A spatiotemporal chaotic system based on pseudo-random coupled map lattices and elementary cellular automata

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Introduction
In the past decades, the chaotic system has become a research hotspot in nonlinear systems. Since the chaotic system which is a deterministic system possesses many particular dynamical properties such as ergodicity, unpredictability, pseudorandomness and initial sensitivity, etc. [1][2][3][4][5][6], it has been utilized in many fields, including mathematics [7][8], biology [9][10], physics [11][12], cryptology [3][4][5][13][14], and even social science [15][16]. Nevertheless, dynamical degradation will occur when the chaotic system operates in digital computers with finite computing precision [17]. Due to the inherent drawback that digital chaos systems have, quite a few schemes have been proposed to alleviate the dynamical degradation: using higher precision digital system [18], cascading multiple chaotic systems [19][20], randomly perturbing the chaotic systems [21]. According to the theoretical analysis in literature [17], the best of the above-mentioned scheme is randomly perturbing the system. And the most effective perturbation-based solution is perturbing the output of iteration in the chaotic system, hence it attracts much more attention than other solutions [22][23][24]. In literature [25][26][27], the authors use the delay-introducing method as a novel solution to improve the dynamical degradation. It's essentially perturbing the control parameter. In other words, comparing with perturbing the output of iteration the effect of this approach is limited.
The above-mentioned schemes for alleviating dynamical degradation are lowdimensional chaos systems. Unfortunately, they are vulnerable to attacks based on the phase space reconstruction for crypto-system [28]. In recent years, as the international research on chaotic systems goes further, high-dimensional chaotic systems have attracted more attention, especially the spatiotemporal chaotic systems [3][4][5][13][14] which not only can efficiently alleviate the dynamical degradation but also possess a larger Lyapunov exponent (LE) and more complex dynamical performance [3]. Since the coupled map lattices (CML) [29][30] as an enlightening technique was proposed in 1985 [31], lots of spatiotemporal chaotic systems based on CML are investigated. According to coupling methods, the spatiotemporal chaotic systems can be classified into three categories [4]: spatial adjacent coupling, spatial random coupling, spatial nonlinear coupling. Although the first category is the most popular scheme that regards the output of adjacent lattices as interference to the current lattice [3,14,[32][33], the periodic windows still exist in its bifurcation diagrams which signify that the ergodicity of the system is broken, and the correlation between the output sequences of different lattices is very high that obviously increases the security risk to cryptosystem [13]. The spatial random coupling efficiently reduces the above-mentioned correlation, but it has a fatal flaw for a crypto-system that the sequences generated by such system can't be reproduced [5]. It means that the spatiotemporal chaotic system based on the spatial random coupling just suits for a pseudo-random numbers generator rather than a crypto-system. As for the spatial nonlinear coupling, it takes into account the complexity and reproducibility of the spatiotemporal chaotic systems. Zhang et al. [4][5] developed a nonlinear coupling based on the Arnold cat map [34], and found that the systems they proposed possess new chaotic features which are superior to the conventional CML. It was subsequently employed in image encryption [1][2]. They demonstrated the effectiveness, feasibility and security of the CML based on nonlinear coupling for crypto-system. However, the randomness of the sequences generated by the system was not tested by the NIST SP800-22 suite [35] which is used as an important standard for judging whether the sequences possess random performance. Moreover, according to the analysis in literature [36][37][38], Arnold cat map has the characteristic of periodicity that brings potential flaws for the system.
To remedy the aforementioned problems, a novel pseudo-random CML (PRCML) system with perturbation is proposed in this article, which not only has the advantages of above-mentioned schemes but also significantly reduces the correlation between the sequences generated by any two different lattices.
Moreover, the sequences generated by the proposed spatiotemporal chaotic system have passed the NIST test that proves the outstanding randomness of our scheme. The main contributions of this paper are as follows: Without loss of generality, the logistic map is used as the iterative function of each lattice. And rather than employing the adjacent lattices for coupling as a conventional CML, the PRCML's choice of lattices for coupling is innovatively dependent on the iterative result of the ECA which can be chaotic or complex [39][40][41]. Thus, the dynamical degradation is effectively mitigated. Moreover, because of the chaotic character of ECA, the periodicity as Arnold cat map is avoided.
The iterative result of the ECA is employed for disturbing the PRCML which leads to the following advantages: Firstly, the periodic windows in bifurcation diagram fade out obviously. Secondly, dynamical degradation is further alleviated according to literature [17]. Thirdly, owing to the different pseudo-random perturbation for each lattice, the correlation between any two lattices is significantly reduced. Above all, the perturbation based on the ECA improves the ergodicity, unpredictability and complexity of the PRCML which is more suitable for crypto-system.
The remaining part of this paper is organized as follows. In Sect. 2, the preliminary knowledge about CML and ECA is introduced, and the PRCML with perturbation is presented in Sect. 3. Then, simulation results and performance analyses are reported in Sect. 4. Finally, the conclusion is drawn in Sect. 5.

