chuits4

1. A random number generator based on Chua’s circuit

The aim of research is to obtain
a pseudo-random binary
sequence using a hardware
and software implementations
of Chua’s circuit.
Oleg Opyakin, 2-nd year DREC student
Konstantin Lishik, 2-nd DASR year
Daniil Vikultsev, 2-nd year LPR student
Scientific advisor : Stanislav Vinogradov

2. Plan of Research

1. Objectives
2. Setup & mathematical model
3. Hardware implementation
4. Obtaining random numbers
5. Processing & results
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3. Objectives

• Describe the behavior of Chua’s circuit
• Develop a method to obtain random sequences
• Prove the randomness of obtained sequence
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4. SETUP & MATHEMATICAL MODEL

SETUP & MATHEMATICAL MODEL
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5. Chua’s diode

Voltage-current characteristics
of Chua’s diode
Elecrtonic scheme
of Chua’s diode
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6. Chua’s Circuit

• Chaotic oscillations:
– Voltages on C1 and C2
– Current through L
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7. Mathematical model

- Kirchhoff law
- dimensionless coefficients
- function of Chua’s diode
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8. Positions of equilibrium

Positions of equilibrium
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9. Phase portraits

Unstable “3-D focuses”
Stable “3-D focus”
Finally:
Unstable 2-D focus
• Positions of equilibrium of experimental setup will match E1, E2 and E3
• The system will evolve from E1 and E2, towards E3
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10. Numerical solution

Euler’s method implemented with Python:
• Positions of equilibrium match E1, E2 and E3
• The system evolves from E1 and E2
• Unclear behavior near E3
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11. Measuring setup

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12. Circuit parameters

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13. Hardware implementation

A double-scroll attractor
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14. OBTAINING RANDOM NUMBERS

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15. Obtaining random numbers

State vector:
x > 0 – the right state
x < 0 – the left state
- characteristic time of the system
In our system:
= 5 ms
= 10
We will monitor the system states
.
Hypothesis:
Probabilities of finding the dynamic system in the left or right states are equal
Thus:
Right state - 0
Left state - 1
Binary sequence
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16. RESULTS & PROCESSING

RESULTS & PROCESSING
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17. Sequence visualization

Example:
100101011
1
0
0
1
0
1
0
1
1
NO visible patterns
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18. NIST tests

Experimental data
Experimental satistics
Probabity theory
Etalon satistics
Comparing experimental and
etalon statistics
P - value
Probability that generator generates true random numbers
P > 0.01
the sequence is random
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19. NIST tests results

All of P – values are more than 0.01, thus, the generated sequence is random
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20. Results

1. The behavior of the system has been described
mathematically and studied numerically and experimentally.
Theoretical assumptions match with the experiment.
2. Random numbers can be obtained by the developed
method.
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21.

ADDITIONAL SLIDES
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22. Literature

• [1] L. Chua, “The genesis of Chua’s sircuit”, 1992
• [2] D. V. Sivukhin, “Electricity” 6th Edition, FITMAZLIT,
Moscow, 2019.
• [3] V.I. Arnold, “Ordinary Differential Equations”, 4 th edition,
Izhevsk, 2000
• [4] Y.S. Ilyashenko, “Attractors of Dynamic Systems”, 2008
• [5] A.S. Dmitriev, “Chaos generators”, Moscow
• [6] NIST, “A Statistical Test Suite for Random and
Pseudorandom Number Generators for Cryptographic
Applications ”, 2010
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23. Equilibrium positions

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24. NIST test example

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