Dynamic Analysis and Cryptographic Application of a Sinusoidal-polynomial Composite Chaotic System

Owning to complex properties of ergodici-ty, non-periodic ability and sensitivity to initial states, chaotic systems are widely used in cryptography. In this paper, we propose a sinusoidal–polynomial composite chaotic system (SPCCS), and prove that it satisﬁes Devaney’s deﬁnition of chaos: the sensitivity to initial conditions, topological transitivity and density of periodic points. The experimental results show that the SPCCS has better unpredictability and more complex chaotic behavior than the classical chaotic maps. Furthermore, we provide a new image encryption algorithm combining pixel segmentation operation, block chaotic matrix confusing operation, and pixel diﬀusion operation with the SPCCS. Detailed simulation results verify eﬀectiveness of the proposed image encryption algorithm.

. However, there hasn't been a strict and uniform mathematical definition of chaos yet. The most famous measure of chaos is Li-Yorke's chaos and Devaney's chaos, which are equivalent in some sense [5,22]. With the development of chaos, people proposed many chaotic maps, such as Sine map, Chebyshev map, Logistic map and Tent map. Based on these classical chaotic maps, many kinds of variants were constructed [1,2,4,6,15,17,33]. Specifically, in [1], Ashish et al. provided rich dynamical chaotic behaviors by using a Mann iterative operation to Logistic map. In [13], Hua et al. generated 1D chaotic framework, which had better chaotic property than the used seed system. Furthermore, Hua et al. proposed the nonlinear chaotic structure containing six basic nonlinear operations with existing chaotic maps to generate a huge number of new chaotic maps in [14]. In these papers, the authors often tried to indicate the system's chaotic behavior by analyzing many mathematical experiments, however, the proof with computer simulation is not mathematically strict.
Moreover, using the definition of Li-Yorke's chaos or Devaney's chaos, many researchers have achieved significant results on dynamic analysis. For instance, Wu et al. proved that the lattice dynamical system was Li-Yorke chaotic for coupling constant 0 < ε < 1 [34]. Luo et al. presented a homeomorphism and showed that an invertible mapping on a compact metric space was Li-Yorke chaotic [25]. Wang et al. provided two chaotic systems satisfying Devaney's definition of chaos in [29,30]. In [38], Zheng et al. solved the chaotic dynamical degradation problem in real domain by providing an effective schemes with symbolic dynamics.
In fact, the improvement of chaos theory also has benefited image secure communication. The output of a chaotic system is sensitive to change of its parameter, non-periodic and unpredictable, which are consistent with the sensitivity requirement of a cryptographic system on secret key and the plaintext. Since Fridrich's chaotic encryption scheme analyzed in [36] was proposed in 1998, the corresponding research achievements have been growing rapidly [2,8,11,12,16,19,27,31,33,37,40,41]. In [8], Guesmi et al. used DNA sequences and Lorenz chaotic system to enhance the information entropy of the proposed algorithm. Wang et al. proposed a high degree of security image communication scheme with Arnold coupled logistic map lattice model [31]. Due to the dynamics degradation of chaotic systems in finite-precision digital computer, the randomness of pseudo-random number sequence generated by iterating a chaotic system may be very low [18,20,28,38]. In [9], Hu et al. proposed a new encryption scheme combining chaos with cycle operation for DNA sequences and obtained excellent diffusion properties. In [39], Zhou et al. proposed a novel 1D compound chaotic map with several simple chaotic maps. Ge et al. in [7] studied security of a feedback image encryption algorithm with rigorous mathematical proof. Besides, in [42], Zhu et al. also proposed an efficient image encryption algorithm with 2D hyper-chaotic system and local binary pattern, which had better security than ordinary chaotic-based encryption system.
In this paper, we propose a 1D sinusoidal polynomial composite chaotic system (SPCCS) combining Sine function with some special polynomial functions, and prove that the SPCCS satisfies Devaney's definition of chaos: the sensitivity to initial conditions, topological transitivity and periodic point density. To analyze the chaotic property of SPCCS, mathematical indexes such as bifurcation diagram, Lyapunov exponent and Correlation dimension are realized. The results show that SPCCS has excellent chaotic characteristics, and it is consistent with the theoretical analysis. The chaotic sequences generated by SPCCS are used for two rounds pixel segmentation operation, block chaotic matrix confusing operation, and pixel diffusion operation in the proposed image encryption algorithm. Experiments and simulation are provided to show the effectiveness of the theoretical analysis of the proposed algorithm.
The rest of the paper is organized as follows. A 1-D sinusoidal-polynomial composite chaotic system (SPCC-S) is proposed in Sec. 2. Then, the proof of Devaney's definition of chaos of SPCCS is given in Sec. 3. Sec. 4 illustrates the chaotic performance of SPCCS with Chebyshev map and Sine map by some mathematical experiments. The proposed SPCCS-based image encryption algorithm is provided in Sec. 5. The encrypted performance and security analysis are presented in Sec. 6 and Sec. 7, respectively. Sec. 8 concludes the paper.

