Abstract
Based on the robust chaos theorem of S-unimodal maps, this paper studies a kind of cubic polynomial discrete maps (CPDMs) and sets up a novel theorem. This theorem gives general conditions for the occurrence of robust chaos in the CPDMs. By using the theorem, we construct a CPDM. The parameter regions of chaotic robustness of the CPDM are larger than these of Logistic map. By using a fixed point arithmetic, we investigate the cycle lengths of the CPDM and a Logistic map. The results show that the maximum cycle lengths of chaotic sequences with length generated by different initial value conditions exponentially increase with the resolutions. When the resolutions reach , the maximum cycle lengths of the cubic polynomial chaotic sequences are significantly greater than these of the Logistic map. When the resolution reaches , there is the situation without cycle for cubic polynomial chaotic sequences with length . By using the CPDM and Logistic map, we design four chaos-based pseudorandom number generators (CPRNGs): CPRNGI, CPRNGII, CPRNGIII, and CPRNGIV. The randomness of two key streams consisting of bits is tested, respectively, generated by the four CPRNGs. The result suggests that CPRNGIII based on the cubic polynomial chaotic generalized synchronic system has better performance.
1. Introduction
Chaos is one type of complex dynamic behaviors displaying similarly random happenings within a determined nonlinear system or process. Chaotic systems are mainly defined and analyzed in continuous or discrete phase spaces. And they exhibit properties such as sensitive dependent on initial conditions and system parameters and ergodicity and long time’s chaotic behaviors are not predictable [1]. In 1975, Li-Yorke first formally introduced the term chaos into mathematics [2]. They established a criterion for the existence of chaos in one-dimensional difference equations and the famous example is that “period three implies chaos”.
Pseudorandom numbers have wide range of applications in many fields, such as physical systems simulation [3–6], information encryption [7–13], entertainment [14, 15], and computer simulation [16–20]. In the practical applications, pseudorandom algorithm has almost replaced the stochastic indicator and random number generator based on the hardware. John von Neumann was the first scholar who has made outstanding contributions to designing computer-based pseudorandom number generator. Currently, the known statistical test criterions of random number generator/pseudorandom number generators (RNGs/PRNGs) contain DIEHARD test [21] and FIPS 140 test [22] and SP 800-22 test [23] released by the National Institute of Standards and Technology (NIST).
In order to construct a chaotic quadratic polynomial, Zhou and Song set up a necessary and sufficient condition to determine the 3-periodic points of a quadratic polynomial [24], based on Li-Yorke’s criterion. This research is a perfect application of Li-Yorke’s theorem, but it is not suitable for researching the chaotic properties of cubic polynomial. Based on the theorem given in [25], Andrecut and Ali derived some general conditions and practical procedures for generating robust chaos in smooth unimodal maps [26].
Based on the robust chaos theorem of S-unimodal maps, this paper sets up a chaos criterion theorem for cubic polynomial discrete maps and constructs a novel cubic polynomial discrete map. Using this theorem and numerical simulations verifies that the parameter regions of chaotic robustness of the cubic polynomial discrete map are larger than these of Logistic map. The maximum cycle lengths of the cubic polynomial chaotic sequences are significantly greater than these of Logistic map, when the resolutions reach . Using the cubic polynomial discrete map and Logistic map constructs two -dimensional chaotic generalized synchronic systems (6DCGSS). Four chaos-based pseudorandom number generators CPRNGI, CPRNGII, CPRNGIII, and CPRNGIV, respectively, are designed by the cubic polynomial discrete map, Logistic map, and two 6DCGSS. The results of pseudorandomness tests suggest that CPRNGIII has better performance.
The rest of this paper is organized as follows. Section 2 introduces the robust chaos theorem of S-unimodal maps, and sets up a chaos criterion theorem of cubic polynomial discrete maps. Section 3 constructs a novel cubic polynomial discrete map. Numerical simulations analyze bifurcation and cycle lengths of the novel cubic polynomial discrete map and Logistic map. Section 4 presents two 6DCGSS based on the GS theorem. Section 5 designs four chaos-based pseudorandom number generators CPRNGI, CPRNGII, CPRNGIII, and CPRNGIV and makes the statistic tests for the four CPRNGs. Finally, Section 6 concludes the article.
2. Robust Chaos of Cubic Polynomial Maps
Robust chaos refers that, during the system parameter disturbances in a certain range, the system can guarantee the characteristics of chaos and still does not change the overall performance of the system. The robust chaos theorem of S-unimodal maps is introduced as follows.
Theorem 1 (see [26]). Letting be a parametric S-unimodal map with the unique maximum at and , , then generates robust chaos for .
Based on the above theorem, we set up the robust chaos theorem of cubic polynomial discrete maps.
Theorem 2. Let a kind of cubic polynomial discrete maps have the following general form: If the parameters set this satisfies one of the following cases:
Case (1)Case (2)Case (3)Then is the S-unimodal map. And it has the unique maximum at , and , , . satisfies Theorem 1. That is, is robust chaos. The proof progress is shown in the Appendix.
Three cases of robust chaos, respectively, correspond to the three surfaces. The robust chaos regions of case (1), (2) and (3) of the cubic polynomial discrete map are shown in Figure 1. The three surfaces are the robust chaos regions.

