Signal Processing 149 (2018) 148–161
Contents lists available at ScienceDirect
Signal Processing
journal homepage: www.elsevier.com/locate/sigpro
2D Logistic-Sine-coupling map for image encryption
Zhongyun Hua, Fan Jin, Binxuan Xu, Hejiao Huang∗
School of Computer Science and Technology, Harbin Institute of Technology Shenzhen Graduate School, Shenzhen 518055, China
article info
Article history:
Received 20 November 2017 Revised 4 March 2018 Accepted 16 March 2018 Available online 16 March 2018
Keywords:
Chaos
Cryptography
Image encryption Multimedia security
1. Introduction
With the rapid development of digital technology, more and more multimedia information is generated and spread in the In- ternet [1]. Among all these multimedia information, digital image is an information format that can carry information with visual- ized way. For these digital images transmitted in networks, many of them are private images. For example, the personal medical im- ages are usually private images, as they contain the information of personal healthy conditions. If these private images are obtained by some unauthorized ways, serious security disasters may hap- pen. Thus, it is important to protect these private images [2–4] and image encryption is one efficient technology to protect them [5–8].
One strategy of encrypting image is to treat an image as a binary data stream and then use the developed data encryption algorithms to encrypt the data stream. These algorithms include the well-known data encryption standard [9], advanced encryption standard [10]. However, image data has many unique character- istics such as large data volume, high correlation and strong re- dundancy [11,12]. Treating an image as a binary stream will miss these characteristics, and thus may make the encryption inefficient.
∗ Corresponding author.
E-mail addresses: huazyum@gmail.com, huazhongyun@hit.edu.cn (Z. Hua),
huanghejiao@hit.edu.cn, hjhuang@aliyun.com (H. Huang).
https://doi.org/10.1016/j.sigpro.2018.03.010
0165-1684/© 2018 Published by Elsevier B.V.
abstract
Image encryption is a straightforward strategy to protect digital images by transforming images into un- recognized ones. The chaos theory is a widely used technology for image encryption as it has many significant properties such as ergodicity and initial state sensitivity. When chaotic systems are used in image encryption, their chaos performance highly determines the security level. This paper presents a two-dimensional (2D) Logistic-Sine-coupling map (LSCM). Performance estimations demonstrate that it has better ergodicity, more complex behavior and larger chaotic range than several newly developed 2D chaotic maps. Utilizing the proposed 2D-LSCM, we further propose a 2D-LSCM-based image encryption algorithm (LSCM-IEA), which adopts the classical confusion-diffusion structure. A permutation algorithm is designed to permutate image pixels to different rows and columns while a diffusion algorithm is devel- oped to spread few changes of plain-image to the whole encrypted result. We compare the efficiency of LSCM-IEA with several advanced algorithms and the results show that it has higher encryption efficiency. To show the superiority of LSCM-IEA, we also analyze the security of LSCM-IEA in terms of key security, ability of defending differential attack, local Shannon entropy and contrast analysis. The analysis results demonstrate that LSCM-IEA has better security performance than several existing algorithms.
© 2018 Published by Elsevier B.V.
To address this issue, many image encryption schemes consider- ing image features have been proposed using various technolo- gies, such as the chaos theory [13–16], DNA coding [17,18], quan- tum theory [19,20], compressive sensing [21,22] and some mathe- matics models [23,24]. Among these technologies, chaos theory is the most popular one. This is because chaotic behavior has many unique properties that are similar with the principles of image en- cryption [25–27]. Specifically, the ergodicity and initial state sen- sitivity of chaos theory correspond to the confusion and diffusion properties of encryption [28]. Some examples of chaos-based en- cryption schemes are as follows. In [29], Zhou et al. first proposed a new chaotic system that can use existing chaotic maps as seed maps to generate new chaotic maps, and then used one newly generated chaotic map to design an image encryption algorithm. In [30], Pak and Huang proposed a new color image encryption algorithm using the combination of Logistic, Sine and Chebyshev maps. In [21], Zhou et al. proposed a new image security scheme using hyperchaotic system and compressive sensing technology. This scheme can perform image encryption and image compres- sion simultaneously.
For these chaos-based image encryption algorithms, their se- curity is determined by the structure of encryption algorithms and the chaos performance of their used chaotic maps. On one hand, if the designed encryption structures are not secure enough, the encryption algorithms can be successfully broken using dif- ferent security attacks [31–33]. On the other hand, with the fast
development of discerning chaos methodology, researchers found that some existing chaotic maps have security problems if they have weak chaos performance [34–36]. This will also cause secu- rity problems to the corresponding chaos-based encryption algo- rithms [37,38]. Thus, designing encryption structures with higher security and developing new chaotic systems with better chaos performance can significantly promote the chaos-based image en- cryption.
To design new chaotic map with better chaos performance for image encryption, this paper presents a two-dimensional (2D) Logistic-Sine-coupling map (2D-LSCM). It is generated by first cou- pling the Logistic and Sine maps, and then extending the dimen- sion from one-dimensional (1D) to 2D. Chaos performance estima- tions demonstrate that 2D-LSCM has better ergodicity, more com- plex chaotic behavior and wider chaotic interval than several newly designed 2D chaotic maps. Using 2D-LSCM, we further present a 2D-LSCM-based image encryption algorithm (LSCM-IEA). The se- cret key is to obtain the initial states of the 2D-LSCM, and then produce chaotic sequences. The chaotic sequences are used to do permutation and diffusion operations to the plain-image. Simula- tion results prove the ability of LSCM-IEA. Efficiency evaluation shows that it can achieve faster encryption speed than several other algorithms. The security analysis demonstrates that LSCM- IEA can outperform several advanced image encryption algorithms in security performance.
We organize the rest of this paper as follows. Section 2 in- troduces the proposed 2D-LSCM and evaluates its chaos perfor- mance. Section 3 presents the developed image encryption algo- rithm, LSCM-IEA. Section 4 simulates LSCM-IEA and analyzes its efficiency. Section 5 analyzes the security level of LSCM-IEA and Section 6 concludes this paper.
2. 2D Logistic-Sine-coupling map
This section presents a novel 2D chaotic map, called 2D Logistic-Sine-coupling map (2D-LSCM), and then discusses its chaotic complexity.