Sinusoidal-Polynomial Composite Chaotic System
As we know, Sine map can be denoted by If µ ∈ [0.87, 1], Sine map is chaotic. Especially, when µ = 1, it is a self-map defined in [0,1]. Similarly, Chebyshev map can be defined by If λ > 1, Chebyshev map has chaotic behaviors, and it is also a self-map defined in [0, 1]. Sine map and Chebyshev map are usually used as seed maps to apply various compound chaotic systems and chaotic encryption [39]. Then, we propose the sinusoidal-polynomial composite chaotic system (SPCCS) where a > 0. Moreover, we assume that g(x) satisfies the following assumptions: is a polynomial function with only odd terms and positive coefficients. Namely, g(x) = n i=0 p i x i , where p 2i = 0 and p 2i+1 ≥ 0.
-2) a is a positive real number and satisfies g(a) = kπ (k = 2, 3, · · · ) and Especially, for the inequality (2), we can flexibly adjust the value of p 1 to make the value of a wider.
Obviously, f (x) is a self-map in [0, 1]. It should be noted that due to the mathematic complexity of the definition of Li-Yorkes chaos, Devaney's chaos is better than Li-Yorkes chao in many special sense. In addition, we give the Theorem 1 and Theorem 2 to prove the Devaney's chaos of SPCCS x n+1 = sin(g(ax n )) in Sec. 3.
where f n (x) denotes the n-th iteration of f (x) = sin(g(ax)).
, then due to a > 0, one has Similarly, for the different extreme points x i and x i+1 , one can obtain from |f ′ (x + i )| = 1 and |f ′ (x − i+1 )| = 1. Especially, x 1 and x n are respectively the first and last extreme points of f (x) in [0,1], so g(ax 1 ) = π 2 and g(ax n ) = kπ − π 2 . To facilitate the above the discussion, draw the situation in the neighborhood of the first two extreme points and the last extreme point f (x) shown in Fig. 1. According to Eqs. (4) and (5), one has where i = 2, 3, · · · , n. For any x i , according to Lagrange's median value theorem and [ If we can proof x 1 − x − 1 < 1 − x + n , then the above equation becomes Meanwhile, one can obtain f (x − i ), 1 ⊂ (x + n , 1) and f (x + i ), 1 ⊂ (x + n , 1) ⊂ X 1 , so the theorem is proved. Now, we turn to proof x 1 −x − 1 < 1−x + n . Setting i = x 1 and i = x n in Eq. (3), one can get | cos(g(ax − 1 ))| > | cos(g(ax + n ))|, and From g ′′ (x) > 0, one can obtain holds, one has It can be seen that the Eq. (8) is equivalent to Referring to Eq. (6) and Eq. (7), one has According to the inequality (2), one can know that Eq. (7) holds, thus

The Proof of SPCCS by Devaney's Definition of Chaos
In this Section, we will prove SPCCS satisfies Devaney's definition of chaos.

Sensitivity dependence on initial conditions
Firstly, we divide the interval [−1, 1] into two disjoint sets X 1 and X 2 : Then, we discuss the sensitivity dependence on the interval X 1 and X 2 , respectively.
(1) For every x 0 ∈ X 1 , ε > 0, there exists an integer , then there is Thus, for any given ε > 0, when k > k * , one can get , then there must be Thus, for any given ε > 0, when k > k ′ * , one has In conclusion, for any x ∈ [−1, 1], the function f (x) satisfies the sensitivity dependence on initial conditions.