3. Comparison of Bifurcation and Cycle Lengths between the Cubic Polynomial Chaotic Map and Logistic Map
Based on the case (1) of Theorem 2, we construct a novel cubic polynomial discrete chaotic map: where , , , .
We consider Logistic map: where .
The evolution of state variables of the novel cubic polynomial discrete chaotic map with is shown in Figure 2. The evolution of state variables of the Logistic map is shown in Figure 3. Extensive numerical simulations show that the dynamic behaviors of the chaotic map demonstrate chaotic attractor features.


3.1. Comparison of Bifurcation between the Cubic Polynomial Chaotic Map and Logistic Map
The calculated Lyapunov exponent of the chaotic maps (5) is , which is bigger than , and the Lyapunov exponent of Logistic map (6).
The chaotic map (5) is robust chaos for parameter with from the case (1) of Theorem 2. The bifurcation diagram of the as a function of the parameter is shown in Figure 4.

In 2009, Zhou and Song set up a theorem for the necessary and sufficient condition of determination 3-periodic points of a quadratic polynomial.
Theorem 3 (see [24]). A quadratic polynomial has real 3-periodic points if and only if .
Now we use the theorem to determine the parameter regions of robust chaos of Logistic map.
Proof. Let Logistic map , , and . Based on Theorem 3 and Li-Yorke’s criterion that period three implies chaos, Logistic map is chaos if and only if Thence Logistic map is chaos if and only if .
The bifurcation diagram of Logistic map about the parameter is shown in Figure 5. Logistic map is robust chaos for the parameter regions . Compared with Logistic map, the bifurcation of chaotic map (5) is uniform distributed. And the cubic polynomial map is robust chaos for the parameter regions with from the case (1) of Theorem 2. Clearly, the parameter regions of robust chaos of the cubic polynomial discrete map are larger than these of Logistic map.

3.2. Comparison of Cycle Lengths between the Cubic Polynomial Chaotic Map and Logistic Map
When chaotic systems are realized under finite precision, the periodic cycles will occur due to rounding errors. Reference [27] investigates the maximum cycle lengths of Logistic map with respect to different initial condition values.
Using the algorithm proposed in [27] for fixed point realizations analyzes the cycle lengths of the cubic polynomial chaotic map (5) and Logistic map (6). During analysis, uniformly distributed random initial condition values are used to generate chaotic sequences with length , where the resolutions are from to , and the rounding type selects rounded towards zero (fix) and regardless of the iteration to fixed point.
The maximum, mean, and minimum of the cycle lengths for the cubic polynomial chaotic sequences and Logistic sequences are listed in Table 1. The cycle lengths of chaotic sequences increase with the resolutions. When the resolution reaches , there is the situation without cycle for cubic polynomial chaotic sequences with length . That is, the maximum of the cycle lengths for cubic polynomial chaotic sequences is larger than .
The change diagrams of maximum, mean, and minimum of cycle lengths are shown in Figures 6, 7, and 8, respectively. The red line represents the change of cycle lengths for the cubic polynomial chaotic map (5), and the blue line represents the change of cycle lengths for the Logistic map (6). According to the three figures, the maximum of cycle lengths exponentially increases with the resolutions. And the maximum of cycle lengths of cubic polynomial chaotic sequences is significantly larger than these of the Logistic sequences when resolutions are from to .