2.1. Definition of 2D-LSCM
The 2D-LSCM is derived from two existing 1D chaotic maps, namely the Logistic map [39] and the Sine map [29]. The Logistic map is defined as
2.2. Performance evaluation
The proposed 2D-LSCM can inherently enhance the chaos per- formance of the Logistic and Sine maps. To show its superiority, we evaluate its chaos performance and compare it with several newly generated 2D chaotic maps. The evaluations are performed in terms of chaos trajectory, Lyapunov exponent [41], and Kol- mogorov entropy [42].
2.2.1. Chaos trajectory
Trajectory demonstrates the motion starting from a given initial state with the time increases. The trajectory of a periodic motion is a closed curve and the trajectory of a chaotic behavior will never close or repeat in theory. Thus, the chaos trajectory usually occu- pies a part of phase space and it can reflect the randomness of the outputs of a chaotic system. A chaotic system has better random outputs if its chaos trajectory can occupy a larger phase space.
Fig. 1shows the trajectories of four 2D chaotic maps. When plotting these trajectories, all the initial states are set as (0.8,0.5) and the control parameters are selected as the settings that can make the corresponding chaotic maps obtain their best chaos per- formance. Specifically, the control parameters of the 2D Logis- tic map, 2D Sine Logistic modulation map (2D-SLMM) [40], 2D Logistic-adjusted-Sine map (2D-LASM) [43], and 2D-LSCM are set as 1.19, 1, 0.9 and 0.99, respectively. To show the actual behaviors of chaotic systems in stable state, we plot the iteration points from 5000 to 35,000 in each trajectory. One can see from Fig. 1 that the trajectories of the 2D Logistic map and 2D-SLMM only occupy a small space in the phase plane, while that of the 2D-LASM and 2D-LSCM can occupy all phase plane. Besides, It is obvious that the points of the 2D-LSCM distribute more uniform than that of the 2D-LASM. Thus, the proposed 2D-LSCM can generate more random output sequences than other three chaotic maps.
2.2.2. Lyapunov exponent
The initial state sensitivity is the most obvious feature of chaotic behavior. The Lyapunov exponent (LE) [41] can provide a quantitative description to the initial state sensitivity. For two tra- jectories of a chaotic system beginning with two close initial states, the LE describes their average separation rate. For a differentiable one-dimensional dynamical system xi+1 = f (xi ), its LE can be de- fined as
x =4ηx(1−x), i+1 i i
where its control parameter η ∈ [0, 1]. The Sine map is given as
xi+1 =βsin(πxi),
(1) (2)
λ=lim lnf(xi).
(4)
where β is a control parameter and it also has an interval of [0,1]. The Logistic and Sine maps have many disadvantages such as simple behaviors and frail chaotic intervals, and these disadvan- tages may bring negative effects for some chaos-based applica- tions [40]. However, when coupling the Logistic and Sine maps, we can obtain a new chaotic map with quite complex chaos, namely
2D-LSCM, which can be defined as
xi+1 =sin(π(4θxi(1−xi)+(1−θ)sin(πyi))); yi+1 =sin(π(4θyi(1−yi)+(1−θ)sin(πxi+1))),
A 2D chaotic system has two LEs and Fig. 2 plots the two LEs of different 2D chaotic maps with the change of their con- trol parameters. One can observe that the 2D-SLMM has chaotic behavior when α ∈ (0.84, 1), and has hyperchaotic behavior when α ∈ (0.91, 1), the 2D-LASM has chaotic behavior when μ ∈ (0.32, 1), and has hyperchaotic behavior when μ ∈ (0.45, 1), the 2D-LSCM has chaotic behavior when θ ∈ (0, 1), and has hyperchaotic behav- ior when θ ∈ (0, 0.34) ∪ (0.67, 1). This shows that the proposed 2D- LSCM has much wider chaotic range and hyperchaotic range than
where θ is the control parameter and θ ∈ [0, 1]. As can be observed from its definition, the 2D-LSCM is obtained by first coupling the Logistic and Sine maps together, and then performing a sine trans- form to the coupling result, and last extending the dimension from 1D to 2D. By this way, the complexity of the Logistic map and Sine map can be sufficiently mixed, which can obtain complex chaotic behavior.
Z. Hua et al./Signal Processing 149 (2018) 148–161 149
n→∞ n
1 n−1 ′
i=0
A high-dimensional dynamical system has more than one LE and the maximum LE (MLE) determines whether a high-dimensional system has chaotic behavior or not. A positive MLE means that the close trajectories of a dynamical system diverge in each unit time and will evolve to completely different trajectories with the in- creasement of time. Thus, a dynamical system is chaotic if its MLE is positive and larger MLE means better performance. If a dynam- ical system can obtain more than one positive LE, its trajectories will diverge in multi-directions, which makes it has hyperchaotic behavior. The hyperchaotic behavior is a much more complicated motion than the chaotic behavior.
(3)
150 Z. Hua et al./Signal Processing 149 (2018) 148–161
Fig. 1. Trajectories of four 2D chaotic maps: (a) the 2D Logistic map with parameter r = 1.19; (b) the 2D-SLMM with parameter α = 1; (c) the 2D-LASM with parameter μ = 0.9; (d) the 2D-LSCM with parameter θ = 0.99.
Fig. 2. The two LEs of different 2D chaotic maps: (a) the 2D-SLMM; (b) the 2D-LASM; (c) the 2D-LSCM; (d) the MLE comparison of 2D-SLMM, 2D-LASM and 2D-LSCM.
the other chaotic maps. Besides, Fig. 2(d) compares the MLEs of different chaotic maps. It shows that 2D-LSCM has the largest MLE in most parameter settings. This further demonstrate that the pro- posed 2D-LSCM has more complex chaotic behavior.
2.2.3. Kolmogorov entropy
The Kolmogorov entropy (KE) is a type of entropy that describes the state evolution of dynamical system [42]. It can be used to measure the degree of chaos by testing the needed extra informa- tion of predicting the future trajectory using the previous states. Dividing the n-dimensional phase space into a number of boxes (i0,i1,…,in) with ε size, the KE can be described as
we can see that the proposed 2D-LSCM has positive KEs in the whole parameter range. It can achieve larger KEs than 2D-LASM in most parameter settings and can outperform 2D Logistic map and 2D-SLMM in all the parameter ranges. This sufficiently proves that the proposed 2D-LSCM has good unpredictability.