Topological transitivity
Let X 1 k and X 2 k be the maximum and minimum points set of k-th (k = 1, 2, · · ·) iteration of f (x), respectively. Then X k The elements of X 1 k and X 2 k are arranged in ascending order, we have X k because the elements in X k 1 and X k 2 are origin-symmetric. Without loss of generality, we only consider d k 2 corresponding to X k 2 in the following.
According to Theorem 1 and f (±1) = sin(g(±a)) = 0, there are only two types curves of f k (x) between x k 2i0 and x k 2,i0+1 which are shown in Fig. 2 With the increasing of iterations times k, the So, it implies that lim

Numerical simulations
In this section, we propose two specific examples of SPCCS called S-P1 map and S-P2 map with a ∈ [0, 3] in Table 1. The chaotic behaviors are compared with Chebyshev chaotic map and Sine chaotic map in Subsection 4.1, Subsection 4.2 and Subsection 4.3, respectively.

Bifurcation diagram
According to the dynamical system theory, the bifurcation phenomenon is a symbol of chaos. In this subsection, the bifurcation diagrams of Sine map, Chebyshev map, S-P1 map and S-P2 map are illustrated in Fig. 4. As can be seen from Fig. 4, the output of S-P1 map and S-P2 map have larger chaotic area than Sine map and Chebyshev map.

Lyapunov exponent
The Lyapunov exponent describes the orbit produced by two very similar initial values in the phase space [19,34]. The average rate of dispersion or convergence of Lyapunov exponent (LE) is an important index to judge chaos. Here, we depict the Lyapunov exponent spectrum of the Sine map, Chebyshev map, S-P1 map and S-P2 map in Fig. 5 with a ∈ [0, 3]. Morover, the corresponding maximum Lyapunov exponent and average Lyapunov calculation results are also listed in Table 2. All these results demonstrate that the S-P1 map and S-P2 map have larger Lyapunov exponent and average Lyapunov exponent, and have better chaotic and random performance.

Correlation dimension
Correlation dimension (CD) is applied to illustrate the singularity of attractors in dynamical system [25]. Let S = {s 1 , s 2 , · · · } be a time series and e be embedded dimension, then CD of the S is computed by log C e (r) log r , where θ(·) is the Heaviside step function and ζ is time delay. Ands t = (s t , s t+ζ , s t+2ζ , · · · , s t+(e−1)ζ ), t = 1, 2, · · · , N − (e − 1)ζ. Generally, the embedding dimension e = 2 and time delay ζ = 1 for a 1-D system [26]. Then the CD values of different systems are drawn in Fig. 6. From Fig. 6, it can be seen that the S-P1 map and S-P2 map have more stable correlation dimension values and larger parameter change fields. Meanwhile, the S-P1 map and S-P2 map have higher maximum and average correlation dimension values with a ∈ [0, 3]. Therefore, the phase space of the S-P1 map and S-P2 map have excellent chaotic performance.

The encryption algorithm
In this section, we propose a SPCCS-based image encryption algorithm, which includes pixel segmentation operation, and two rounds of block chaotic matrix confusing operation and Pixel diffusion operation. Its detailed contents are described as follows: For an image P with size of M ×N , firstly, we divide it into an image H with size of 2 k × 2 l (k ≤ l) by the pixel segmentation operation. Then, the image H is divided into blocks of size 2 k−1 ×2 k−1 . And the confused image U is obtained by block chaotic matrix generated with SPCCS for each block of image H. Finally, the cipher image W generated by the first round of the pixel diffusion operation is used as the H of the second round of the block chaotic matrix confusing operation, and the final W is obtained after the second round of pixel diffusion operation. The encryption algorithm flow chart is shown in Fig. 7.  Fig. 7 The image encryption algorithm.

Pixel segmentation
The detailed steps of the pixel segmentation operation are shown as follows.
Step 1: For a plain image P with size of M × N , choose the integers k, l,(k ≤ l) that 2 k−1 < M ≤ 2 k , 2 l−1 < N ≤ 2 l .
Step 2: Randomly select the initial values x 1 , x 2 and the control parameters a 1 , a 2 of SPCCS that satisfy condition 2), and iterate 2 k × 2 l + 2000 times to obtain two chaotic sequences s 1 , s 2 , where the first 2000 outputs are discard to avoid the harmful effect of SPCCS, then obtain two integer sequences S 1 , S 2 by Step 3: Suppose m = 2 k − M , so m < M . Firstly, for each pixel value x from M −m+1 to M row of plain image P , we turn it into a 8 bits binary sequence and divide it into two pixel values by t 1 = CL(x, S 1 (i)), t 2 = CR(x, S 1 (i)).
Then we concatenate t 1 , t 2 vertically, and get an intermediate image T with size of 2 k ×N , where CL(x, S 1 (i)) represents to circleleft shift x with chaotic number S 1 (i), CR(x, S 1 (i)) represents to circle right shift x with chaotic number S 1 (i).
Step 4: Suppose n = 2 l − N , so, n < N . For each pixel value y from column N − n + 1 to N of image T , we divide it into two pixel values by y 1 = CL(x, S 2 (i)), y 2 = CR(x, S 2 (i)).

