and set c0 = [c1 (0), c2 (0), c3 (0), c4 (0)], then other terms are ⎧𝑐11 ⎪ 1 ⎪𝑐 2 ⎨ 1 ⎪𝑐 3 ⎪ 1 ⎩𝑐 4
= 𝜎(𝑐20 − 𝑐10 ) = −𝑐20 − 𝛿𝑐30 + (𝛾 − 𝑐40 )𝑐10 = 𝛿𝑐20 − 𝑐30 = −𝑏𝑐40 + 𝑐10 𝑐20
,
(7)
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 2. Bifurcation diagram with different parameter intervals: (a) 𝜎 ∈ [3, 7], (b) b ∈ [1, 2], (c) 𝛾 ∈ [10, 35], (d) 𝛿 ∈ [0, 2.5], (e) q ∈ [0.7, 1.03].
Fig. 3. LEs with different parameter intervals: (a) 𝜎 ∈ [3, 7], (b) b ∈ [1, 2], (c) 𝛾 ∈ [10, 35], (d) 𝛿 ∈ [0, 2.5], (e) q ∈ [0.7, 1.03].
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 4. Complexity of the time sequences for q ∈ [0.7, 1.03]: (a) 𝜎 ∈ [3, 7], (b) b ∈ [1, 2], (c) 𝛾 ∈ [10, 35], (d) 𝛿 ∈ [0, 2.5].
⎧𝑐12 ⎪ 2 ⎪𝑐 2 ⎨ 2 ⎪𝑐 3 ⎪ 2 ⎩𝑐 4
= 𝜎(𝑐21 − 𝑐11 )
⎧𝑐 1 ⎪ 3 ⎪𝑐 2 ⎨𝑐 3 ⎪ 3 ⎪𝑐 3 ⎩ 4
= 𝜎(𝑐22 − 𝑐12 )
⎧𝑐 1 ⎪ 4 ⎪𝑐 2 ⎨ 4 ⎪𝑐 3 ⎪𝑐 4 ⎩ 4
= 𝜎(𝑐23 − 𝑐13 )
𝑞+1) = −𝑏𝑐43 + 𝑐13 𝑐20 + (𝑐11 𝑐22 + 𝑐12 𝑐21 ) Γ(𝑞Γ(2 + 𝑐10 𝑐23 +1)(2𝑞 +1)
2.3. Dynamic performances of the improper fractional-order laser chaotic system
⎧𝑐15 ⎪ 5 ⎪𝑐2 ⎨𝑐 5 ⎪ 3 ⎪𝑐 5 ⎩ 4
= 𝜎(𝑐24 − 𝑐14 )
The dynamic characteristics of the improper fractional-order laser chaotic system are analyzed by attractor phase, bifurcation diagram and Lyapunov exponents spectrum (LEs). Setting the parameters of the system 𝜎 = 4, 𝛿 = 0.5, 𝛾 = 27, b = 1.8 and q = 1.005, initial condition x1 (0) = 2, x2 (0) = 1, x3 (0) = 1 and x4 (0) = 2, iteration time step h = 0.001, then the system (2) is iterated 7000 times. The attractor
= 𝛿𝑐21 − 𝑐31 =
−𝑏𝑐41
+
⎧𝑐 1 ⎪ 6 ⎪𝑐 2 ⎨𝑐 6 ⎪ 3 ⎪𝑐 6 ⎩ 4 6
= −𝑐21 − 𝛿𝑐31 + 𝛾𝑐11 − 𝑐41 𝑐10 − 𝑐40 𝑐11 𝑐11 𝑐20
+
,
(8)
𝑐10 𝑐21
= 𝜎(𝑐25 − 𝑐15 ) 𝑞+1) 𝑞+1) = −𝑐25 − 𝛿𝑐35 + 𝛾𝑐15 − 𝑐45 𝑐10 − (𝑐41 𝑐14 + 𝑐44 𝑐11 ) Γ(𝑞Γ(5 − (𝑐43 𝑐12 + 𝑐43 𝑐12 ) Γ(2Γ(5 − 𝑐40 𝑐15 +1)(4𝑞 +1) 𝑞 +1)(3𝑞 +1)
= 𝛿𝑐25 − 𝑐35
𝑞+1) 𝑞+1) = −𝑏𝑐45 + 𝑐15 𝑐20 + (𝑐11 𝑐24 + 𝑐14 𝑐21 ) Γ(𝑞Γ(5 + (𝑐13 𝑐22 + 𝑐13 𝑐22 ) Γ(2Γ(5 + 𝑐10 𝑐25 +1)(4𝑞 +1) 𝑞 +1)(3𝑞 +1)
(12) 3
4
= −𝑐22 − 𝛿𝑐32 + 𝛾𝑐12 − 𝑐42 𝑐10 − 𝑐41 𝑐11 Γ(2𝑞+1)2 − 𝑐40 𝑐12 Γ(𝑞+1)
= 𝛿𝑐22 − 𝑐32
,
(9)
𝑥̃ 𝑗 (𝑡) = 𝑐𝑗0 + 𝑐𝑗1
= −𝑏𝑐42 + 𝑐12 𝑐20 + 𝑐11 𝑐21 Γ(2𝑞+1)2 + 𝑐10 𝑐22 Γ(𝑞+1)
+𝑐𝑗4
𝑞+1) = −𝑐23 − 𝛿𝑐33 + 𝛾𝑐13 − 𝑐43 𝑐10 − (𝑐41 𝑐12 + 𝑐42 𝑐11 ) Γ(𝑞Γ(3 − 𝑐40 𝑐13 +1)(2𝑞 +1)
= 𝛿𝑐23 − 𝑐33
,
𝑞+1) 𝑞+1) = −𝑐24 − 𝛿𝑐34 + 𝛾𝑐14 − 𝑐44 𝑐10 − (𝑐41 𝑐13 + 𝑐43 𝑐11 ) Γ(𝑞Γ(4 − 𝑐42 𝑐12 Γ(3 − 𝑐40 𝑐14 +1)(3𝑞 +1) Γ(𝑞+1)3
= 𝛿𝑐24 − 𝑐34 =
−𝑏𝑐44
+
𝑐14 𝑐20
Therefore, the solution of the improper fractional-order laser chaotic system (2) is expressed as
+
(𝑐11 𝑐23
+
𝑞+1) 𝑐13 𝑐21 ) Γ(𝑞Γ(4 +1)(3𝑞 +1)
+
𝑞+1) 𝑐12 𝑐22 Γ(3 Γ(𝑞+1)3
+
(10)
,
𝑐10 𝑐24
(11)
ℎ𝑞 ℎ2𝑞 ℎ3𝑞 + 𝑐𝑗2 + 𝑐𝑗3 Γ(𝑞 + 1) Γ(2𝑞 + 1) Γ(3𝑞 + 1)
ℎ4𝑞 ℎ5𝑞 ℎ6𝑞 + 𝑐𝑗5 + 𝑐𝑗6 , Γ(4𝑞 + 1) Γ(5𝑞 + 1) Γ(6𝑞 + 1)
(13)
where, j=1,2,3,4, h is the time step.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 5. Sequences time domain waveform: (a) sequence x1 , (b) sequence x2 , (c) sequence x3 , (d) sequence x4 .