2.2.4. Dynamical degradation analysis
For a dynamical system with chaotic behavior, its trajectory will never close or repeat in theory. However, as the finite precision domain cannot own infinite states, the close states in the phase plane will overlap when a chaotic map is digitalized in the finite precision platforms. This phenomenon is known as the dynami- cal degradation [44]. The dynamical degradation is unavoidable for digitalized chaotic maps and it causes many negative effects for chaos-based applications. However, many chaos-based applications only use finite states of a chaotic trajectory. Chaotic maps with dy- namical degradation are still available to these applications if the cycle lengths of the digitalized chaotic maps are larger than the required cycle lengths.
To investigate the dynamical degradation of different chaotic maps, we calculate the cycle lengths of these chaotic maps using different precisions. For each chaotic map, we first randomly gener- ate a number groups of initial states, where the control parameters are all within the chaotic ranges, and then generate trajectories us- ing these initial states under different precisions, and finally calcu- late the average cycle lengths of these trajectories. Table 1 lists the average cycle lengths of different chaotic maps under various pre- cisions. One can observe that our proposed 2D-LSCM can obtain the largest average cycle lengths under most precisions. Its cycle length fast increases with the increasement of precision and it can achieve 4,455,734 under the precision 10−8. As the precisions of the commonly used platforms are usually much higher than 10−8, the cycle lengths of 2D-LSCM in these platforms are far larger than 4455734. On the other hand, when chaotic map is used in im-
K =−limlim lim 1 p(i1,…,in)lnp(i1,…,in),
i0 ,i1 ,…,in
(5)
τ→0ε→0n→∞ nτ
where n is the embedding dimension, τ is the time delay, and p(i1 , . . . , in ) represents the joint probability when the trajectory of system is in i0 at the starting time, in i1 at the time τ, …, and in in at the time nτ. A positive KE indicates that extra informa- tion is required to predict the trajectory of the dynamical system and larger KE demonstrates more required information. Thus, a dy- namical system is unpredictable if it has a positive KE and larger KE indicates better unpredictability.
Our experiment uses the Grassberger method provided in [42] to calculate the KEs of different chaotic systems and Fig. 3 plots the obtaining results. One can see that Fig. 3(a) plots the KEs of the Logistic map, Sine map and 2D-LSCM, while Fig. 3(b) compares the KEs of the 2D Logistic map, 2D-SLMM, 2D-LASM and 2D-LSCM. To provide a better comparison envi- ronment, we shift the parameter of the 2D Logistic map when plotting its KEs. As can be seen from Fig. 3(a), although 2D-LSCM is derived from the Logistic and Sine maps, it has much better unpredictability than the Logistic and Sine maps. From Fig. 3(b),
Z. Hua et al./Signal Processing 149 (2018) 148–161 151
Fig. 3. KE comparison of (a) the Logistic map, Sine map and 2D-LSCM; (b) the 2D Logistic map, 2D-SLMM, 2D-LASM and 2D-LSCM.
Table 1
The average cycle lengths of different chaotic maps with different precisions.
Table 2
NIST SP800-22 test results of binary sequences generated using 2D-LSCM. P-value Result
Precisions
2D Logistic 2D-SLMM 2D-LASM 2D-LSCM
10−4 10−5 10−6 10−7 324 1432 8215 45,266
796 4858 36,959 339,576 74 555 4474 39,522 666 5431 40,544 413,618
10−8 259,535
3,127,743 195,389 4,455,734
Sub-tests
Approximate Entropy(m = 10)
Block Frequency(M = 128) Cumulative Sums
FFT
Frequency
Linear Complexity(M = 500) Longest Run
Non-Overlapping Template(m = 9)a Overlapping Template(m = 9) Random Excursionsa
Random Excursions Varianta
Rank
Runs
Serial(m = 16)
Universal
a The average values of multiple tests.
Forward Reverse
P-value1 P-value2
≥ 0.01
0.704009 Pass
0.993633 Pass 0.069202 Pass 0.077280 Pass 0.840006 Pass 0.076727 Pass 0.504113 Pass 0.447729 Pass 0.472143 Pass 0.936519 Pass 0.217525 Pass 0.416696 Pass 0.151412 Pass 0.740543 Pass 0.163838 Pass 0.400104 Pass 0.958368 Pass
age encryption, the number of required chaotic outputs approxi- mates to the size of the image, e.g. almost 1,000,000 chaotic out- puts for an image of size 1000×1000. Thus, in the commonly used platforms, the cycle lengths of 2D-LSCM are larger than the cycle lengths required in image encryption.
To further show that the proposed 2D-LSCM is suitable for de- signing image encryption algorithm, we use the National Institute of Standards and Technology (NIST) SP800-22 [45] to test the ran- domness of the output sequences of 2D-LSCM. The NIST SP800- 22 has 15 sub-tests and each sub-test can generate a P-value. Ac- cording to the recommendation of Bassham et al. [45], 100 binary streams with 1,000,000 bits are suggested as input and the gen- erated P-value is expected to fall into the range [0.01,1] to pass the corresponding sub-test. Our experiment uses the double float data format to present the iterative outputs of 2D-LSCM. For each output of 2D-LSCM, we transform its fractional part to be a bi- nary stream with 49 bits. The input binary streams are obtained by combining these binary streams from the outputs. Table 2 shows the test results and one can see that binary streams obtained from the outputs of 2D-LSCM can pass all the sub-tests. This indicates that 2D-LSCM can generate a long sequence of aperiodic outputs, which are suitable for image encryption.
3. 2D-LSCM-based image encryption algorithm
Using the developed 2D-LSCM, this section presents a 2D- LSCM-based image encryption algorithm (LSCM-IEA) and its struc- ture is shown in Fig. 4. The secret key is to generate initial state of the 2D-LSCM, and the chaotic matrices generated by 2D-LSCM are used to do 2D-LSCM permutation and 2D-LSCM diffusion. The 2D- LSCM permutation can efficiently shuffle pixel positions and the 2D-LSCM diffusion can completely change pixel values and spread few changes in plain-image to the whole cipher-image. As the 2D- LSCM permutation can achieve excellent confusion property and the 2D-LSCM diffusion can obtain good diffusion, two rounds of permutation and diffusion can obtain a high security encryption
result in theory. However, more encryption rounds can achieve higher security results. Our proposed LSCM-IEA uses four encryp- tion rounds, as four encryption rounds can obtain high security encryption results and can balance the trade-off between the effi- ciency and security. Next, we will describe each of the encryption processes in detail.