Block chaotic matrix confusing operation
After the pixel segmentation operation, we do block chaotic matrix confusing operation in this subsection, the detailed steps are illustrated as follows.
Step 1: For image H obtained in Subsection 5.1, we truncate it into q = 2 l−k+2 image blocks [K t ] q t=1 with size of 2 k−1 × 2 k−1 from left to right and top to bottom.
Step 2: Randomly select the initial value x 3 and the control parameter a 3 of SPCCS that satisfy condition 2). Then we iterate 2 k × 2 l + 2000 times to obtain a chaotic sequence s 3 , and obtain an integer sequence S 3 by S 3 = f loor(s 3 × 2 16 ) mod 256.
Step 3: Use the sequence S 3 to get the initialized image matrix [V t ] q t=1 from left to right and top to bottom. Then, we sort each row of V t and obtain corresponding index matrices [D t ] q t=1 .
Step 4: Construct a new index matrix Q t with the index matrix D t by Then, we can obtain the blocked confused image with size of 2 k−1 × 2 k−1 by G t (D t (i, j), i) = K t (D t (i + 1, j), i + 1) Step 5: For i = 1 to q, repeat Step 4, and generate the confused image matrix U with size of 2 k × 2 l .

Pixel diffusion
After the block chaotic matrix confusing operation, in this subsection, we will show the detailed pixel diffusion procedure.
Step 3: Randomly select integer r 1 ∈ [0, 255], and do Step3 and Step 4, then where U is the confused image matrix with size of 2 k × 2 l .
Step 4: For j=2 to 2 l − 1, we calculate Step 5: For i = 2 to 2 k , j = 1 to 2 l , we diffuse the next 2 k − 1 rows of image U by Step 6: Compute W = W ⊕ J 2 and obtain W .

Simulation results
In this section, to illustrate the encryption performance of the proposed SPCCS-based encryption algorithm. Many different kinds of images such as gray value images, color images and binary images are chosen from USB-SIPS image database and Brown Univ Large Binary image database to do the encryption quality and encryption time analysis.

Encryption quality
Obviously, the proposed image algorithm should have outstanding encryption performances in different kinds of images. Fig. 11 shows the encryption effect of binary image, gray image, monochrome image and color images, respectively. Fig. 11c shows that all the plain images are encrypted to a random-like image. Moreover, as show in Fig. 11d, the histogram of cipher images are uniformly distributed that verify the encrypted effect, and the attackers can't obtain meaningful information about the plain images. Fig. 11 Simulation results. (a) the binary, gray, monochrome and color images; (b) histograms of (a); (c) encrypted results of (a); (d) histograms of (c); (e) decrypted results of (c).

Encryption time performance
The running time of a good image encryption algorithm must be acceptable for different scenario [21]. From Table 3, the encryption time of the proposed algorithm is much lower than that of the three comparable algorithms for different scenario, it shows the proposed encryption algorithm can be a good choice for practical application.

Security analysis
An ideal image encryption algorithm must can pass several security analysis, and if SPCCS-based image encryption algorithm fails one of the security index, it means the SPCCS-based algorithm cannot be used for real application. Here, different kinds of security analysis for the SPCCS-based algorithm are depicted.

Histogram analysis
From the perspective of statistics, the histogram of cipher image must be flat. Fig. 11 shows the histograms of the white-black image, gray image, monochrome image and color image, respectively. As shown in Fig. 11, the histograms of their encrypted images are all uniform distributions, so, it is enough to resist statistic attacks for universal image.

Correlation analysis
Generally, image data has high redundancy of data and strong correlation among neighboring pixels, which is meaningful for attackers. Then, we choose 1000 pairs of adjacent pixels from plain and encrypted images, and analyze the correlations at horizontal, vertical and diagonal directions, respectively. The correlation coefficient is calculated by where E(•) and D(•) are the expectation and variance operation of a random variable. The correlation diagram and correlation coefficients are shown in Fig. 12 and Table 4, respectively. In Fig. 12 and Table 4, the correlation coefficient of cipher image is very close to 0. It implies that the correlation of the plain image is eliminated in encrypted image.