4. 6-Dimensional Chaotic Generalized Synchronic Systems
Based on the cubic polynomial map (5) and the Logistic map (6), we consider the following -dimensional discrete chaotic map as the driving part: where , , , and .
The calculated Lyapunov exponents of the chaotic map (8) are . The calculated Lyapunov exponents of the chaotic map (9) are .
In order to construct a GS driven system, define an invertible transformation : where is an invertible matrix. Now let the driven part have the form:
From (13) and (14), it follows that the error equations and can be represented by and . It guarantees that the zero solution of the error equation is asymptotically stable. From the chaos generalized synchronization (GS) theorem [28], systems (8) and (13) as well as (9) and (14) are GS with respect to the transformation for any initial value and . Since is invertible, systems (13) and (14) are also chaotic.
4.1. Numerical Simulations
Select the following initial conditions:
The chaotic orbits of the state variables for the first iterations are shown in Figures 9(a)–9(d). The evolutions of the state variables, , , and , are shown in Figures 10(a)–10(c). The chaotic orbits of the state variables for the first iterations are shown in Figures 11(a)–11(d). The evolutions of the state variables, , , and , are shown in Figures 12(a)–12(c). The dynamic behaviors of the chaotic map demonstrate chaotic attractor characteristics. Figures 13(a)–13(c) show that although the initial condition (16) has a perturbation, and are rapidly converting into generalized synchronization as the chaos GS theorem predicts.

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The chaotic orbits of the state variables for the first iterations are shown in Figures 14(a)–14(d). The evolutions of the state variables, , , and , are shown in Figures 15(a)–15(c). The chaotic orbits of the state variables for the first iterations are shown in Figures 16(a)–16(d). The evolutions of the state variables, , , and , are shown in Figures 17(a)–17(c). The dynamic behaviors of the chaotic map demonstrate chaotic attractor characteristics. Observe that Figures 18(a)–18(c) show that although the initial condition (17) has a perturbation, and are rapidly converting into generalized synchronization as the chaos GS theorem predicts.