3.1. Initial state generation
According to the discussion in [46], the key length of a chaos- based encryption algorithm should be larger than 100 bits to resist brute-force attack. We set the length of secret key as 256 bits in LSCM-IEA, considering the rapid enhancement of computer com- puting ability. Specially, K = {x0, y0, r, a1, a2, a3, a4}, where (x0, y0) are the initial values, r is the control parameter and a1 ∼ a4 are the perturbation coefficients to change r in the four encryption rounds. The x0, y0 and r have size of 52 bits, and they can be converted to float numbers using the IEEE 754 Floating-Point standard. Suppose b1 b2 . . . b52 is a 52-bit binary string, the conversion equation is as follows,
52
v=bi2−i. (6)
i=1
152 Z. Hua et al./Signal Processing 149 (2018) 148–161
The ai (i = 1, 2, 3, 4) is an integer that can be obtained by directly transforming a 25-bit binary string to a decimal integer.
The secret key is to generate four initial states for 2D-LSCM. The
initial value of the first encryption round (x(1), y(1)) is directly set 00
as (x0, y0), and the initial values of the second, third and fourth encryption rounds are set as the last iteration state of 2D-LSCM in previous encryption round. The control parameters in four encryp- tion rounds can be generated as
(7)
Using the four initial states (x(i),y(i),r(i)) (i=1,2,3,4), the 2D- 00
LSCM can generate chaotic matrices for the following 2D-LSCM permutation and 2D-LSCM diffusion.
3.2. 2D-LSCM permutation
High correlations and data redundancy may exist between ad- jacent pixels of a natural image, as the image pixel is usually rep- resented using 8 or even more bits. An efficient image encryption algorithm should de-correlate these high correlations. Pixel permu- tation can randomly shuffle adjacent pixels to different positions and it can de-correlate their high correlations.
Most of the existing permutation operations shuffle image pix- els row-by-row or column-by-column. Then each operation can only change a pixel’s row position or column position. Multiple permutation operations are required to obtain a totally shuffled re- sult. To obtain better shuffling efficiency, we designed a new 2D- LSCM permutation that can simultaneously shuffle the image’s row and column positions in one operation. The detail procedure can be described as follows,
• Step 1: Suppose the plain-image P is of size M×N, a chaotic matrix S of size M×N is generated using the 2D-LSCM with the initial state;
• Step 2: Sort each column of S and obtain the index matrix O;
• Step 3: Set row index m = 1;
• Step 4: Select the pixels in P with positions
{(Om,1,1),(Om,2,2),…,(Om,N,N)};
• Step 5: Sort the values in S with positions
{(Om,1,1),(Om,2,2),…,(Om,N,N)} and obtain an index vector
v;
• Step 6: Shuffle these selected pixels in P using v;
• Step 7: Iterate Step 3 to Step 6 for m=2∼M.
To better explain the procedure of 2D-LSCM permutation, we provide a numeral example with the image size of 4×5 and it is shown in Fig. 5. Fig. 5(a) shows the generation of permutation ma- trix PM from the chaotic sequence S. First, sort each column of
S with ascending order to obtain the sorted result S′ and an in- dex matrix O, where S′i, j = SOi, j , j . Using the index matrix O as the row position, we can obtain a position matrix PM. Fig. 5(b) shows the detail pixel shuffling using PM and S. The detail pixel shuffling procedure can be described as follows.
• The 1-st row of PM is {(3, 1), (2, 2), (4, 3), (4, 4), (4, 5)}. Se- lect the values in S with these positions and sort them with as- cending order to obtain the index vector v = {2, 1, 3, 4, 5}. Then use the obtained v to shuffle the pixels in P with these po- sitions, namely T3,1 = P2,2 , T2,2 = P3,1 , T4,3 = P4,3 , T4,4 = P4,4 , T4,5 = P4,5.
• The 2-nd row of PM is {(2, 1), (4, 2), (1, 3), (3, 4), (2, 5)}. Se- lect the values in S with these positions and sort them with as- cending order to obtain the index vector v = {3, 5, 1, 2, 4}. Then use the obtained v to shuffle the pixels in P with these po- sitions, namely T2,1 = P1,3, T4,2 = P2,5, T1,3 = P2,1, T3,4 = P4,2, T2,5 = P3,4.
• The 3-rd row of PM is {(4, 1), (1, 2), (2, 3), (2, 4), (3, 5)}. Se- lect the values in S with these positions and sort them with as- cending order to obtain the index vector v = {5, 2, 1, 4, 3}. Then use the obtained v to shuffle the pixels in P with these po- sitions, namely T4,1 = P3,5 , T1,2 = P1,2 , T2,3 = P4,1 , T2,4 = P2,4 , T3,5 = P2,3.
• The 4-th row of PM is {(1, 1), (3, 2), (3, 3), (1, 4), (1, 5)}. Se- lect the values in S with these positions and sort them with as- cending order to obtain the index vector v = {2, 4, 1, 5, 3}. Then use the obtained v to shuffle the pixels in P with these po- sitions, namely T1,1 = P3,2 , T3,2 = P1,4 , T3,3 = P1,1 , T1,4 = P1,5 , T1,5 = P3,3.
Algorithm 1shows the pseudo-code of the 2D-LSCM permuta-
Algorithm 1 The 2D-LSCM permutation.
Input: The plain-image P and the chaotic matrix S. Both have the
size M × N.
1: Sort each column of S with ascending order and obtain O and
S′, where S′i,j = SOi,j,j;
2: Set T∈NM×N, b∈N1×N, t∈N1×N; 3: for i = 1 to M do
4: for j = 1 to N do
5: tj = POi,j,j, bj = SOi,j,j;
6: end for
7: Sort b with ascending order and obtain v and b′, where b′ =
bv;
8: for j = 1 to N do 9: TOi,j,j = tvj ;
10: end for
11: end for
Output: The permuted result T.
⎧⎪ r ( 1 ) = ( r × a 1 ) m o d 1 ;
⎨r(2) =(r×a2)mod1;
⎪r(3) = (r × a3) mod 1;
⎩r(4) =(r×a )mod1. 4
Fig. 4. The structure of LSCM-IEA.
tion.
3.3. 2D-LSCM diffusion
An image encryption algorithm should have diffusion property, which means that slight change in plain-image can cause total dif- ference in cipher-image. In the proposed LSCM-IEA, we designed a 2D-LSCM diffusion to achieve the diffusion property. Using the chaotic sequence generated by 2D-LSCM, the image pixels can be randomly changed. Using the two previous pixel values to change
the current one, the 2D-LSCM diffusion can efficiently spread few changes of plain-image to the whole cipher-image. Suppose both the permutation result T and chaotic matrix R are with the size of M×N, the 2D-LSCM diffusion is described as
Z. Hua et al./Signal Processing 149 (2018) 148–161 153
Fig. 5. An example of 2D-LSCM permutation using the image P of size 4×5: (a) the generation procedure of permutation matrix PM from chaotic sequence S; (b) permu- tation to P using PM and S.