phases of the system (2) with different planes are shown in Fig. 1. As we can see, the attractor trajectories of the improper fractional-order laser chaotic system are distributed over a large area, which means that the improper fractional-order laser chaotic system has good ergodic property. To explore the parameter intervals of the system (2) dynamics characteristics, we fix 𝜎 ∈ [3, 7], b ∈ [1, 2], 𝛾 ∈ [10, 35], 𝛿 ∈ [0, 2.5] and q ∈ [0.7, 1.03], and other conditions as above. The bifurcation diagram and LEs with different parameter intervals as shown in Figs. 2 and 3. The bifurcation diagrams and LEs illustrate that the improper fractionalorder laser chaotic system has complex dynamic performances. In addition, the randomness of time sequences is tested by C0 complexity algorithm. The C0 complexity algorithm is described as follows. Step 1 Discrete fourier transformation of the sequence 𝑥(𝑛), 𝑛 = 0, 1, 2, … 𝑁 − 1 is 𝑋(𝑘) =
𝑁−1 ∑
2𝜋
𝑥(𝑛)−𝑗 𝑁 𝑛𝑘 =
𝑛=0
𝑁−1 ∑ 𝑛=0
𝑥(𝑛)𝑊𝑁𝑛𝑘 ,
(14)
where k = 0, 1, 2, … , 𝑁 − 1. Step 2 The X(k) is processed by ⎧ ⎪ 𝑋(𝑘) ⎪ 𝑋̃ (𝑘) = ⎨ ⎪ 0 ⎪ ⎩
|𝑋(𝑘)|2 > 𝑟 𝑁1 |𝑋(𝑘)| ≤ 2
𝑟 𝑁1
𝑁−1 ∑ 𝑛=0 𝑁−1 ∑ 𝑛=0
|𝑋(𝑘)|2 (15) |𝑋(𝑘)|
2
where k = 0, 1, 2, … , 𝑁 − 1, r is the introduced parameter.
Step 3 Inverse Fourier transform of X(k) is 𝑥̃ (𝑘) =
𝑁−1 𝑁−1 2𝜋 1 ∑ ̃ 1 ∑ ̃ 𝑋 (𝑘)𝑒𝑗 𝑁 𝑛𝑘 = 𝑋 (𝑘)𝑊𝑁−𝑛𝑘 , 𝑁 𝑛=0 𝑁 𝑛=0
where n = 0, 1, 2, … , 𝑁 − 1. Step 4 The C0 complexity is expressed as ∑𝑁−1 |𝑥(𝑛) − 𝑥̃ (𝑛)|2 𝐶0 (𝑟, 𝑁) = 𝑛=0 . ∑𝑁−1 2 𝑛=0 |𝑥(𝑛)|
(16)
(17)
Keeping the above parameters and initial values, then according to the C0 complexity algorithm, the complexity of chaotic sequences is shown in Fig. 4. This shows that sequences generated by the improper fractional-order laser chaotic system have good randomness. The sensitivity of the improper fractional-order laser chaotic system to the initial values are analyzed by time domain waveform. Keeping the other initial values, a initial value is respectively changed 10−5 and 10−7 . Then getting the sequences time domain waveform is Fig. 5. The results indicate that the sequences generated by the improper fractional-order laser chaotic system when initial value slight change are significantly different. Therefore, the improper fractional-order laser chaotic system to the initial values are extremely sensitive. 2.4. DSP Implementation of the improper fractional-order laser chaotic system Due to DSP digital signal processor has the advantages of fast processing speed, strong programmable, high anti-interference and easy to
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 6. Attractors phase with different planes: (a) 𝑥1 − 𝑥2 plane, (b) 𝑥1 − 𝑥4 plane, (c) 𝑥2 − 𝑥4 plane.
