A Pseudorandom Number Generator Based on the Chaotic Map and Quantum Random Walks
<p>Conical section in unit region.</p> "> Figure 2
<p>Lyapunov exponent of the map (<a href="#FD4-entropy-25-00166" class="html-disp-formula">4</a>).</p> "> Figure 3
<p>Bifurcation diagram of the map (<a href="#FD4-entropy-25-00166" class="html-disp-formula">4</a>).</p> "> Figure 4
<p>Iterative trajectory of map (<a href="#FD4-entropy-25-00166" class="html-disp-formula">4</a>).</p> "> Figure 5
<p>Lyapunov exponent and Bifurcuresation diagram of sys (<a href="#FD9-entropy-25-00166" class="html-disp-formula">9</a>) for <math display="inline"><semantics> <msub> <mi>k</mi> <mn>14</mn> </msub> </semantics></math>.</p> "> Figure 6
<p>Block diagram of the optimization scheme.</p> "> Figure 7
<p>Trajectories of different chaotic systems.</p> "> Figure 8
<p>Approximate entropy.</p> "> Figure 9
<p>Histograms of the sequences generated by the digital chaotic system.</p> "> Figure 9 Cont.
<p>Histograms of the sequences generated by the digital chaotic system.</p> "> Figure 10
<p>Autocorrelation analysis.</p> "> Figure 11
<p>RQA measures for the proposed PRNG.</p> "> Figure 12
<p>Normalized Shannon entropy <math display="inline"><semantics> <msub> <mi>H</mi> <mi>s</mi> </msub> </semantics></math> and intensive SCM <math display="inline"><semantics> <msub> <mi>C</mi> <mi>j</mi> </msub> </semantics></math> for the proposed PRNG.</p> "> Figure 13
<p>The Scale index of the Henon map, the logistic map and the proposed PRNG.</p> "> Figure 14
<p>Windowed Scale index of the Henon map, the logistic map and the proposed PRNG.</p> ">
Abstract
:1. Introduction
2. A Internal Randomness System Is Constructed in Unit Region
Lyapunov Exponent, Trajectory Iteration Diagram and Bifurcation Diagram
3. Design and Performance Analysis of a New Compound Chaotic System
3.1. Optimization Algorithm Based on Two-Dimensional Chaotic Map and Quantum Random Walk
3.1.1. Two-Dimensional Hyper-Chaotic System
- a .
- F is continuously differentiable in some fields of z and the absolute values of all eigenvalues of are greater than 1. Thus, there exists a normal number r and a norm of , so that F can expand on under , is a closed sphere of space centered on z;
- b .
- z is the return-expansion fixed point of F, that is, it exists a point and positive integer m such that , where is the opening ball of space centered on z. F is continuous and differentiable in a field of and , where .
3.1.2. Sequence Generation Algorithm Based on Quantum Random Walk
- 1.
- The coin operator is as following:Let the position shift operator be , and the expression is as following:Assuming that the initial state of the system is , and after t steps, according to Equation (11), the joint state isThen, the probability of stopping at point in graph G after step t isIn order to design a sequence generation algorithm, the following theorem based on Theorem 3.6 and Theorem 4.1 in literature [23] is given.
Algorithm 1 Sequence generator algorithm. |
Input: Output: 1. , ; 2. , ; , , , , ,ultimately , , , . 3. |
3.1.3. Optimization Scheme and a Compound Chaotic System
3.1.4. The Digital Compound Chaotic System Expression
3.2. Performance Evaluation of the Compound Chaotic System
3.2.1. Analysis under Finite Precision
3.2.2. Time Series Complexity
3.2.3. Histogram Analysis
3.2.4. Autocorrelation Analysis
4. Design and Performance Analysis of Pseudo Random Number Generator
4.1. Design of PRNG Based on the Proposed Compond Chaotic System
- 1.
- Import the keys: initialize , and , which are the control parameters and initial conditions as shown in Figure 6.
- 2.
- Iterate the proposed compound chaotic map 1000 times and the output is discarded, where i end at 999.
- 3.