Ci =
(T1 +TG +TG−1 +⌊Ri ×232⌋)modF (T2 +C1 +TG +⌊Ri ×232⌋)modF (Ti +Ci−1 +Ci−2 +⌊Ri ×232⌋)modF
if i=1;
if i=2; (8) if i∈[3,G],
where F is the number of allowed pixel values in plain-image P, e.g. F = 256 if P is 8-bit grayscale image, and the operation ⌊x⌋ is to obtain the largest integer that is smaller than or equals to x.
154 Z. Hua et al./Signal Processing 149 (2018) 148–161
Fig. 6. Demonstration of 2D-LSCM diffusion: (a) plain-image I1; (b) 2D-LSCM diffusion result of I1; (c) the difference of 2D-LSCM diffusion results to I1 and I2, where I2 is another plain-image that has one pixel difference with I1 in position (64,64); (d) the difference of two rounds of 2D-LSCM diffusion to I1 and I2.
The 2D-LSCM diffusion can be divided into two steps: row diffu- sion and column diffusion. When doing the row diffusion, G = N and Eq. (8) is applied to each row. When performing the column diffusion, G = M and Eq. (8) is applied to each column.
The 2D-LSCM diffusion in the decryption process is the inverse of the forward operation. The inverse 2D-LSCM diffusion is defined
4.2. Efficiency analysis
The high efficiency of image encryption is required, as a large number of digital images with high resolutions are generated ev- ery moment. The proposed LSCM-IEA has low time complexity, as it can achieve the following properties: (1) the used chaotic map is a 2D discrete-time map and has low implementation cost; (2) the 2D-LSCM permutation can shuffle the pixel column and row po- sitions simultaneously, and thus has high shuffling efficiency; (3) four rounds of encryption processes can guarantee a high secu- rity level. Table 3 compares the time complexity and encryption time of several advanced image encryption algorithms using im- ages of different sizes. For Xu et al. and Diaconu et al.s’ [47] algo- rithms, the time complexity is reported in their papers and thus we directly refer their results. To provide a fair comparison, we adopt the same principle with Diaconu et al.’s algorithm to cal- culate the time complexity of our proposed LSCM-IEA and Zhou et al.’s [29] algorithm. The second column of Table 3 lists the time complexity of different algorithms and the results show that our proposed LSCM-IEA has the lowest time complexity. To compare the actual encryption time of these encryption algorithms, we im- plement these algorithms using Matlab R2015b and use images of different sizes to test their actual encryption time. The experimen- tal environments are as follows: Intel(R) Core(TM) i5-3320M CPU @ 2.6 GHz with 8GB memory, Windows 7 Operation system. One can see that the proposed LSCM-IEA requires the least time when encrypting images with different sizes. This further indicates that it has the higher encryption efficiency than other three algorithms.
5. Security analysis
The security performance is the most important indictor of an image encryption algorithm. This section analyzes the security of the proposed LSCM-IEA in terms of key security, ability of defend- ing differential attack, local Shannon entropy and contrast analysis.
5.1. Key security
The secret key plays an important role in an encryption algo- rithm. On one hand, the secret key should have proper size to re- sist the brute-force attack. As mentioned in Section 3.1 that the length of secure key should be bigger than 100 bits. Considering the rapid enhancement of computer computing ability, we set the length of the secret key of LSCM-IEA as 256 bits. On the other hand, the secret key must be very sensitive. If a secret key isn’t sensitive, an equivalent secure key can be obtained and this will greatly reduce the actual key space.
as
(Ci −Ci−1 −Ci−2 −⌊Ri ×232⌋)modF
Ti = (C2 −C1 −TG −⌊Ri ×232⌋)modF (C1 −TG −TG−1 −⌊Ri ×232⌋)modF
if i∈[3,G]; if i=2;
if i=1.
(9)
To show the performance of the 2D-LSCM diffusion, we pro- vide an image example, which is shown in Fig. 6. One can see that the 2D-LSCM diffusion can randomly change pixel values, which is shown in Fig. 6(b). When using the same secret key to do the 2D- LSCM diffusion to two plain-images with only one bit difference, the difference can be spread to all the pixels behind the different pixel, which is shown in Fig. 6(c). After two rounds of diffusion, the change of one pixel can be spread all over the image, which can be seen from Fig. 6(d). Thus, the 2D-LSCM diffusion can achieve good diffusion property.
4. Simulation results and efficiency analysis
This section simulates the proposed LSCM-IEA and analyzes its efficiency. Most of test images in our experiments are selected from the USC-SIPI image dataset1 (grayscale images and color im- ages) and Brown Univ Large Binary image database2(binary im- ages).
4.1. Simulation results
An efficient image encryption algorithm must have the abil- ity to encrypt different types of digital images into unrecognized cipher-images. Only with the correct key, one can completely de- crypt the cipher-image. Without key or with a wrong key, one can’t obtain any useful information about the original image. Fig. 7 shows the encryption procedures of the binary, grayscale and color images. One can observe that all the plain-images have many pat- terns that make them hard to be processed. However, their cipher- images are all random-like and their pixel values distribute very randomly, which can be seen from Fig. 7(c) and (d). Attackers can’t obtain any useful information about the original images from their pixel distributions. Using the same secret key, the decryption pro- cess can totally recover the original images, which can be seen in Fig. 7(e).
1 http://sipi.usc.edu/database/ .
2 http://vision.lems.brown.edu/content/available- software- and- databases/ .
Image size Xu’s [25]
Diaconu’s [47] Zhou’s [29] LSCM-IEA
Time complexity
O(M log(8N) + 8N log M + M + 8N)
O(9MN)
O(8MN)
O(4(M log N + M + N))
128×128 256×256 0.0321 0.1484
0.0687 0.2637 0.0814 0.3042 0.0196 0.0800
512×512 1024×1024 0.6921 2.8115
1.1003 4.3618 1.2030 4.8264 0.4842 2.2848
Z. Hua et al./Signal Processing 149 (2018) 148–161 155
Fig. 7. Simulation results of LSCM-LEA: (a) the binary, 8-bit grayscale, and 24-bit color images; (b) histograms of (a); (c) encrypted results of (a); (d) histograms of (c); (e) decrypted results of (c).
