Analysis of the Time Series Generated by a New High-Dimensional Discrete Chaotic System

. The chaotic behavior of low-dimensional digital chaotic systems is seriously degraded, and the output sequence has a short period. In this study, a digital sequence generator based on a high-dimensional chaotic system is proposed to ensure performance and security. The proposed generator has low resource consumption, and the digital pseudo-random output sequence has a large period. To avoid the nonchaotic state, the multistability in the high-dimensional discrete chaotic system is analyzed. The statistical performance of the output sequence of the proposed digital high-dimensional chaotic system is evaluated, and the results demonstrate that it is a suitable candidate for a long-period pseudo-random sequence generator.


Introduction
Random sequence generators are used in several engineering applications including compressed sensing, image encryption, secure communication, spread-spectrum communication, and distance measurement [1][2][3][4][5][6][7].They can be classified into true random sequence generators and pseudo-random sequence generators.True random sequence generators are based on physical sources, such as resistor thermal noise, atmospheric noise, and race hazard circuits.Although true random sequence generators are highly secure, their implementation is overly complex.Moreover, they are difficult to control.By contrast, pseudo-random sequence generators are based on seeds (initial values).A given seed completely determines the behavior of the pseudo-random sequence generator.Pseudo-random sequence generators are designed by using certain mathematical algorithms, such as linear feedback shift register (LFSR), nonlinear feedback shift register (NLFSR), linear congruence, nonlinear congruence, and BBS (Blum Blum Shub).These design methods are quite limited because they depend on the corresponding algorithm.LFSR is a linear function.It can be quickly reconstructed by the Berlekamp-Massey algorithm without prior knowledge of the seed.NLFSR, linear congruence, nonlinear congruence, and BBS are one-dimensional discrete maps.They generate a largeperiod pseudo-random sequence at high computational cost.With the rapid development of networks, the speed and period of pseudo-random sequence generators have attracted increasing attention.Therefore, the design method of pseudo-random sequence generators should be improved to meet the requirements of fast big data processing.
Chaos is a universal phenomenon in nonlinear systems.Chaotic systems exhibit a large number of special behaviors, such as initial value sensitivity, orbital ergodicity, and aperiodicity [8].These behaviors are in accordance with the confusion and diffusion proposed by Shannon [9].Therefore, chaotic systems are considered a new method for constructing pseudo-random sequence generators.A large number of chaotic systems have been discovered, and several chaos control methods have been proposed.Compared with other pseudo-random sequence generators, chaotic pseudo-random sequence generators allow a large variety of design choices.Therefore, the standards can be focused not only on the randomness of the output sequence but also on speed, period, and resource consumption.A chaotic pseudo-random sequence generator should be appropriately optimized to meet the requirements of different engineering applications, particularly big data processing.
Chaotic systems can be classified into continuous and discrete systems.For digital applications, continuous chaotic systems should first be discretized and then digitized.Discretization methods include the Euler method [10] and the Runge-Kutta method [11].By contrast, discrete chaotic systems require digitizing only and are thus more appealing in digital applications.However, the chaotic behavior of digital chaotic systems gradually degenerates owing to the finite precision effect.Digital chaotic systems are not aperiodic but periodic [12][13][14][15].
Chaotic systems can also be classified into highdimensional and low-dimensional systems.Low-dimensional chaotic systems have high efficiency and low resource consumption.The most commonly used low-dimensional systems are the logistic map [16], the Henon map [17], and the Sawtooth chaotic map [18][19][20][21].The chaotic behavior of these systems is highly degenerate.It is difficult to ensure that the output sequence has a large period.By contrast, high-dimensional chaotic systems have a more complex nonlinear dynamic behavior.However, they have the disadvantages of high resource consumption and lowspeed performance.Therefore, it is very necessary to design a large-period high-dimensional digital chaotic system with high-speed performance and low resource consumption.
Compared with other low-dimensional discrete chaotic systems, the Sawtooth chaotic map has a particularly simple form.It is easy to be digitized, as its output consists of positive decimals.There are several pseudo-random sequence generators based on the Sawtooth chaotic map.When the parameter satisfies a certain condition, the period of the output sequence can reach 2 N−1 2 n − 1 [22].With parameter perturbations, the period can reach 2 2N−1 2 n − 1 [23].The variables N and n represent the precision length and the dimension, respectively.Although the period of the output sequence is large at the same precision, the operating efficiency is not satisfactory.
In this study, a digital pseudo-random sequence generator based on a high-dimensional discrete chaos map is proposed.For low computing complexity, the values of all the parameters of the high-dimensional discrete chaos are set as powers of two.Compared with the period of other digital chaotic pseudo-random sequence generators, the period of the output sequence of the proposed generator is closer to the upper limit of the maximum period.