Fig. 7. The flowchart of encryption algorithm.
realize, in this paper, DSP platform is used to digital hardware implementation of the improper fractional-order laser chaotic system. DSP platform and the corresponding of results are shown in Fig. 6. The results illustrate that the chaotic signal generated by DSP platform is consistent with the MATLAB simulation results under the same system parameters and initial condition, which realizes the digital hardware implementation of the improper fractional-order laser chaotic system.
3. Image encryption algorithm of the improper fractional-order laser chaotic system 3.1. Encryption algorithm In this section, we designed an image encryption algorithm based on the improper fractional-order laser chaotic system. The flowchart of the proposed image encryption algorithm is given in Fig. 7. The corresponding detailed process is expressed as follows.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 8. The flowchart of decryption algorithm.
Step 1: Inputting a plain image I and it’s size is defined to M × N. Step 2: Setting the parameter and initial values of the improper fractional-order laser chaotic system. The system Eq. (2) is iterated (𝐿 = 𝑀 × 𝑁) times. Then improper fractional-order laser chaotic sequences {x1 (i), x2 (i), x3 (i), x4 (i)} are obtained, here they both have length L. Step 3: The sequences x1 (i) and x2 (i) are processed by ⎧ 16 ⎪𝑏𝑟 = mod (𝑓 𝑙𝑜𝑜𝑟(abs(𝑥1 (𝑖)) × 10 , 𝑀∕2) ⎪𝑏𝑐 = mod (𝑓 𝑙𝑜𝑜𝑟(abs(𝑥2 (𝑖)) × 1016 , 𝑁∕2) , ⎨𝑋 = 𝑥 (1 ∶ 𝑀) 1 ⎪ ⎪𝑌 = 𝑥2 (1 ∶ 𝑁) ⎩
(18)
where br, bc, X and Y are used to scramble the plane image I. Step 4: According to the br and X, the scrambled row image T is generated through ⎧𝑋(𝑖) > 0 ⎪ ⎪𝑇 (𝑖, end − 𝑏𝑟(𝑖) + 1 ∶ end) = 𝐼(𝑖, 1 ∶ 𝑏𝑟(𝑖)) ⎪𝑇 (𝑖, 1 ∶ end − 𝑏𝑟(𝑖)) = 𝐼(𝑖, 𝑏𝑟(𝑖) + 1 ∶ end) , ⎨𝑋(𝑖) < 0 ⎪ ⎪𝑇 (𝑖, 1 ∶ 𝑏𝑟(𝑖)) = 𝐼(𝑖, end − 𝑏𝑟(𝑖) + 1 ∶ end) ⎪𝑇 (𝑖, 𝑏𝑟(𝑖) + 1 ∶ end) = 𝐼(𝑖, 1 ∶ end − 𝑏𝑟(𝑖)) ⎩ where 𝑖 = 1, 2, … , 𝑀.
Step 5: Based on bc and Y, the scrambled column image T1 is obtained by
⎧𝑌 ( 𝑖 ) > 0 ⎪ ⎪𝑇 1(end − 𝑏𝑐(𝑗) + 1 ∶ end, 𝑖) = 𝑇 (1 ∶ 𝑏𝑟(𝑗 ), 𝑗 ) ⎪𝑇 1(1 ∶ end − 𝑏𝑐(𝑗 ), 𝑗 ) = 𝑇 (𝑏𝑐(𝑗) + 1 ∶ end, 𝑗) , ⎨𝑌 ( 𝑖 ) < 0 ⎪ ⎪𝑇 1(1 ∶ 𝑏𝑐(𝑗 ), 𝑗 ) = 𝑇 (end − 𝑏𝑐(𝑗) + 1 ∶ end, 𝑗) ⎪𝑇 1(𝑏𝑐(𝑗) + 1 ∶ end, 𝑗) = 𝑇 (1 ∶ end − 𝑏𝑟(𝑗), 𝑗) ⎩
where 𝑗 = 1, 2, … , 𝑁. Step 6: Two matrixes Z and W based on two chaotic sequences x3 (i) and x4 (i) are given as {
(19)
(20)
𝑍(𝑖, 𝑗) = mod (𝑓 𝑙𝑜𝑜𝑟( mod (𝑥3 ((𝑖 − 1) × 𝑁 + 𝑗 + 500, 1) × 1016 )), 256) , 𝑊 (𝑖, 𝑗) = mod (𝑓 𝑙𝑜𝑜𝑟( mod (𝑥4 ((𝑖 − 1) × 𝑁 + 𝑗 + 500, 1) × 1016 )), 256) (21)
where 𝑖 = 1, 2, … , 𝑀. 𝑗 = 1, 2, … , 𝑁.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 9. Encryption and decryption test results: (a) “Couple”, “Camera”, “Lake” and “Boat” of original image, (b) “Couple”, “Camera”, “Lake” and “Boat” of encrypted image, (c) “Couple”, “Camera”, “Lake” and “Boat” of decrypted image.