Table 3
Time complexity and encryption time (second) of different image encryption algorithms for images with different sizes.
To visually display the key sensitivity of LSCM-IEA, we first ran- domly generate a secret key K1,
K1 = EFC796D47FDFFFE9AB7DF3DFFF3CE7AFDEFEFC6977757 FC9DA69D93F4D76FC7F,
and then change one bit of K1 to obtain two other keys, K2 and K3. Fig. 8 shows the key sensitivity in the encryption process and Fig. 9 demonstrates the key sensitivity in decryption process. One can see that when encrypting a plain-image using two secret keys with only one bit difference, the two obtained cipher-images are completely different (see Fig. 8(d)). Only the correct key can to- tally recover the original image (see Fig. 9(b)). When decrypting a cipher-image with two slightly different keys, the obtained two decrypted results are random-like (see Fig. 9(c) and (d)), and also totally different (see Fig. 9(e)).
To quantitatively test the key sensitivity, we use the number of bit change rate (NBCR) to calculate the difference of images. For two sequences S1 and S2 with the same length, their NBCR can be described as
NBCR = Hm[S1, S2] × 100%, (10) Lb
where Lb is the length of S1 or S2 and Hm[S1, S2] is to calcu- late their Hamming distance [48]. If S1 and S2 are two statistic- independent data sequences, their NBCR will approach to 50%.
For each of the 256 bits in K1, we set the experiments as fol- lows. (1) Change the bit to obtain a slightly different key; (2) use the two secret keys to encrypt a same plain-image and calculate the NBCR of the two encrypted results; (3) use the two secret keys to decrypt a same cipher-image and calculate the NBCR of two de-
156 Z. Hua et al./Signal Processing 149 (2018) 148–161
Fig. 8. Key sensitivity analysis in encryption process: (a) plain-image P; (b) cipher-image C1 = Enc(P, K1 ); (c) cipher-image C2 = Enc(P, K2 ); (d) the difference between C1 and C2, |C1 − C2|.
Fig. 9. Key sensitivity analysis in decryption process: (a) cipher-image C1 ; (b) decrypted result D1 = Dec(C1 , K1 ); (c) decrypted result D2 = Dec(C1 , K2 ); (d) decrypted result D3 = Dec(C1 , K3 ); (e) the difference between D2 and D3 , |D2 − D3 |.
crypted results. Fig. 10 shows the test results. One can see that when changing any one bit of a randomly generated secret key, the two obtained encrypted results are totally different (see Fig. 10(a)) and the two obtained decrypted results are also independent (see Fig. 10(b)). This means that LSCM-IEA has quite sensitive encryp- tion and decryption keys.
5.2. Ability of defending differential attack
The differential attack is a kind of chosen-plaintext attacks. By tracing how the slight change in plaintexts can affect the cipher- texts, the differential attack tries to find the connections between the plaintexts and ciphertexts, and uses the built connections to recover the ciphertext without secret key. For an image encryption algorithm, its ability of defending differential attack can be tested
using the number of pixel changing rate (NPCR) and the unified averaged changed intensity (UACI). For two cipher-images, C1 and C2, encrypted from two plain-images with one bit difference, their NPCR and UACI are defined as
(11)
× 100%, (12)
NPCR(C1, C2 ) = and
UACI(C1, C2 ) =
M N W ( i , j )
H × 100%, M N | C ( i , j ) − C ( i , j ) |
i=1 j=1
1 2
i=1 j=1
H×Q
Nα∗ is obtained by
Z. Hua et al./Signal Processing 149 (2018) 148–161 157
Fig. 10. NBCR in encryption and decryption processes: (a) NBCR between C1 and C2, which C1 and C2 are two cipher-images encrypted from a same plain-image and two secret keys with only one bit difference; (b) NBCR between D1 and D2, which D1 and D2 are two decrypted images from cipher-image C1 and two secret keys with only one bit difference.
respectively, where H is the total number of pixels in an im- age, Q represents the largest allowed pixel value in the image, and
W(i,j)= 0 ifC1(i,j)=C2(i,j); (13) 1 ifC1(i,j)̸=C2(i,j).
Recently, more strict criterions about the NPCR and UACI were developed in [49]. For a significance level α, a critical NPCR score
Fig. 11 plots the NPCR scores of different image encryption algo- rithms and Fig. 12 shows their UACI scores. The LSCM-IEA can ob- tain NPCR and UACI scores that are all within the accepted inter- vals. On the other hand, other image encryption schemes fail to pass some tests. This indicates that the proposed LSCM-IEA can achieve higher ability of defending differential attack than these other encryption algorithms.
5.3. Local Shannon entropy
The pixels of a cipher-image are expected to randomly dis- tribute to resist various security attacks. The local Shannon entropy (LSE) can provide a strict description to the randomness of image pixel [50]. For an image I, randomly select k non-overlapping im- age blocks S1, S2, . . . , Sk with TB pixels, the LSE can be defined as
k H(Si)
Q−−1(α) Q/H N α∗ = Q + 1 .
( 1 4 )
An image encryption scheme can be considered to pass the
NPCR if the obtained NPCR is larger than Nα∗. The critical UACI in-
terval (U∗−,U∗+) can be calculated by αα
U∗− =μ −−1(α/2)σ ; αUU(15)
(16)
(17)
A number of 28 grayscale images in USC-SIPI image database
are selected in our experiment. Among these 28 images, six
images have size of 256 × 256; eighteen images have size of
512 × 512 and four images have size of 1024 × 1024. According
to the discussions in [49], we set the significance level α=
0.05, then for images of size 256 × 256, Nα∗ = 99.5693% and
(U∗−,U∗+) = (33.2824%,33.6447%); for images of size 512×512, αα
N∗ = 99.5893% and (U∗−,U∗+) = (33.3730%,33.5541%); and for ααα
images of size 1024×1024, N∗=99.5994% and (U∗−,U∗+)= ααα
(33.4183%,33.5088%). In each test, we randomly change one bit of an image to obtain another image, and then encrypt the two images using a same secret key to get two encrypted results, and calculate the NPCR and UACI scores of the two encrypted results.
Hk,TB (I) =
where H(Si) is the Shannon entropy of image block Si and can be
defined as
L
H(Si)− P(l)log(P(l)), (19)
l=1
where L is the total number of pixel values and P(l) is the proba-
bility of lth values.