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5. Chaotic Pseudorandom Number Generator and Pseudorandomness Tests
5.1. Chaotic Pseudorandom Number Generator
Denote where , and are defined by (8) and (13), and and are defined by (9) and (14).
First, introduce transformations , , , and : which transform the chaotic streams of GS systems (18) and (19) as well as (20) and (21) into key streams. Here where , represents the dot product of vectors , and represents the dot product of vectors .
Second, construct a transform : which is defined bys.t. Letting , then where dec2bin and are both Matlab commands.
Finally, transformations , , , and are defined viaNow, based on the cubic polynomial chaotic map (5) and Logistic map (6), we design CPRNGI, CPRNGII, CPRNGIIII, and CPRNGIV. are the key streams, respectively, generated via CPRNGI, CPRNGII, CPRNGIIII, and CPRNGIV.
5.2. Pseudorandomness Tests
The FIPS 140-2 test consists of four subtests: Monobit Test, Poker Test, Run Test, and Long Run Test. Each test needs a single stream of 20,000 one and zero bits from the keystream generator. Any failure in the first three tests means that the corresponding quantity of the sequences falls out the required intervals listed in the second column in Table 2. The Long Run test is passed if there are no runs of length 26 or more.
It has been pointed out that the required intervals of the Monotone test and the Pork test correspond significantly to for the normal cumulative distribution and the distribution, respectively, and the required intervals of the Run tests correspond approximately the significant for the normal cumulative distribution [29, 30]. If we select the significant of all tests, the corresponding accepted intervals are listed in the third column in Table 2. According to Golomb’s three postulates on the randomness that ideal pseudorandom sequences should satisfy [31], and the ideal values of the first three tests should be those listed in the 4th column in Table 2.
The FIPS 140-2 test is used to check keystreams randomly generated, respectively, by CPRNGI, CPRNGII, CPRNGIIII, and CPRNGIV, with perturbed randomly the parameters, the initial conditions , , , and , and the parameters of matrix in the range . The test results are listed in the 2/3/4/5th column in Table 3. The statistic test results are listed in the 3/4/5/6th column in Table 4, in which the statistic results are described by mean values and standard deviation (MeanSD).
The RC4 was designed by Rivest of the RSA Security in 1987, which has been widely used in popular protocols such as Secure Sockets. The RC4 Algorithm PRNG can be designed via Matlab commands:where randi([0 ],1,) generates a vector of uniformly distributed random integers of dimension ; mod means modulus after division; zeros is a zero raw vector of dimension N.
Consequently, the RC4 Algorithm based PRNG is designed. The FIPS 140-2 test is used to test the keystreams randomly generated by RC4. The test result is listed in the 6th column in Table 3. The statistic test results are listed in the 7th column in Table 4. Observe that the statistical properties of the pseudorandomness of the sequences generated via the four CPRNGs and RC4 do not have significant differences. And compared with the other three CPRNSs, CPRNGIII based on the cubic polynomial chaotic generalized synchronic system has better performance.
5.3. Key Space
The key set parameters of CPRNGs includes the parameters , the initial conditions , , , and , and the matrix . It can be proved that if the perturbation matrix satisfies the matrix is still invertible. Therefore CPRNGI, CPRNGII, CPRNGIIII, and CPRNGIV, respectively, have key parameters denoted by
Let the key set be perturbed by where
Therefore the key spaces of the four CPRNGs are, respectively, , , , and . The key space of CPRNGIII is larger than .
6. Conclusions
First, based on the robust chaos theorem of S-unimodal maps, this paper sets up a robust chaos theorem on a kind of cubic polynomial discrete maps. This theorem provides parameter inequalities to determine the robust chaos regions.
Second, using the Theorem 2 constructs a cubic polynomial map. The analysis results of the cycle lengths of cubic polynomial chaotic sequences show that when the resolutions reach , the maximum of cycle lengths of the cubic polynomial chaotic sequences is significantly greater than these of Logistic map. When the resolution reaches , there is the situation without cycle for cubic polynomial chaotic sequences with length . The maximum of cycle lengths of Logistic sequences is less than .
Third, combining the robust chaos Theorems 2 and 3 and GS theorem proposes two 6DCGSS. The numerical simulations of two 6DCGSS have verified the effectiveness of theoretical results.
Finally, design four chaos-based pseudorandom number generators CPRNGI, CPRNGII, CPRNGIII, and CPRNGIV. Comparing the results of the FIPS 140-2 test for the keystreams generated via the four CPRNGs with the RC4 PRNG shows that the randomness of the sequences generated via the CPRNGIII has better performance. The simulations also suggest that the key space of the CPRNGIII is larger than , which is large enough to against brute-force attacks.
Appendix
Proof of Theorem 2.
Case(1). First, we show that has three order derivative function and is the unique maximum point.
We now judge the symbol of .(1)If implies then .(2)If implies then the judgement of is equivalent to the judgement of . Letting then .From , , , and , then Thence .
Clearly, has three-order derivative function: and is the solution of .
Since then is the maximum point.
Second, we proof , and is strictly increasing on and strictly decreasing on .
Letting it has three solutions: and , then .
Solving has the other solution: Since , then thence . Since is the unique maximum at and , then is strictly increasing on and strictly decreasing on .
Third, we proof has negative Schwarzian derivative on . For every and ; thence has negative Schwarzian derivative on .
Finally, we proof the maximum value on . Substituting into , Since , then Therefore is the S-unimodal map and satisfies Theorem 1.
Case(2). First, we show that has three order derivative function and is the unique maximum point.
Since , obviously .
From , then thence .
Similar to the proof given in case (1), has first-order derivative function (A.5), second-order derivative function (A.6), and third-order derivative function (A.7). Solving has one solution . From (A.8), and thus is the maximum point.
Second proof that and is strictly increasing on and strictly decreasing on . From (A.10), has three solutions: , and , and then . Letting , we get the other solution (A.11) and . Thence is strictly increasing on and strictly decreasing on .
Third, form (A.13) and , we proof that has negative Schwarzian derivative on .
Finally, from (A.14), and , then Therefore is the S-unimodal map and satisfies Theorem 1.
Case(3). First, we show that has three-order derivative function, and is the unique maximum point.
We now judge the symbol of .
By , then and thence the judgement of is equivalent to the judgement of . From (A.3), clearly , and then .
From , then and thus .
Similar to the proof given in case (1), has first-order derivative function (A.5), second-order derivative function (A.6), and third-order derivative function (A.7). Solving has one solution . From (A.8), and thus is the maximum point.
Second, we proof that , and is strictly increasing on and strictly decreasing on . From (A.10), has three solutions: , and , and then . Letting , we get the other solution (A.11), and . Thence is strictly increasing on and strictly decreasing on .
Third, form (A.13) and , we proof that has negative Schwarzian derivative on .
Finally, from (A.14) and , then Therefore is the S-unimodal map, and satisfies Theorem 1. This completes the proof.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This project is jointly supported by the National Natural Science Foundation of China (Grant No. 61170037) and Research Award Fund for Natural Science Foundation of Shandong Province No. ZR2017BA022.