Step 7: The diffused image T2 is obtained by using matrix Z and Eq. (22). ⎧ ⎪𝑇 2(𝑀 , 𝑁 ) = 𝑇 1(𝑀 , 𝑁 ) + 𝑍(𝑀 , 𝑁 ) mod 256 ⎪𝑇 2(𝑀, 𝑗) = 𝑇 1(𝑀, 𝑗) + 𝑇 2(𝑀, 𝑗 + 1) + 𝑍(𝑀, 𝑗) mod 256 , ⎨𝑇 2(𝑖, 𝑁) = 𝑇 1(𝑖, 𝑁) + 𝑇 2(𝑖 + 1, 𝑁) + 𝑍(𝑖, 𝑁) mod 256 ⎪ ⎪𝑇 2(𝑖, 𝑗) = 𝑇 1(𝑖, 𝑗) + 𝑇 2(𝑖 + 1, 𝑗) + 𝑇 2(𝑖, 𝑗 + 1) + 𝑍(𝑖, 𝑗) mod 256 ⎩
(22)
where 𝑖 = 𝑀 − 1, 𝑀 − 2, … , 2, 1. 𝑗 = 𝑁 − 1, 𝑁 − 2, … , 2, 1. Step 8: The diffused image T2 is scrambled by Zigzag algorithm. Then we get a confused image B. Step 9: The image B is changed into C through matrix W in step 6. The specific operation is expressed as follows: If 𝑖 = 𝑀, 𝑗 = 𝑁. 𝐶(𝑖, 𝑗) = 𝐵(𝑖, 𝑗) + 𝑊 (𝑖, 𝑗) mod 256.
(26)
When 𝑗 = 𝑗 − 1, j > 1. 𝐶1(𝑖, 𝑗) = 𝐵(𝑖, 𝑗) − 𝐶1(𝑖, 𝑗 + 1) − 𝑊 (𝑖, 𝑗) mod 256.
(27)
Other, 𝑗 = 𝑁, 𝑖 = 𝑖 − 1 and i ≥ 1. 𝐶1(𝑖, 𝑗) = 𝐵(𝑖, 𝑗) − 𝑠𝑢𝑚(𝐶1(𝑖 + 1, 1 ∶ 𝑁)) − 𝑊 (𝑖, 𝑗) mod 256.
(24)
Step 5: The scrambled image C1 is recovered to T1 by inverse zigzag algorithm. Step 6: The image T1 is restored by
Other, 𝑗 = 𝑁, 𝑖 = 𝑖 − 1 and i ≥ 1. 𝐶(𝑖, 𝑗) = 𝐵(𝑖, 𝑗) + 𝑠𝑢𝑚(𝐶(𝑖 + 1, 1 ∶ 𝑁)) + 𝑊 (𝑖, 𝑗) mod 256.
𝐶1(𝑖, 𝑗) = 𝐶(𝑖, 𝑗) − 𝑊 (𝑖, 𝑗) mod 256.
(23)
When 𝑗 = 𝑗 − 1, j > 1. 𝐶(𝑖, 𝑗) = 𝐵(𝑖, 𝑗) + 𝐶(𝑖, 𝑗 + 1) + 𝑊 (𝑖, 𝑗) mod 256.
Step 1: Inputting the cipher image C(M × N). Step 2: As in step 2 of encryption algorithm, four chaotic sequences {x1 (i), x2 (i), x3 (i), x4 (i)} are obtained. Step 3: According to step 6 of encryption algorithm, two matrixes Z and W are generated. Step 4: The restored diffusion image C1 is obtained based on the following principles. If 𝑖 = 𝑀, 𝑗 = 𝑁.
(25)
Step 10: The matrix C is cipher image. 3.2. Decryption algorithm Fig. 8 is a flowchart of decryption algorithm. Decryption algorithm refer to restor the cipher image process. In this paper, the detailed steps of the decryption algorithm is expressed as follows.
(28)
⎧ ⎪𝑇 2(𝑀 , 𝑁 ) = 𝑇 1(𝑀 , 𝑁 ) + 256 × 2 − 𝑍(𝑀 , 𝑁 ) mod 256 ⎪𝑇 2(𝑀, 𝑗) = 𝑇 1(𝑀, 𝑗) + 256 × 2 − 𝑇 2(𝑀, 𝑗 + 1) − 𝑍(𝑀, 𝑗) mod 256 , ⎨𝑇 2(𝑖, 𝑁) = 𝑇 1(𝑖, 𝑁) + 256 × 2 − 𝑇 2(𝑖 + 1, 𝑁) − 𝑍(𝑖, 𝑁) mod 256 ⎪ ⎪𝑇 2(𝑖, 𝑗) = 𝑇 1(𝑖, 𝑗) + 256 × 3 − 𝑇 2(𝑖 + 1, 𝑗) − 𝑇 2(𝑖, 𝑗 + 1) − 𝑍(𝑖, 𝑗) mod 256 ⎩ (29) where 𝑖 = 𝑀 − 1, 𝑀 − 2, … , 2, 1. 𝑗 = 𝑁 − 1, 𝑁 − 2, … , 2, 1. Step 7: br, bc, X and Y are obtained based on step 3 of encryption algorithm.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 10. Histograms: (a) “Couple”, “Camera”, “Lake” and “Boat” of original image,(b) “Couple”, “Camera”, “Lake” and “Boat” of encrypted image, (c) “Couple”, “Camera”, “Lake” and “Boat” of decrypted image.