- Generate and output the random number using the following equation from the :
4.2. Analysis and Test of Security for the Proposed PRNG
4.2.1. Key Space Analysis
4.2.2. Correlation Analysis
- 1.
- and , a sequence with numbers is generated. If the initial condition changes small () and the others remain unchanged, a new sequence will be generated.
- 2.
- Let the initial condition u is changed () and the others remain unchanged, a new sequence with the same length will be generated.
- 3.
- Let the initial condition v is changed () and the others remain unchanged, a new sequence with the same length will be generated.
4.2.3. Recurrence Plots Analysis
4.2.4. Information Entropy
4.2.5. Statistical Complexity Measure
4.2.6. Degree of Non-Periodicity
4.2.7. Differential Attack
4.2.8. Random Tests
4.2.9. Speed Performance Analysis
5. Conclusions and Discussion
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
PRNG | Pseudorandom number generator |
RP | Recurrence plot |
CWT | Continuous Wavelet Transform |
BCR | Bit Change Rate |
SCM | Statistical complexity measure |
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p | 1 | 4 | 8 | 16 | 32 |
---|---|---|---|---|---|
m | ≥3 | ≥2 | ≥2 | ≥2 | ≥1 |
5000 | 10000 | 20000 | |||
---|---|---|---|---|---|
7.9904252203 | 7.9953919349 | 7.9975813926 | 7.9994779223 | 7.9999951973 | |
7.9899963471 | 7.9949233070 | 7.9974845070 | 7.9994868500 | 7.9999952355 | |
7.9895789685 | 7.9947618385 | 7.9970912252 | 7.9995475969 | 7.9999954916 |
Test Name | p-Value | Pass Rate | Results |
---|---|---|---|
Frequency | 0.657933 | 98/100 | Pass |
Block Frequency (m = 128) | 0.051942 | 100/100 | Pass |
Cumulative Sums (Forward) | 0.224821 | 98/100 | Pass |
Cumulative Sums (Reverse) | 0.455937 | 98/100 | Pass |
Runs | 0.514124 | 100/100 | Pass |
Longest Run of Ones | 0.262249 | 100/100 | Pass |
Rank | 0.955835 | 98/100 | Pass |
FFT | 0.883171 | 98/100 | Pass |
Non-Overlapping Templates | 0.867692 | 100/100 | Pass |
(m = 9, B = 000000001) | |||
Overlapping Templates (m = 9) | 0.574903 | 100/100 | Pass |
Universal | 0.474986 | 99/100 | Pass |
Approximate Entropy (m = 10) | 0.574903 | 100/100 | Pass |
Random-Excursions (data3) | 0.422034 | 67/67 | Pass |
Random-Excursions Variant Serial (data5) | 0.922036 | 67/67 | Pass |
Serial Test 1 (m = 16) | 0.883171 | 99/100 | Pass |
Serial Test 2 (m = 16) | 0.350485 | 100/100 | Pass |
Linear complexity (M = 500) | 0.275709 | 99/100 | Pass |
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Zhao, W.; Chang, Z.; Ma, C.; Shen, Z. A Pseudorandom Number Generator Based on the Chaotic Map and Quantum Random Walks. Entropy 2023, 25, 166. https://doi.org/10.3390/e25010166
Zhao W, Chang Z, Ma C, Shen Z. A Pseudorandom Number Generator Based on the Chaotic Map and Quantum Random Walks. Entropy. 2023; 25(1):166. https://doi.org/10.3390/e25010166
Chicago/Turabian StyleZhao, Wenbo, Zhenhai Chang, Caochuan Ma, and Zhuozhuo Shen. 2023. "A Pseudorandom Number Generator Based on the Chaotic Map and Quantum Random Walks" Entropy 25, no. 1: 166. https://doi.org/10.3390/e25010166
APA StyleZhao, W., Chang, Z., Ma, C., & Shen, Z. (2023). A Pseudorandom Number Generator Based on the Chaotic Map and Quantum Random Walks. Entropy, 25(1), 166. https://doi.org/10.3390/e25010166