Local Shanny entropy
Except for information entropy, Local Shanny entropy (LSE) is another norm for measuring randomness [35]. It is computed by

Key sensitivity analysis
Apparently, the security level of key is the most important factor of an encryption algorithm. So if there is a slightly change in the key, it must lead to totally different plain or cipher image [19]. In this subsection, we do the encryption and decryption simulation with very slightly change of key. From Fig. 13 and Fig. 14, it can be known that the proposed SPCCS-based algorithm is very sensitive to tiny change of key, which is most important for image communication. Furthermore, we use the number of bit change rate (NBCR) to quantitatively compute the difference between encrypted and decrypted images. NBCR can be described as where L represents the length of Q 1 , Q 2 , and H[Q 1 , Q 2 ] represents the Hamming distance [3]. If S 1 and S 2 are two independent sequences, the ideal NBCR is close to 50%. Next, we provide two group slightly keys to encrypt and decrypt the plain image and cipher image, respectively. The NBCR results are calculated in Table 6 and Table 7. In Table 6 and Table 7, if a slightly change appeared in key, then it will lead to totally different encrypted and decrypted results. So,the SPCCSbased encryption algorithm is quite sensitive to key. 7.5 Capability of defending differential attack Except for the key sensitivity, the SPCCS-based algorithm must be sensitive to the slightly change in  the plain image. The number of pixels change rate (N-PCR)and unified average changing intensity (UACI) are two crucial indictors to evaluate this sensitivity. The detailed definition of NPCR and UACI are shown by Note that G is the total number of pixels of encrypted image, and L is the gray level of the image. Recently, the expected scores of NPCR and UACI were shown in [10], which are given by If L =256, then N P CR e = 99.6094% and U ACI e = 33.4635%. It means that if the NPCR and UACI value of cipher image are close to 99.6094% and 33.4635%, the cipher image will have stronger ability to resist differential attack. The results are shown in Table 8 and Table 9, respectively. In Table 8 and Table 9, the NPCR  and UACI of SPCCS-based algorithm has the smallest difference to the ideal value than that of Refs. [2,39,42], Thus, the proposed SPCCS-based scheme is very sensitive to slightly modification in plain image.

Ability of resisting chosen-plaintext and chosen-ciphertext attacks
Chosen-plaintext and chosen-ciphertext attacks are widely used for attacking in cryptanalysis. For the SPCCSbased encryption algorithm, specific operations can be used to resist chosen-plaintext and chosen-ciphertext attacks: -The size of plain image is changed by pixel segmentation operation, and chaotic sequences are used to control the pixel value after segmentation. Then, the pixel value of the plain image is segmented for several times. Even if a attacker use a chosen-plaintext and chosen-ciphertext attacks, it is difficult to obtain the complete pixel value information from the cipher image. -In the pixel diffusion operation, there is an operation to accumulate pixel values, which affects the subsequent diffusion process. Furthermore, the block chaotic matrix confusion operation disturb the distribution of pixel values in a plain image. After two rounds confusion and diffusion operations, each pixel value information of plain image is expanded to the whole cipher image. Therefore, slight changes in the plain images can result in huge differences in cipher images.
-The pixel segmentation operation makes the size of the cipher image different from the size of the corresponding plain image, and links the size of the cipher image and the plain image.

Data loss attack analysis
The noise and data loss can influence digital images during transmission through the network and storage in physical media. An image encryption algorithm should resist these abnormal phenomena. Then, we provide some experiments on data loss and noise analyses. Observing Fig. 15, the SPCCS-based algorithm can decode the experimental cipher images with a good visual effect by 3%, 10%, 30% data loss, respectively. Moreover, even the encrypted image is added Salt & Pepper noise, the SPCCS-based algorithm also can decrypt the cipher images with a good visual effect, which can be seen in

Conclusion
In this paper, we proposed a one-dimensional sinusoidal polynomial composite chaotic system satisfying Devaney's definition of chaos. Its dynamics properties were analyzed comprehensively. Moreover, a new image encryption algorithm combining pixel segmentation operation and block chaotic matrix confusing operation with the chaotic system was established and analyzed by various metrics, including key size analysis, histogram analysis, correlation analysis, local Shanny entropy, key sensitivity analysis, ability of defending differential attack, ability of resisting chosen-plaintext and chosen-ciphertext attacks, and data loss attack. This work will promote application of cascading multiple chaotic system in protecting image data communication.