Step 8: The scrambled image T2 of rows and columns is restored through ⎧𝑌 ( 𝑖 ) < 0 ⎪ ⎪𝑇 1(end − 𝑏𝑐(𝑗) + 1 ∶ end, 𝑖) = 𝑇 2(1 ∶ 𝑏𝑟(𝑗 ), 𝑗 ) ⎪𝑇 1(1 ∶ end − 𝑏𝑐(𝑗 ), 𝑗 ) = 𝑇 2(𝑏𝑐(𝑗) + 1 ∶ end, 𝑗) , ⎨𝑌 ( 𝑖 ) > 0 ⎪ ⎪𝑇 1(1 ∶ 𝑏𝑐(𝑗 ), 𝑗 ) = 𝑇 2(end − 𝑏𝑐(𝑗) + 1 ∶ end, 𝑗) ⎪𝑇 1(𝑏𝑐(𝑗) + 1 ∶ end, 𝑗) = 𝑇 2(1 ∶ end − 𝑏𝑟(𝑗), 𝑗) ⎩
4. Experiment test results and security performance analysis
(30)
where 𝑗 = 1, 2, … , 𝑁. ⎧𝑋(𝑖) < 0 ⎪ ⎪𝑇 (𝑖, end − 𝑏𝑟(𝑖) + 1 ∶ end) = 𝑇 1(𝑖, 1 ∶ 𝑏𝑟(𝑖)) ⎪𝑇 (𝑖, 1 ∶ end − 𝑏𝑟(𝑖)) = 𝑇 1(𝑖, 𝑏𝑟(𝑖) + 1 ∶ end) , ⎨𝑋(𝑖) > 0 ⎪ ⎪𝑇 (𝑖, 1 ∶ 𝑏𝑟(𝑖)) = 𝑇 1(𝑖, end − 𝑏𝑟(𝑖) + 1 ∶ end) ⎪𝑇 (𝑖, 𝑏𝑟(𝑖) + 1 ∶ end) = 𝑇 1(𝑖, 1 ∶ end − 𝑏𝑟(𝑖)) ⎩ where 𝑖 = 1, 2, … , 𝑀.
Step 9: The ciphertext image C is restored to plane image T.
(31)
To prove the effectiveness and the security of the image algorithm based on improper fractional-order laser chaotic system, the corresponding of simulation experiments are realized on MATLAB (version R2018a). In this experiment, the “Couple”, “Camera”, “Lake” and “Boat” of size 256 × 256 gray images are tested images of the proposed image encryption algorithm. The parameters of the improper fractional-order laser chaotic are selected 𝜎 = 4, 𝛿 = 0.5, 𝛾 = 27, 𝑏 = 1.8 and 𝑞 = 1.005, initial values are 𝑥1 (0) = 2, 𝑥2 (0) = 1, 𝑥3 (0) = 1 and 𝑥4 (0) = 2. In decryption algorithm, the parameter and initial values are the same as the encryption algorithm. The original grayscale images “Couple”, “Camera”, “Lake” and “Boat” are Fig. 9(a). The ciphertext images are shown in Fig. 9(b). The corresponding of restor ciphertext images as shown in Fig. 9(c). As we can see from Fig. 9, the visual information of the plain-
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 11. The correlation distribution of images in horizontal, vertical and diagonal directions: (a) Plaintext “Couple” image, (b) Ciphertext “Couple” image, (c) Plaintext “Camera” image, (d) Ciphertext “Camera” image, (e) Plaintext “Lake” image, (f) Ciphertext “Lake” image, (g) Plaintext “Boat” image, (h) Ciphertext “Boat” image.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
text image cannot be seen in the ciphertext image, in addition, there is no significant difference between the restored ciphertext images and the plaintext images. Therefore, the proposed encryption algorithm could effectively hide image information. 4.1. Histogram analysis Histogram is an important tool to analyze statistical attack. Fig. 10(a) are the histograms of plaintext images “Couple”, “Camera”, “Lake” and “Boat”. The corresponding of the histograms of ciphertext images are Fig. 10(b). The histograms of restored ciphertext images as shown in Fig. 10(c). The results indicate that the pixel values distribution of the plaintext images are concentrated in some certain ranges, while the pixel values distribution of encrypted images are uniform, in addition, pixel values distribution of the recovered ciphertext images are the same as original images. So the ciphertext images can resist the histogram attack. 4.2. Chi-square test To further verify uniform distribution of the histograms with encrypted image, the Chi-square test is adopted. The computational formula of 𝜒 2 test is expressed by 255 ∑ (𝐶𝑛 − 𝐺)2 𝜒2 = , 𝐺 𝑛
Table 1 Chi-square test results of images. Images
Couple
Plaitext Ciphertext Decision
where Cn represents actual amount of each grayscale level, G means the desired number of each grayscale level. If the histogram distribution of ciphertext image more uniformly is, the Chi-square value is smaller. It shows that it can pass the Chi-square test when the confidence level is
5.4617 × 10 236.3203 pass
Lake