Our experiment also uses the images from USC-SIPI image
dataset to do the simulation. According to the recommendation in [50], we set the parameters (k, TB ) = (30, 1936) and significance α = 0.05, then the ideal LSE is 7.902469317 and an image is con- sidered to pass the test if the obtained LSE falls into the interval (7.901901305, 7.903037329). Table 4 lists the LSE scores of cipher- images encrypted by several image encryption schemes. One can see that LSCM-IEA has 20 cipher-images that are within the ac- cepted interval and its pass rate is the highest. This means that the proposed LSCM-IEA can encrypt images into cipher-images with high randomness.
5.4. Contrast analysis
Contrast feature is a kind of statistical texture characteristic and it can reflect the clarity degree of image and the texture of the
U∗+ =μ +−1(α/2)σ , αUU
k
i=1
, (18)
where μU=Q+2,
3Q + 3 and
2 (Q + 2)(Q2 + 2Q + 3) σU= 18(Q+1)2QH .
If the obtained UACI falls into
encryption algorithm is considered to have high security level.
range
(U ∗− , U ∗+ ), αα
the
corresponding
158
Z. Hua et al./Signal Processing 149 (2018) 148–161
Table 5
Table 4
The LSE scores of cipher-images encrypted by different image encryption schemes.
Images LSE scores
Wu’s [51]
5.1.09 7.903223
5.1.10 7.903087
5.1.11 7.906766
5.1.12 7.903390
5.1.13 7.899016
5.1.14 7.901087
5.2.08 7.900827
5.2.09 7.903732
5.2.10 7.901648
7.1.01 7.898618
7.1.02 7.904654
7.1.03 7.901633
7.1.04 7.905116
7.1.05 7.902414
7.1.06 7.901472
7.1.07 7.902247
7.1.08 7.903583
7.1.09 7.905126
7.1.10 7.904126
boat.512 7.902755 elaine.512 7.902115 gray21.512 7.904832 numbers.512 7.901345 ruler.512 7.902244
5.3.01 7.899751
5.3.02 7.903496
7.2.01 7.903118 testpat.1k 7.901350 Mean 7.902599 Std 0.0019 Pass/All 5/28
Zhou’s [26] 7.903595
7.902314
7.903901 7.900834 7.902525 7.903649 7.901765 7.900433 7.901966 7.901058 7.903413 7.904178 7.902179 7.902124 7.904908 7.903210 7.904767 7.902820 7.904401 7.900889 7.902934 7.902972 7.900308 7.900604 7.904291 7.903330 7.902309 7.903963 7.902701 0.0014 9/28
Wang’s [4] 7.902682
7.903397 7.904131 7.902789 7.903841 7.901668 7.903854 7.905012 7.902882 7.902966 7.906349 7.900470 7.900964 7.901991 7.902182 7.900828 7.901676 7.901032 7.903549 7.901836 7.902525 7.901614 7.901442 7.915057 7.902397 7.902727 7.897698 7.905429 7.902964 0.0029 9/28
Liu’s [5] 7.901914
7.900288 7.900441 7.900276 7.904302 7.899342 7.902088 7.905057 7.900481 7.901860 7.901194 7.901148 7.903347 7.901434 7.903007 7.901576 7.901945 7.903082 7.902345 7.902716 7.904935 7.900796 7.901988 7.901759 7.899133 7.903761 7.903574 7.900457 7.901937 0.0016 6/28
Zhou’s [29] 7.903032
7.901979
7.901630 7.904191 7.901417 7.905588 7.904109 7.899799 7.900969 7.900260 7.900644 7.901404 7.902802 7.900019 7.904535 7.903704 7.901698 7.901889 7.901574 7.903704 7.901016 7.901009 7.901481 7.900907 7.906108 7.903088 7.901969 7.900552 7.902181 0.0017 4/28
Xu’s [25] 7.903543
7.903137 7.905119 7.904896 7.901933 7.901559 7.902714 7.898275 7.904609 7.903507 7.902496 7.899554 7.900657 7.902717 7.903967 7.902364 7.902537 7.902012 7.900796 7.899492 7.902890 7.901349 7.904014 7.902885 7.900850 7.904658 7.902648 7.902370 7.902412 0.0017 11/28
LSCM-IEA
7.902281 7.902198 7.899982 7.902827 7.902281 7.903117 7.902304 7.902022 7.906701 7.902191 7.902047 7.902584 7.901913 7.902392 7.902565 7.904015 7.901096 7.902933 7.902534 7.901782 7.902569 7.902593 7.902295 7.904102 7.902119 7.902658 7.902529 7.904472 7.902611 0.0012 20/28
The average contrast feature scores of plain-images and cipher-images encrypted by different encryption schemes.