Coupled map lattices system
Coupled map lattices system is a spatiotemporal chaotic system that can alleviate the dynamical degradation and enhance the complexity of digital chaotic system. In the CML system, the current lattice is determined by the two adjacent coupling lattices, which is defined as Eq. (1): where n (n = 0, 1, 2, …) is the number of iterations and ε (0 ≤ ε ≤ 1) is the coupling coefficient which represents the strength of coupling. i (i = 1, 2, …, L) is the index of current lattice, and L is the total number of lattices in the system in which Lth lattice is regarded as the left adjacent lattice of the 1-th. In generally, f(x) is the logistic map as follows: where μ (0 < μ ≤ 4) is the control parameter. When μ is in interval [3.6, 4], the system is chaotic.

Elementary cellular automata
The Cellular automata (CA) was proposed by Stanislaw M. Ulam and John von Neumann in 1948 [42], which was initially utilized for simulating the selfreproduction of bio-systems. And it was also a simplified mathematical model of complex phenomena in nature.
The CA is made up of the cell, cell-space, neighbor and rule that can be defined as follows: ( , , , ), where A N represents cell-space, and N is the dimension of cell space. ∑ is the set of cell states. f is local transition rule. E denotes the boundary condition of CA.
As for the elementary CA (ECA), it's a special one-dimension CA in which the number of cell states is 2. And the set of cell states ∑ can be represented as {0, 1}.
The radius of cellular automata is 1 which means that the neighbors of current cell are the adjacent two cells. And the boundary condition of ECA is cyclic boundary condition which is shown as Fig. 1. of C1, and C1 is the right neighbor of CL. In the ECA, the next state of a cell is absolutely determined by the current states of itself and its two neighbors that is expressed as Eq. (4). 1 11 ( , , ), in which t i S is the current state of i-th cell which can be represented as "0" or "1", t is the number of iterations, and i is the index of cells. f is the local transition rule that is essentially a mapping from the set {000, 001, 010, 011, …, 111} to the set {0, 1}, and it is easy to know that there are 256 ECAs. For instance, the local transition rule of ECA No. 105 is shown in Table 1. lattice represents the cell with the status value "1" ("0"). It is obvious that the iterative result is pseudo-random, aperiodic, etc. 3 The proposed pseudo-random CML system with perturbation The proposed system coupled by pseudo-random links as follows: where f(x) represents the logistic map as Eq. (2). ε is the coupling coefficient. i (i = 1, 2, …, L) denotes the index of the current lattice, and L is the total number of lattices, L = 100 in proposed system. n (n = 0, 1, 2, …) is the number of iterations.
a and b are the indexes of lattices for coupling, which are defined by the ECA as follows: in which S n is the iterative result of the ECA as Eq. (4), n is the number of iterations in Eq. (5). And the total number of cells is 100 in the proposed system. i is the index of current lattices in Eq. (5). search(S n , i) is a function for finding two cells, the status value of which must be "1", and they are the nearest two of the ith cell in S n . The return values of search(S n , i) are the indexes of above two cells. For example, the process of this function is shown as Fig. 3. As for p(S n ) in Eq. (5), it is the function to calculate the perturbation. S n can be regarded as a binary number S n =b1b2b3…b100, and bi denotes the status value of Ci.
The middle 32 bits of S n are employed for perturbing the proposed system. p(S n ) is calculated as 35 and the operation x mod 1 is to preserve the fractional part of x, which ensures that the result of Eq. (5) is always in the interval (0, 1).

The dynamic properties of PRCML system
For comparison analysis, the control parameters and initial value of PRCML where S0 is represented as a hexadecimal number. And No. 105 is selected as the local transition rule of the ECA, which insures that the ECA is chaotic, aperiodic and pseudo-random.
The CML system, the mixed linear-nonlinear CML (MLNCML) system and the Logistic-dynamic mixed linear-nonlinear CML (LDCML) system [13] are chosen to compare with the proposed system. The initial values of them are set as former, and η=0.8 in the MLNCML and LDCML.