Sawtooth Chaotic Map
The Sawtooth chaotic map is also called Bernoulli shift or Renyi map.It is defined as where 1 < β ∈ R and x n ∈ 0, 1 .It can be digitized by either fixed-point or float-point representation.Compared with floating-point computing, fixed-point computing is faster, and hardware implementation is smaller.Thus, for more efficient hardware implementation, fixed-point representation is chosen.For the decimal, the fixed-point representation of x n is where 2 N x n represents the integer part of 2 N x n .Therefore, (1) can be transformed into where β is an integer.Multiplying both sides by 2 N , (3) is transformed into The maximum period of the digital Sawtooth chaotic map (4) can reach 2 N−2 for a certain parameter β.There are two most commonly used parameters: one is 30517578125 (N = 35), and the other is 1220703125 (N = 31).The output sequence of the digital Sawtooth chaotic map consists of integers.They should be quantified as binary sequences before being tested by the NIST SP800-22 test suite.To ensure large periodicity, the quantified binary output sequences are the most significant bits (MSB) of the digital Sawtooth chaotic map output.When the parameter β is 30517578125, the period of the output sequence can reach 2 33 .The NIST SP800 test suite can reflect the statistical properties of the randomness of the pseudo-random sequence [24].It consists of 15 different tests.Each test accepts as input a sequence of L bits and returns a P value.A sequence will pass the test if the corresponding P value is greater than 0.01.The NIST SP800 test suite may also be used as follows: each test accepts as input K sequences of L bits.The test results contain two indicators: U value and ratio.The ratio changes with K.The U value is the uniformity indicator of the P value.If the U value is greater than 10 −4 , the P values are uniformly distributed.100 output sequences of the digital Sawtooth chaotic map of length L = 2 × 10 6 bits were tested by the NIST SP800 test suite, and the results are shown in Table 1.
The digital Sawtooth chaotic map fails the randomness test because the U value of the nonoverlapping template is significantly less than 10 −4 .The resource consumption of the hardware implementation by FPGA (field-programmable gate array) is shown in Table 2, and the block diagram of the digital Sawtooth chaotic map in FPGA is shown in Figure 1.

Digital Pseudo-Random Sequence Generator
Although the randomness of the digital Sawtooth chaotic map is not satisfactory, its form is quite simple.Accordingly, a high-dimensional discrete chaotic system based on the Sawtooth chaotic map is proposed, and its digital model is analyzed in detail.

Fast Arithmetic Operation on Fixed-Point Computing.
The arithmetic operations are multiplication, division, addition, and subtraction.Division is difficult to implement on FPGA; thus, it should be avoided in the formulation of the chaotic equation.The maximum frequency of FPGA is seriously affected by multiplication.In Table 2, the maximum frequency of the hardware implementation of the digital Sawtooth chaotic map by FPGA is not high for multiplication.For fast arithmetic operations in fixedpoint computations, the values of all parameters are set as powers of two in this study.Therefore, division and multiplication are easier to implement.When the multiplier is a power of two, the function of 2 n is to shift n bits of the multiplicand to the right, and the missing low position bits are filled by 0. When the divisor is a power of two, the function of 2 n is to shift n bits of the dividend to the left, and the missing high position bits are filled by 0. The complex multiplication and division operations are thus reduced to the right and left shift operations, respectively.The shift operation is easier to implement on FPGA.Compared with subtraction, addition is easier to implement.Therefore, optimized multiplication, optimized division, and addition are selected in the design of the high-dimensional discrete chaotic system.