1.1097 × 10 289.5234 pass
5
4.8806 × 10 249.3359 pass
Boat 4
1.0067 × 105 242.0234 pass
0.05 and the Chi-square value not exceed 293.2478. In this experimental, the Chi-square test results of the different images as Table 1. The results indicate that histogram distribution of the ciphertext image are uniformly. 4.3. Correlation of adjacent pixels The correlation coefficient of two pixels refer to the degree of correlation for the between different pixels. The correlation coefficient of the adjacent pixels for the image is calculated by 𝑛 ∑
𝑟= √
𝑖=1 𝑛 ∑ 𝑖=1
(32)
Camera 4
(𝑥𝑖 −
(𝑥𝑖 −
1 𝑛
1 𝑛
∑𝑛
𝑖=1
∑𝑛
𝑥𝑖 )(𝑦𝑖 − 𝑛 2 ∑
𝑖=1 𝑥𝑖 )
𝑖=1
1 𝑛
∑𝑛
(𝑦𝑖 −
𝑖=1 𝑦𝑖 )
1 𝑛
∑𝑛
,
(33)
2
𝑖=1 𝑦𝑖 )
where xi and yi represent the gray value of the two adjacent pixels. n is the number of the different pixels pairs. To measure the pixel correlation of encrypted image, in this test, we randomly select 2000 pixels in plaintext image and ciphertext image to calculate the correlation coefficients in horizontal, vertical and diagonal directions. The correlation distribution of images is Fig. 11. The Fig. 12. Key sensitivity analysis results: (a1) 𝜎 + 10−15 , (a2) 𝛿 + 10−15 , (a3) 𝛾 + 10−15 , (a4) 𝑞 + 10−15 , (a5) 𝑏 + 10−15 , (a6) 𝑥1 (0) + 10−15 , (a7) 𝑥2 (0) + 10−15 , (a8) 𝑥3 (0) + 10−15 , (a9) 𝑥4 (0) + 10−15 .
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Table 2 Correlation coefficients of different images. Images
Horizontal
Plain image Vertical
Diagonal
Horizontal
Cipher image Vertical
Diagonal
Couple Camera Lake Boat
0.9337 0.9597 0.9568 0.9451
0.9255 0.9342 0.9584 0.9268
0.8773 0.9062 0.9295 0.8835
−0.0095 −0.0003 0.0044 −0.0037
−0.0044 −0.0052 −0.0047 −0.0009
0.0070 −0.0027 −0.0021 −0.0027
Table 3 Correlation coefficients of different algorithms. Algorithm
Images
Horizontal
Vertical
Diagonal
Ours Ref [14] Ref [15] Ref [16] Ref [17] Ours Ref [18] Ref [19] Ref [20] Ref [21] Ours Ref [18] Ref [19] Ref [20] Ref [21] Ours Ref [14] Ref [15] Ref [16] Ref [17]
Encrypted “Couple”
−0.0095 0.0086 −0.0382 0.1539 −0.0290 −0.0003 0.0079 −0.0264 0.0341 0.0685 0.0044 −0.0012 −0.0306 0.0600 0.2071 −0.0037 0.0086 0.0250 0.1830 −0.0037
−0.0044 0.0060 0.0083 0.1342 −0.0166 −0.0052 −0.0005 0.26375 −0.0509 0.0821 −0.0047 0.0095 0.3188 0.0196 0.2128 −0.0009 0.0060 0.0186 0.1316 −0.0177
0.0070 0.0017 0.0277 −0.0029 0.0281 −0.0027 −0.0044 0.0272 −0.0216 0.0821 −0.0021 −0.0093 −0.0137 −0.0231 0.2102 −0.0027 0.0017 −0.0051 0.0599 −0.0131
Encryption “Camera”
Encrypted “Lake”
Encrypted “Boat”
Table 5 Information entropy of ciphertext images with different algorithms.
Images
Couple
Camera
Lake
Boat
Plaitext Ciphertext
7.4000 7.9974
7.0097 7.9968
7.4584 7.9973
7.1587 7.9973
The information entropy reflect the uncertainty of image information. It is calculated by 𝑝(𝑦𝑗 )log2 (𝑝(𝑦𝑗 )),
Ref [16]
Ref [17]
7.9953 7.9948 7.9948 7.9956
7.4101 7.4740 7.7877 7.4232
7.9937 7.9938 7.9939 7.9944
4.6. Key sensitivity analysis
4.4. Information entropy
𝑗=1
Ref [15]
7.9974 7.9968 7.9973 7.9973
The size of the key space shows the ability of the resist violent attacks for encryption algorithm. Theoretically, for a good encryption algorithm, its key space should be larger than 2100 . In this paper, the key space of the proposed encryption algorithm is composed of 𝜎, 𝛿, 𝛾, q, b, x1 (0), x2 (0), x3 (0) and x4 (0). The experimental results find that the key space for 𝜎, 𝛿, 𝛾, q, b, x1 (0), x2 (0), x3 (0) and x4 (0) is about 1015 . So all key space of the proposed encryption algorithm is 2449 much large than 2100 , which shows that the key space of the proposed encryption algorithm is large enough to resist violent attacks.
