Modification of Intertwining Logistic Map and a Novel Pseudo Random Number Generator
Abstract
:1. Introduction
- 1.
- A scheme of modifying the local structure of the Intertwining logic map, is proposed;
- 2.
- The new map is evaluated through methods and analyses, such as Scale index and Fuzzy entropy. The evaluation results show an improvement in the efficiency of software calculation and a reduction of the cost of hardware implementation while maintaining the complex dynamic behavior of the original map.
- 3.
- Based the new map, a novel pseudo random number generator (PRNG) is proposed;
- 4.
- The proposed PRNG is evaluated through methods and tests, such as statistical complexity measure and a test suite named NIST sp800-22. The evaluation results show that the proposed PRNG is safe and efficient.
2. Simple Intertwining Logistic
2.1. Processing for Transcendental Functions in the Map (1)
2.2. Degree of Non-Periodicity and Input Parameters Setting
2.3. The Proposed Map “Simple Intertwining Logistic”
2.4. Analysis of Computational Efficiency and Hardware Implementation
3. Chaotic Properties of Simple Intertwining Logistic
3.1. Lyapunov Exponent and Bifurcation Diagram
3.2. Comparison of the Non-Periodicity
3.3. The Fuzzy Entropy of Chaotic Sequence with Finite Precision
4. The PRNG Based on Simple Intertwining Logistic
- 1.
- Import the keys: initialize and , which are the initialize arguments and control parameter. Set the required length of sequence with ;
- 2.
- To avoid transient effect, iterate the map Equation (2) 1000 times and the outputs are discarded.
- 3.
- A final number x based on is generated by the following equation:
- 4.
- If the length of current generated sequence does not reach , return to the step 3, otherwise stop.
5. Analysis and Test of Security for the Proposed PRNG
5.1. Key Space Analysis
5.2. Correlation Analysis
- 1.
- Let
- 2.
- Let
- 3.
- Let
5.3. Recurrence Plots Analysis
5.4. Information Entropy
5.5. Statistical Complexity Measure
5.6. Differential Attack
5.7. Randomness Analysis
5.7.1. NIST SP 800-22 Test
5.7.2. DIEHARD Test Suite
5.8. Analysis of Speed
6. Conclusions and Future Works
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
PRNG | Pseudo random number generator |
CWT | Continuous Wavelet Transform |
SCM | Statistical complexity measure |
RP | Recurrence plot |
RQA | Recurrence quantification analysis |
BCR | Bit Change Rate |
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the proposed map | 0.021130440 | 0.211738380 | 2.152552600 | 21.714554600 |
original map | 0.062234550 | 0.622838910 | 6.297382210 | 63.018479200 |
10,000 | 20,000 | ||||
---|---|---|---|---|---|
7.8682369570 | 7.9330325313 | 7.9845058647 | 7.9998646492 | 7.9999855595 | |
7.8681921385 | 7.9291288733 | 7.9825195201 | 7.9998442316 | 7.9999858863 | |
7.8652402717 | 7.9228013409 | 7.9851102583 | 7.9998350428 | 7.9999845185 |
Test Name | p-Value | Pass Rate | Result |
---|---|---|---|
Frequency | 0.350485 | 100/100 | Success |
Block Frequency (m = 128) | 0.851383 | 100/100 | Success |
Cumulative Sums (Forward) | 0.494392 | 100/100 | Success |
Cumulative Sums (Reverse) | 0.213309 | 100/100 | Success |
Runs | 0.319084 | 98/100 | Success |
Longest Run of Ones | 0.171867 | 99/100 | Success |
Rank | 0.946308 | 100/100 | Success |
FFT | 0.637119 | 100/100 | Success |
Non Overlapping | 0.816537 | 100/100 | Success |
(m = 9, n = 8) | |||
Overlapping Templates (m = 9) | 0.236810 | 99/100 | Success |
Universal | 0.334538 | 98/100 | Success |
Approximate Entropy (m = 10) | 0.437274 | 99/100 | Success |
Random-Excursions (data3) | 0.330628 | 88/88 | Success |
Random-Excursions Variant Serial (data5) | 0.534146 | 87/88 | Success |
Serial Test 1 (m = 16) | 0.699313 | 98/100 | Success |
Serial Test 2 (m = 16) | 0.834308 | 99/100 | Success |
Linear complexity (M = 500) | 0.816537 | 99/100 | Success |
Test Name | p-Value | Result (Assessment) |
---|---|---|
Birthday spacing | 0.94319333 | Passed |
Overlapping permutation | 0.51435951 | Passed |
Binary rank | 0.34466449 | Passed |
Binary rank | 0.68760941 | Passed |
Bitstream | 0.84946357 | Passed |
OPSO | 0.41780931 | Passed |
OQSO | 0.57576741 | Passed |
DNA | 0.28379599 | Passed |
Count ones str | 0.42594465 | Passed |
Count ones byt | 0.14161743 | Passed |
Parking Lot | 0.14887474 | Passed |
2DS spheres | 0.90080180 | Passed |
3DS spheres | 0.35141495 | Passed |
Squeeze | 0.72469462 | Passed |
Runs (up) | 0.91243702 | Passed |
Runs (down) | 0.75463775 | Passed |
Craps for no. of wins | 0.59335173 | Passed |
Craps for throws/game | 0.56547058 | Passed |
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Zhao, W.; Ma, C. Modification of Intertwining Logistic Map and a Novel Pseudo Random Number Generator. Symmetry 2024, 16, 169. https://doi.org/10.3390/sym16020169
Zhao W, Ma C. Modification of Intertwining Logistic Map and a Novel Pseudo Random Number Generator. Symmetry. 2024; 16(2):169. https://doi.org/10.3390/sym16020169
Chicago/Turabian StyleZhao, Wenbo, and Caochuan Ma. 2024. "Modification of Intertwining Logistic Map and a Novel Pseudo Random Number Generator" Symmetry 16, no. 2: 169. https://doi.org/10.3390/sym16020169
APA StyleZhao, W., & Ma, C. (2024). Modification of Intertwining Logistic Map and a Novel Pseudo Random Number Generator. Symmetry, 16(2), 169. https://doi.org/10.3390/sym16020169