High-Dimensional Discrete Chaotic Map Modeling
. By taking into account the form of the Sawtooth chaotic map, a high-dimensional discrete chaotic is proposed as follows: where S n is defined as a state vector x 1 n , x 2 n , x 3 n , … , x m n T , and A is defined by In (6), there are at most two nonzero elements per row to further reduce complexity and improve parallel computing efficiency.Therefore, the following parameter matrix is proposed: where Proof.Using the expansion theorem for determinants, det Proposition 2. For the high-dimensional discrete map (5) and the parameter matrix A (7), there exists at least one positive Lyapunov exponent in (5).
Proof.The Jacobian matrix of ( 5) is the parameter matrix A (7).It is assumed that the eigenvalues of the parameter matrix A are λ i i = 1, 2, 3, … , m .By the fundamental property of the eigenvalues, det A = m i=1 λ i .By Proposition 1, . There are two cases for the eigenvalues of the parameter matrix A: either there exists at least a λ k whose absolute value is greater than 1 or all eigenvalues are equal to 1.The latter is obviously impossible.Therefore, there exists at least one positive Lyapunov exponent in (5).
For at least one positive Lyapunov exponent in (5), the high-dimensional discrete map must be a chaotic system.In practical engineering, the dimension of the chaotic system need not be high.Therefore, a 6-dimensional discrete chaotic system is proposed in this study.The parameter matrix A is defined by A = 2 0 0 0 0 0 2 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 2 −1 0 0 0 2 0 0 8 Combined with (5), the 6-dimensional discrete chaotic system is represented by Then, the six Lyapunov exponents are LE1 = 0 3915, LE2 = −0 0523, LE3 = −0 0523, LE4 = −0 0673, LE5 = −0 1098, and LE6 = −0 1098.Although (9) has only one positive Lyapunov exponent, the output sequence of its digital model has a large period.A discrete chaotic system with a large number of positive Lyapunov exponents does not ensure that the output sequence generated by its digital model will have a larger period.The phase diagram of the chaotic attractors is shown in Figure 2.
An autocorrelation algorithm can be used to detect the periodicity of the time series.It is defined as follows: From Figure 3, it can be seen that the triangular wave is considerably smooth.This indicates that the output sequence is aperiodic.
Multistability is present in various chaotic systems.The parameter and the initial value seriously affect the stability of the chaotic system.In ( 5) and ( 9), the fixed parameter has no effect on stability.The Jacobian matrices of ( 5) and ( 9) are ( 7) and ( 8), respectively, which are constant matrices.Therefore, the Lyapunov exponents depend only on the constant matrices ( 7) and ( 8) and are the invariant constants.In (9), the six Lyapunov exponents are 0.3915, −0.0523, −0.0523, −0.0673, −0.1098, and −0.1098 for the initial value S 0 .For the positive Lyapunov exponent 0.3915, the discrete map ( 9) is a chaotic system.Propositions 1 and 2 prove that in (5), there exists at least one positive Lyapunov exponent.Therefore, ( 5) is also a chaotic system for all initial values.4 Complexity Equation ( 9) is transformed into

High-Dimensional
Although the Jacobian matrix of ( 12) is also (8), the value spaces of x i ∈ ℕ i = 1, 2, 3, 4, 5, 6 are limited.Therefore, ( 12) is periodic and can be described by the finite state machine in Figure 4.
The phase diagram of the attractors is also shown in Figure 5.
Compared with Figure 2, the phase diagram of the attractors shows a significant change: it is sparse because the value space of the variables x i i = 1, 2, 3, 4, 5, 6 is limited.In Figure 5, the attractors are obviously periodic; thus, the chaotic attractors degenerate into periodic attractors.
Compared with Figure 3, the triangular wave is not quite smooth and has a large number of sharp peaks.This indicates that the output sequence is not aperiodic.From Figures 6(a) and 6(c), it is obvious that the output sequence is periodic.