correlation coefficients are shown in Table 2. The results show that the correlation adjacent pixels of original image at horizontal, vertical and diagonal directions are almost close to 1, distribution is linear 𝑥 = 𝑦. However, the correlation coefficients adjacent pixels of ciphertext image are almost 0, distribution in the whole plane. Therefore, the proposed algorithm can effectively destroy correlation of the between adjacent pixels. In addition, comparison results of correlation coefficients with existing algorithms [14–21]are listed in Table 3.
𝑀 ∑
Ours
Couple Camera Lake Boat
4.5. Key space analysis
Table 4 Information entropy test results of images.
ℎ=−
Images
(34)
where yj represent the jth grayscale value of the M-level grayscale image, p(yj ) is the probability of the yj . The information entropy value is 8 for an ideal encrypted image, which indicates that the ciphertext image is entirely random. The Table 4 listed information entropy values of the different images. The Table 4 illustrate that information entropy values of encrypted images are very close to 8, therefore, the proposed encryption algorithm can resist entropy value attacks. In addition, comparison results of information entropy with existing algorithms [15–17]are listed in Table 5.
For the chaotic cryptosystem, the sensitivity of chaotic system to initial value and parameter value determines the sensitivity of encryption system to the key. In this experimental, the decryption key 𝜎, 𝛿, 𝛾, q, b, x1 (0), x2 (0), x3 (0) and x4 (0) are respectively changed 10−15 , then decryption “Boat” images as Fig. 12. It shows that the decryption image has not any valuable information of original image, which indicate that the proposed image algorithm is extremely sensitivity for the key. 4.7. Differential attacks The Number of Pixels Change Rate (NPCR) and the Unified Average Changing Intensity (UACI) are two methods to test the ability of the differential attack. The corresponding of calculation equation is { 0, 𝐶1 (𝑖, 𝑗) = 𝐶2 (𝑖, 𝑗) 𝐷(𝑖, 𝑗) = , (35) 1, 𝐶1 (𝑖, 𝑗) ≠ 𝐶2 (𝑖, 𝑗) ∑ D(i,j) ⎧ ⎪UPCR = i,j M×N × 100% , ⎨ ∑ |𝐶1 (𝑖,𝑗 )−𝐶2 (𝑖,𝑗 )| ⎪UACI = 𝑖,𝑗 255×𝑀×𝑁 × 100% ⎩
(36)
where C1 (i, j) represents the pixel value of the normal ciphertext image. C2 (i, j) is the pixel value of the encrypted image for the changed pixel value of plaintext image. Recently, Wu et al. [49] proposed the criterion values of NPCR and UACI. If a significance level is 𝛼, its critical NPCR value 𝑁𝛼∗ is calculated by √ 𝐹 − Φ−1 (𝛼) 𝐹 ∕𝐿 𝑁𝛼∗ = . (37) 𝐹 +1 For an image encryption algorithm, if the NPCR value is more than 𝑁𝛼∗ , which shows that it has a high ability to resist differential attack.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Fig. 13. The Anti-shear attack analysis results: (a) Ciphertext “Couple” image of data loss, (b) Restored “Couple” image, (c) Ciphertext “Camera” image of data loss, (d) Restored “Camera” image, (e) Ciphertext “Lake” image of data loss, (f) Restored “Lake” image, (g) Ciphertext “Boat” image of data loss, (h) Restored “Boat” image.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Table 6 The theoretical values of critical NPCR and UACI for different size of images. Images size
NPCR (%) 𝑁0∗.05
𝑁0∗.01
𝑁0∗.001
UACI(%) (𝑈0∗− , 𝑈0∗+ ) .05 .05
(𝑈0∗− , 𝑈0∗+ ) .01 .01
(𝑈0∗− , 𝑈0∗+ ) .001 .001
256 × 256 512 × 512 1024 × 1024
99.5693 99.5893 99.5994
99.5527 99.5810 99.5952
99.5341 99.5717 99.5906
(33.2824, 33.6447) (33.3730, 33.5541) (33.4183, 33.5088)
(33.2255, 33.7016) (33.3445, 33.5826) (33.4040, 33.5231)
(33.1594, 33.7677) (33.3115, 33.6156) (33.3875, 33.5396)
The critical UACI values (𝑈𝛼∗ −, 𝑈𝛼∗ +) for the significance level 𝛼 are obtained as { ∗− 𝑈𝛼 = 𝜇𝑢 − Φ−1 (𝛼∕2)𝜎𝑢 , (38) 𝑈𝛼∗+ = 𝜇𝑢 + Φ−1 (𝛼∕2)𝜎𝑢
NPCR and UACI are shown in Tables 7 and 8. What’s more, Table 9 listed the comparison results of NPCRs and UACIs with existing algorithms [15,16,18,22]. The analysis results illustrate that the proposed encryption scheme has ability to resist differential attacks.
where 𝐹 +2 𝜇𝑢 = , 3𝐹 + 3
4.8. Robustness analysis (39)
in addition 𝜎𝑢 =
(𝐹 + 2)(𝐹 2 + 2𝐹 + 3) 18(𝐹 + 1)2 𝐹 𝐿
.