Image names
5.1.09
5.1.10
5.1.11
5.1.12
5.1.13
5.1.14
5.2.08
5.2.09
5.2.10
7.1.01
7.1.02
7.1.03
7.1.04
7.1.05
7.1.06
7.1.07
7.1.08
7.1.09
7.1.10 boat.512 elaine.512 gray21.512 numbers.512 ruler.512 5.3.01
5.3.02 7.2.01 testpat.1k Mean
Std
Plain-images
146.58 643.59 162.52 319.59 2156.50 373.42 365.48 495.37 500.50 112.71 66.00 111.90 87.06 223.11 214.00 169.91 80.87 173.58 82.16 264.54 118.06 32.12 2412.02 9887.66 180.32 294.80 65.67 3004.42
Cipher-images
Wu’s [51] 10,990.18
10,928.06 10,970.74 10,947.53 10,970.09 10,915.49 10,924.60 10,874.32 10,883.53 10,939.47 10,890.05 10,930.22 10,953.86 10,920.42 10,909.55 10,938.58 10,925.85 10,908.33 10,931.56 10,916.86 10,952.38 10,918.52 10,938.82 10,969.83 10,910.96 10,931.86 10,933.63 10,928.63 10,930.50 26.4372
Zhou’s [26] 10,825.09
10,759.94 10,951.97 10,984.96 11,063.32 10,822.34 10,825.03 10,913.67 10,862.59 10,779.52 10,967.52 10,831.80 10,794.12 10,779.67 10,814.31 10,778.70 10,748.99 10,811.99 10,733.89 10,815.90 10,849.81 10,920.27 10,878.45 11,132.50 10,886.72 10,818.72 11,005.99 10,906.06 10,866.56 97.3630
Wang’s [4] 10,893.38
10,899.89 11,208.19 10,878.80 10,673.72 10,882.79 10,950.11 10,620.65 10,927.80 10,828.41 11,053.26 10,799.62 10,783.33 10,727.20 10,983.37 10,816.53 10,689.40 10,800.38 10,756.54 10,808.99 10,873.72 10,780.19 10,842.31 10,964.11 10,893.26 10,765.51 10,888.40 10,785.69 10,849.13 120.0099
Liu’s [5] 10,931.11
10,903.62 11,013.58 10,907.35 10,968.20 10,957.51 10,941.72 10,934.30 10,914.24 10,901.89 10,920.88 10,929.77 10,906.88 10,956.37 10,927.87 10,934.56 10,960.78 10,959.07 10,944.49 10,908.80 10,951.17 10,946.62 10,928.22 10,918.64 10,919.96 10,926.88 10,925.96 10,922.59 10,934.39 24.2624
Zhou’s [29] 10,895.04
10,932.14 10,910.65 10,916.19 10,961.14 10,948.08 10,920.95 10,940.19 10,921.69 10,873.45 10,891.17 10,955.71 10,938.24 10,923.36 10,918.13 10,889.13 10,916.22 10,952.96 10,911.70 10,938.30 10,938.32 10,943.65 10,912.27 10,917.91 10,918.70 10,937.98 10,891.52 10,917.30 10,922.57 21.7385
Xu’s [25] 10,864.31
10,921.68 10,879.77 10,887.21 10,915.41 10,888.47 10,958.19 10,923.55 10,912.08 10,904.39 10,924.32 10,936.30 10,925.36 10,893.59 10,902.22 10,881.50 10,873.91 10,870.78 10,914.01 10,920.49 10,940.97 10,890.10 10,914.78 10,918.19 10,940.02 10,928.71 10,929.63 10,916.52 10,909.87 23.6144
LSCM-IEA
10,918.65 10,971.56 10,892.87 10,906.51 10,918.48 10,935.12 10,930.46 10,912.54 10,894.26 10,869.71 10,925.80 10,934.25 10,927.72 10,944.33 10,908.84 10,939.20 10,879.07 10,901.27 10,958.93 10,908.76 10,913.04 10,932.86 10,922.89 10,915.47 10,931.70 10,919.60 10,911.24 10,908.33 10,919.05 21.7588
Z. Hua et al./Signal Processing 149 (2018) 148–161 159
Fig. 11. NPCR scores of several image encryption schemes using different size of images: (a) images of size 256×256; (b) images of size 512×512; (c) images of size 1024 × 1024.
Fig. 12. UACI scores of several image encryption schemes using different size of images: (a) images of size 256×256; (b) images of size 512×512; (c) images of size 1024 × 1024.
groove depth [52]. It can be used to measure pixel distribution and pixel local variation of an image matrix and its mathematical defi- nition is shown as
C=|i−j|2G′(i,j), (20) i,j
where G′ is gray level co-occurrence matrix, which indicates the probability of fixed patterns with a predefined distance and direc- tion. Fig. 13 shows an example of generating a unnormalized gray
level co-occurrence matrix G from an image Q with 1 distance and horizontal direction. One can see that the size of G is 4×4 as the grayscale level of Q is 4. G(0, 1) = 3 as the number of pattern (0,1) (see the blue cells) in horizontal direction is 3. By this way, we can obtain each value of G, which can be seen from the figure. As the total number of patterns with distance 1 and horizontal direc- tion in Q is 20, we can obtain the gray level co-occurrence matrix, namely G′ = G/20.
Fig. 13. An example of generating the gray level co-occurrence matrix.
160 Z. Hua et al./Signal Processing 149 (2018) 148–161
In our experiments, the test images are all 8-bit grayscale im-
ages, and thus we can obtain the size of G′ as 256×256. For an
ideally random image, each of its patterns is expected to be the
′
same, namely G (i, j) = 1/(256 × 256), where i ∈ [0, 255] and j ∈ [0,
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255]. Thus, the contrast feature of an ideally random image is C = 256 256 |i − j|2/(256 × 256) = 10, 922.50. In each ex-
expected i=1 j=1
periment, we first calculate four gray level co-occurrence matrices
from four directions 0°, 45°, 90°, 135° with 1 distance, and then obtain four feature contrast values and get their average score. Table 5 lists the average feature contrast scores of plain-images and their cipher-images encrypted by different encryption algorithms. One can see that Zhou’s [29] algorithm can achieve the mean value score that is closest to the expected value and our proposed LSCM- IMA can achieve the second-best performance. Besides, LSCM-IMA can obtain a pretty small standard deviation.
6. Conclusion
In this paper, we presented a new chaotic map called 2D-LSCM. It is derived from the existing Logistic and Sine maps. First, couple the outputs of the Logistic and Sine maps using the sine trans- form and then extend the phase plane from 1D to 2D to en- hance the complexity. The chaos performance of 2D-LSCM was an- alyzed using trajectory, Lyapunov exponent, Kolmogorov entropy and dynamical degradation. The analysis results demonstrate that it has better chaos performance than several newly developed 2D chaotic maps, and is suitable for designing encryption algorithms. To show the applications of 2D-LSCM, we further designed a 2D- LSCM-based image encryption algorithm (LSCM-IEA). It has two main components, the 2D-LSCM permutation and 2D-LSCM diffu- sion. The former can fast shuffle pixel row and column positions si- multaneously to achieve confusion property, and the latter is able to spread few changes of plain-image to the whole cipher-image to obtain diffusion property. Simulation results show that LSCM- IEA can encrypt different types of images into unrecognized cipher- images with high efficiency. We have also analyzed the security of LSCM-IEA in terms of key security, ability of defending differential attack, local Shannon entropy and contrast analysis. The analysis results show that LSCM-IEA has a high security level and can out- perform some advanced image encryption algorithms. As the pro- posed LSCM-IEA has high efficiency and security level, our future work will investigate its application in video encryption.
Acknowledgements
This work is financially supported by National Natu- ral Science Foundation of China under grant nos. 61701137, 11371004 and 61672195, National Science and Technol- ogy Major Project under grant nos. 2016YFB0800804 and 2017YFB0803002 and Shenzhen Science and Technology Plan under grant nos. JCYJ20170307150704051, JCYJ20160318094336513, JCYJ20160318094101317 and KQCX20150326141251370.
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