Period Analysis.
Owing to the finite precision effect in the physical device, the chaotic behavior of the digital chaotic system gradually degenerates.The output sequences of the digital chaotic systems are all periodic.Therefore, a large period is an important indicator.For precision length N, the maximum period of the output sequence of the 6dimensional digital map is 2 6 N .The period of the output sequence of the digital chaotic system (12) for various values of N is shown in Table 3.
From Table 3, it can be seen that the period T of the output sequence generated by (12) increases sharply as N increases.Compared with other high-dimensional map periods (T 1 in [22] and T 2 in [23]), T is significantly larger and closer to the maximum period.The initial values are selected as follows: x 1 0 = 2, x 2 0 = 3, x 3 0 = 1, x 4 0 = 3, x 5 0 = 2, and x 6 0 = 2.With computation precision 5, the period of the output sequence can reach 594621509 by using a 5-bit addition operation.It is efficient  6 Complexity to generate a large period pseudo-random sequence with low resource consumption.
, and j = 1, 2, 3, … , N. The low position bits of x i , i = 1, 2, 3, 4, 5, 6, are quantified as the output binary sequences.Therefore, more than one bit can be generated at a time.The throughput of the chaotic pseudo-random sequence generator can be significantly improved.Currently, SCMs (single-chip micyocos), ARMs (advanced RISC machines), CPUs, and FPGAs can process several bytes in one clock cycle, that is, 8 bits, 16 bits, 32 bits, and 64 bits.Therefore, quantification with several output bits is beneficial to information processing.However, it is unsafe to quantify all bits of x i i = 1, 2, 3, 4, 5, 6 as the output binary sequences.If the output sequences contain all the information of x i i = 1, 2, 3, 4, 5, 6 , x i i = 1, 2, 3, 4, 5, 6 can be easily predicted and reconstructed without prior knowledge of the initial seeds.
3.3.3.Quantity Analysis.The output sequences of ( 12) are x 1 n , x 2 n , x 3 n , x 4 n , x 5 n , and x 6 n .x 1 n , x 2 n , x 3 n , x 4 n , and x 5 n are similar.Therefore, two different sequences can be generated by (12).A large number of new sequences can be generated by operations between x 6 n and x 1 n i = 1, 2, 3, 4, 5 .In (12), there are three types of schemes for the output sequence: (a) The output sequence generated by x 1 n .
(b) The output sequence generated by x 2 n .
(c) The output sequence generated by operations between x 6 n and x i n i = 1, 2, 3, 4, 5 , namely, x 6 n + x i n and x 6 n ⊕ x i n .
3.3.4.Randomness Analysis.100 sequences of length L = 2 × 10 6 bits were tested by the NIST SP800 test suite, and the results are shown in Table 4.For comparison with Table 1, N was set to 35.The asterisk " * " indicates that the corresponding test failed.From Tables 4 and 5, the randomness of the low 7 Complexity position bits is better than that of high position bits.The randomness of the 8-bit outputs in x 1 n and x 2 n all passed the NIST SP800 test suit.This is due to the fact that the matrix A involves the division 2 −1 .In fixed-point representations, when the divisor is a power of two, the function of 2 n is to shift n bits of the dividend to the left, and the missing high position bits are filled by 0. Therefore, the low position bits are disturbed by the high position bits.As the missing high position bits are filled by 0, the disturbance of the high position bits is not significant.The test results for the output sequence generated by operations between x 6 n and x i n , i = 1, 2, 3, 4, 5, are shown in Table 6.
By contrast, the randomness of the high position bits is better.The randomness of the output sequence generated by the addition operation "+" is better compared with that by the xor operation " ⊕ ," There are 27 different sequences in Table 6, and 17 different sequences passed the NIST SP800 test suit.

Performance Analysis.
With the same N and m, the proposed pseudo-random sequence generator is faster than others because there are at most two nonzero elements per row in the parameter matrix A, which ensures higher parallel performance in the hardware implementation.The resource consumption is shown in Table 7.
The consumption of memory bits and multiplier 9-bit elements is smaller, and the max frequency and throughput   7.

Figure 1 :
Figure 1: Block diagram of the digital Sawtooth chaotic map in FPGA.

Figure 3 :
Figure 3: Time series plots of x 1 n , x 6 n , and their autocorrelation: (a) x 1 n , (b) autocorrelation of x 1 n , (c) x 6 n , and (d) autocorrelation of x 6 n .

Figure 6 :
Figure 6: Time series plots of x 1 n , x 6 n , and their autocorrelation: (a) x 1 n , (b) autocorrelation of x 1 n , (c) x 6 n , and (d) autocorrelation of x 6 n .

Figure 7 :
Figure 7: Block diagram of a 6-dimensional digital chaotic map in FPGA.

Table 2 :
Resource consumption of the hardware implementation by FPGA.

Table 1 :
Results of the NIST SP800-22 test suite.

Table 4 :
Randomness test of x 1 by NIST SP800.
*The asterisk " * " in Table4indicate that the corresponding test failed.

Table 5 :
Randomness test of x 6 by NIST SP800." in Table 5 indicate that the corresponding test failed.

Table 6 :
Randomness test of the output sequence generated by operations between x 6 n and x i n i = 1, 2, 3, 4, 5 .

Table 7 :
Performance of hardware implementation on FPGA.