Robust Chaos of Cubic Polynomial Discrete Maps with Application to Pseudorandom Number Generators

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 constructa CPDM. The parameter regions of chaotic robustness of the CPDM are larger than these of Logistic map. By using a fixed pointarithmetic, we investigatethe cycle lengths of the CPDMand a Logistic map.The results show thatthe maximum cyclelengths of 1000 chaoticsequenceswith length 3×10 7 generatedby differentinitial valueconditions exponentially increasewith the resolutions. When the resolutions reach 10 −7 ∼ 10 −13 , the maximum cycle lengths of the cubic polynomial chaotic sequences are significantly greater than these of the Logistic map. When the resolution reaches 10 −14 , there is the situation without cycle for 1000 cubic polynomial chaotic sequences with length 3 × 10 7 . 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 1000 key streams consisting of 20000 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.


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".
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].
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.

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  = 3.99.
The evolution of state variables  −   of the novel cubic polynomial discrete chaotic map with V = 5 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.

Comparison of Bifurcation between the Cubic Polynomial
Chaotic Map and Logistic Map.The calculated Lyapunov exponent of the chaotic maps (5) is 0.69135, which is bigger than 0.63927, and the Lyapunov exponent of Logistic map (6).
The chaotic map ( 5) is robust chaos for parameter V > 25/6 with  2 = −3 from the case (1) of Theorem 2. The bifurcation diagram of the  as a function of the parameter V 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.
The bifurcation diagram of Logistic map about the parameter  is shown in Figure 5. Logistic map is robust chaos for the parameter regions  ∈ (1 + 2 √ 2, 4).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 V ∈ (25/6, +∞) with  2 = −3 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.

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, 1000 uniformly distributed random initial condition values are used to generate 1000 chaotic sequences with length 3 × 10 7 , where the resolutions are from 10 −6 to 10 −14 , 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 10 −14 , there is the situation without cycle for 1000 cubic polynomial chaotic sequences with length 3 × 10 7 .That is, the maximum of the cycle lengths for 1000 cubic polynomial chaotic sequences is larger than 3 × 10 7 .
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 7 to 13.

Numerical Simulations.
Select the following initial conditions: ) The chaotic orbits of the state variables { 1 ,  2 ,  3 } for the first 5000 iterations are shown in Figures 9(a  (16) has a perturbation, () and () are rapidly converting into generalized synchronization as the chaos GS theorem predicts.
The chaotic orbits of the state variables { 1 ,  2 ,  3 } for the first 5000 iterations are shown in Figures 14(a
are the key streams, respectively, generated via CPRNGI, CPRNGII, CPRNGIIII, and CPRNGIV.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  = 10 −4 for the normal cumulative distribution and the  2 distribution, respectively, and the required intervals of the Run tests correspond approximately the significant  = 1.6 × 10 −7 for the normal cumulative distribution [29,30].If we select the significant  = 10 −4 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 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:   Therefore the key spaces of the four CPRNGs are, respectively, 10 15×2 , 10 15×2 , 10 15×18 , and 10 15×16 .The key space of CPRNGIII is larger than 2 896 .

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 1000 cubic polynomial chaotic sequences show that when the resolutions reach 10 −7 ∼ 10 −13 , the maximum of cycle lengths of the cubic polynomial chaotic sequences is significantly greater than these of Logistic map.When the resolution reaches 10 −14 , there is the situation without cycle for 1000 cubic polynomial chaotic sequences with length 3 × 10 7 .The maximum of cycle lengths of Logistic sequences is less than 3 × 10 7 .
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 CPRN-GIV.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 2 896 , which is large enough to against brute-force attacks.
Finally, from (A. Therefore () :  →  is the S-unimodal map, and satisfies Theorem 1.This completes the proof.

Figure 1 :Figure 2 :
Figure 1: The robust chaos regions of cubic polynomial discrete map.

Figure 5 :
Figure 5: The bifurcation diagram of Logistic map about the parameter .

Figure 6 :Figure 7 :
Figure 6: The change diagram of maximum of cycle lengths for different decimal resolutions from 6 to 13.

Figure 8 :
Figure 8: The change diagram of minimum of cycle lengths for different decimal resolutions from 6 to 13.

Table 1 :
Cycle lengths (CL) of the cubic polynomial chaotic map and Logistic map about resolutions.

Table 2 :
The required intervals of the FIPS 140-2 Monobit Test, Pork Tests, and Run Test.Here, MT, PT, and LT represent the Monobit Test, the Pork Test, and the Long Run Test, respectively.k represents the length of the run of a tested sequence. 2 DT represents  2 distribution.

Table 4 :
The confident intervals of the FIPS 140-2 tested values of 1,000 key streams generated by CPRNGI/CPRNGII/CPRNGIII/CPRNGIV.Here, SD represents the standard deviation.