(40)
If the UACI value is more than 𝑈𝛼∗− and it is less than 𝑈𝛼∗+ , which shows that encryption algorithm can pass test and resist differential attack. Theoretical values of critical NPCR and UACI for different size of images are calculated in [49], the corresponding of results are listed in Table 6. In this experiment, the pixel of original image is randomly selected and changed its pixel value. For the “Couple”, “Camera”, “Lake” and “Boat” of different images, we test 100 times, calculate the results of
In this section, the anti-shear attack and anti-noise pollution attack are used to analyze the robustness of the proposed algorithm. To measure the Anti-shear attack of the algorithm, the ciphertext images lost with varying degrees of data, then ciphertext images of loss data are restored. The corresponding of results are shown in Fig. 13. The results illustrate that although the restored ciphertext images have noise, however, it does not affect the effect of overall image. So the proposed scheme can resist cropping attack of different degree. To test anti-noise pollution attack ability of the proposed algorithm, the ciphertext images are added by the Gaussian noise with intensity of 0.05, 0.06, 0.07 and 0.08, then to decrypt the ciphertext image of the Gaussian noise with pollution, the results are Fig. 14. The results illustrate that the proposed algorithm can resist the noise pollution. Fig. 14. The anti-noise pollution attack analysis results: (a) Restored “Couple” image, (b) Restored “Camera” image, (c) Restored “Lake” image, (d) Restored “Boat” image.
F. Yang, J. Mou and C. Ma et al.
Optics and Lasers in Engineering 129 (2020) 106031
Table 7 NPCR of the ciphertext images. Critical NPCR(%) Images
Minimum(%)
Maximum(%)
Mean(%)
Couple Camera Lake Boat
98.84 99.54 99.49 99.54
99.95 99.98 99.90 99.68
99.58 99.64 99.64 99.61
𝑁0∗.05 = 99.5693%
𝑁0∗.01 = 99.5527%
𝑁0∗.001 = 99.5341%
pass pass pass pass
pass pass pass pass
pass pass pass pass
Table 8 UACI of the ciphertext images. Critical UACI(%) Images
Minimum(%)
Maximum(%)
Mean(%)
= 33.2824% U∗− 0.05 = 33.6447% U∗+ 0.05
Couple Camera Lake Boat
34.18 32.19 32.46 33.13
33.21 34.88 34.65 33.77
33.55 33.27 33.54 33.43
pass pass pass pass
U∗− = 33.2255% 0.01 U∗+ = 33.7016% 0.01
U∗− = 33.1594% 0.001 U∗+ = 33.7677% 0.001
pass pass pass pass
pass pass pass pass
Table 9 NPCRs and UACIs of the ciphertext images with different encryption methods. Images
Ref[15] NPCRs(%), UACIs(%)
Ref[16] NPCRs(%), UACIs(%)
Ref[18] NPCRs(%), UACIs(%)
Ref[22] NPCRs(%), UACIs(%)
Ours NPCRs(%), UACIs(%)
Couple Camera Lake Boat
99.49, 99.47, 99.50, 99.47,
98.31, 98.43, 98.49, 98.26,
No 99.61, 33.47 99.61, 33.45 No
99.62, 99.59, 99.58, 99.61,
99.58, 99.64, 99.64, 99.61,
33.44 33.31 33.68 33.45
16.78 16.77 18.27 16.17
33.40 33.56 33.57 33.28
33.55 33.27 33.54 33.43
5. Conclusion
Acknowledgements
Dynamic characteristics of an improper fractional-order laser chaotic system and its image encryption application are investigated in this paper. The dynamic analysis illustrate that chaotic state of the improper fractional-order laser chaotic system distribution is in a large parameters range when parameters and order are changed, what’s more, the chaotic sequences are generated by the improper fractional-order laser system have good pseudo-randomness. Therefore, the improper fractional-order laser systems are more suitable for chaotic security communication. The chaotic system is hardware implementation based on DSP platform, which provide theoretical direction for chaotic system in secure communication applications. The encryption algorithm performances results indicate that the proposed encryption algorithm not only has the superior image encryption effect, but also improves the anti-cracking ability, which illustrates that the proposed algorithm more suit for image encryption, further illustrate that the improper fractional-order laser chaotic system is suitable for image encryption and other security communication.
This work is supported by the Basic Scientific Research Projects of Colleges and Universities of Liaoning Province (Grant Nos. 2017J045); Provincial Natural Science Foundation of Liaoning (Grant Nos. 20170540060).
Author Statement Feifei yang designed and carried out experiments, data analyzed and manuscript wrote. Jun Mou made the theoretical guidance for this paper. Chenguang Ma did the DSP implementation of the laser chaotic system. Yinghong Cao improved the algorithm. All authors reviewed the manuscript.
Declaration of Competing Interest No conflict of interest about the publication by all authors.
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