Kolmogorov-Sinai entropy
The Lyapunov exponent (LE) is an important indicator to evaluate the average exponential divergence rate of adjacent orbits in the phase space [14], which is defined as: where F(x) is the mathematical expression of dynamical system. A chaotic system must possess at least one positive LE, and the larger  is, the more chaotic and complex the system is. Without loss of generality, the LEs are calculated by the wolf method [43] as many researchers do [1][2][3]14]. The Kolmogorov-Sinai entropy density is the average of the positive LEs of all lattices [14], which can evaluate the entire multi-dimensional chaotic behavior. It can be described as follows: where h denotes the Kolmogorov-Sinai entropy density,   Fig. 4(a)-(c). Correspondingly, the PRCML system's reaches 0.8 in Fig. 4(d), which is markedly higher than any of the above systems.
It's verified that the chaotic characteristic of PRCML system is much better.
Specifically speaking, as shown in Fig. 4(a)-(c), only when μ >3.6 can the former three systems be in chaos, and the chaotic property is sensitive to the control parameter μ because that the Kolmogorov-Sinai entropy densities increase as μ rises. Furthermore, h is lower around ε = 0.2 in Fig. 4(a)-(b), which indicates that the chaos of CML and MLNCML system is weak when coupling coefficient is around 0.2, and the LDCML system effectively overcomes this drawback as shown in Fig. 4(c). Fortunately, the PRCML system not only avoids above weakness but also has the higher Kolmogorov-Sinai entropy density in entire interval μ ∈ (3, 4] and ε ∈ (0, 1]. It means that the range of control parameters which lead to chaotic behavior is wider than former three systems.
The Kolmogorov-Sinai entropy universality is another important indicator to evaluate the chaotic behavior in spatial level, which is defined as follow: in which L + is the number of lattices with positive LEs in the spatiotemporal chaotic system, and L denotes the total number of lattices. Apparently, hu indicates the percentage of lattices which is in chaos. The Kolmogorov-Sinai entropy universality of above-mentioned systems is illustrated in Fig. 5. (b) hu of the MLNCML (a) hu of the CML Fortunately, the situation has been greatly improved in the PRCML system as shown in Fig. 5(d), from which it's easy to know that all lattices are chaotic in entire interval μ ∈ (3, 4], ε ∈ (0, 1]. In conclusion, the PRCML system possesses stronger chaotic property, and it contains more parameter pairs which lead to chaos. That is, the proposed system provides wider secret key space in parameter (ε , μ) than others so that it is more suitable for building a crypto-system.

Bifurcation diagram
The befurcation diagram evaluates the periodicity and ergodicity of the systems. interval. When μ>2.5, the identifiable orbits have disappeared, and because the ECA and coupled map lattices is both in chaos the PRCML system is thoroughly chaotic and turbulent.

Phase space and time-domain analysis
The uniformity is an important property of a chaotic system when it is employed in crypto-system. The phase space and time-domain analysis are utilized for evaluating the uniformity of the spatiotemporal chaotic system. We assign μ=4.
The chaotic attractors of CML, MLNCML, LDCML and PRCML systems in phase space are listed as follows.

Correlation analysis
Because of the coupling in conventional spatiotemporal chaotic system, the correlation between the outputs of lattices is very high. However, the sequences generated by a same crypto-system should be uncorrelated between each other.
Otherwise, the enemy can easily deduce the output of current lattice according to the outputs of other lattices. In other words, there is a serious security problem in conventional CML system when it is utilized in crypto-system. In this article, Pearson correlation coefficient of each two lattices is employed for correlation analysis, which is defined as follow: Where i, j are the indexes of lattices, and T is the total number of iterations or the length of the sequence which is set to 500. xi denotes the sequence generated by i-

The NIST test
The NIST SP800-22 suite proposed by National Institute of Standards and Technology is the most common statistical test tool, which is employed for evaluating the randomness and unpredictability of the sequences. There are 15 sub-tests in this suite as listed in Table 2, and all of them can be utilized to estimate the randomness of the sequence. For these tests, each P-value is the probability that a perfect random number generator would have produced a sequence less random than the sequence that was tested, given the kind of nonrandomness assessed by the test. If a P-value for a test is determined to be equal to 1, then the sequence appears to have perfect randomness. A P-value of zero indicates that the sequence appears to be completely non-random [36]. In this paper, the significance level α is set to 0.01. If P-value≥ α, the sequence apears to be random, which means that the sequence would be considered to be random with a confidence of 99%. Otherwise, the sequence is non-random with a confidence of 99%. For example, we assign μ=4, ε=0.25, then select the sequence X(x(10001), x(10002), …, x(41250)) generated by lattice No. 60 for the test, which have been converted to 32bit unsigned number as follow: where Y(y(10001), y(10002), …, y(41250)) is the sequence for the test, the length of which is 106 bits. The test results are listed in Table 2.  Table 3. According to literature [44], because of the non-uniformity of conventional chaotic systems such as CML, MLNCML and LDCML system, the sequence generated by them can not be utilized for crypto-system directly. The bitextracting algorithm and some nonlinear functions are usually used in the chaotic crypto system for acquiring better uniformity, randomness and complexity.
Fortunately, without the above methods, the PRCML system still possesses good dynamical properties, which is fully confirmed by that the sequences generated by itself can pass the NIST test.

Conclusion
By utilizing ECA to build the spatiotemporal chaotic system, a novel PRCML system is proposed in this paper. And it possesses two special features: firstly, the coupling of the proposed system is pseudo-random and dynamic, even so, the PRCML system can be reproduced. Secondly, the perturbation for the system is generated by ECA, which is essentially the discrete dynamical system. Therefore, the degeneration of digital chaotic system does not exist in the ECA, when it is in chaos. Furthermore, the analysis results of Kolmogorov-Sinai entropy, bifurcation diagram indicate that the PRCML system owns more complex dynamic behavior, better ergodicity and aperiodicity than other conventional spatiotemporal chaotic systems. Besides, the phase space, time-domain analysis, correlation analysis and NIST test demonstrate that the uniformity, randomness and unpredictability of sequences generated by the PRCML system are outstanding. In summary, these excellent properties of the proposed system are able to contribute more secret keys and larger secret key space for crypto-systems. Thus, the PRCML system is more suitable for building a crypto-system.