# Design and Implementation of a Digital Dual Orthogonal Outputs Chaotic Oscillator

^{*}

## Abstract

**:**

## 1. Introduction

- ■
- jointly generates two orthogonal signals, i.e., the outputs are point by point in phase quadrature and constitute a unit vector.
- ■
- presents statistical characteristics, such as stationary mean and standard deviation, that are also independent of the initial conditions.
- ■
- is robust when one reduces the sample quantification format.

## 2. Usual Digital Chaotic Sequences

#### 2.1. Recurrence Equations

#### 2.2. Influence of the Data Format

#### 2.3. Statistical Variability

## 3. Proposed Orthogonal Chaotic Sequence

#### 3.1. Theoretical Background

_{k}and y

_{k}, given by the following:

#### 3.2. Normalized Recurrence Equations

#### 3.3. Approximation of Normalized Recurrence Equations

#### 3.4. Statistic Properties of the Chaotic Oscillator

#### 3.5. Chaotic Behavior Analysis

- ■
- λ < 0 corresponds to an orbital trajectory attracted by a stable fixed point. This situation is representative of strongly deterministic, harmonic or random sequences.
- ■
- λ very close to 0 represents a steady state close to a chaotic transition.
- ■
- λ > 0 characterizes an unstable and chaotic orbit. The larger λ, the more chaotic character is observed.

## 4. Hardware Implementation

_{s}obtained for three different calculation accuracies, namely, double precision floating point (IEEE754 64-bit), single precision floating point (IEEE754 32-bit) and the custom defined fixed-point format, with two bits for the integer part and 21 bits for the fractional part.

## 5. Application to the Fast Image Encryption

#### 5.1. Image Encryption/Decryption With Bitwise XOR Operation

- ■
- Elaboration of a sequence of N*M numerical coefficients of the chosen chaotic system in which the precise initial conditions have been introduced.
- ■
- Quantification of the coefficients in a format identical to that of the pixels of the original image (eight bits in most cases). Given the variability of the statistical characteristics of the sequence with the initial conditions, this operation can only be carried out after the computation of the entire sequence.
- ■
- Application of a bitwise XOR operation between the pixel values and the quantized coefficients.

#### 5.2. Simultaneous Images Mixing Encryption/Decryption

- Step 1:
- Read the two original images with size N*M, then convert them into one dimensional arrays U and V, and assign the initial conditions x
_{0}and y_{0}. - Step 2:
- For each pair (u
_{k}and v_{k}), compute the following:

- Step 3:
- Convert the two arrays U’ and V’ into two images with the size N*M.

- Step 1:
- Read the two encrypted images with size N*M, then convert them into one-dimensional arrays U’ and V’, and assign initial conditions x
_{0}and y_{0}. - Step 2:
- For each pair (u’
_{k}and v’_{k}), compute the following:

- Step 3:
- Convert the two arrays of U and V into two images with the size N*M.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Oscillator outputs for very close initial conditions (${\theta}_{0}=0.999$ in black and ${\theta}_{0}=1.0$ in red).

Dim. | Model | Recursive Equations | Chaos Conditions |
---|---|---|---|

1D | Logistic | ${x}_{k}=\mu {x}_{k-1}\left(1-{x}_{k-1}\right)$ | ${x}_{0}\in \left]0,1\right[$ $3.57\le \mu <4$ |

1D | Bernoulli | ${x}_{k}=\frac{2}{1-\alpha}\left({x}_{k-1}+1\right)-1\hspace{1em}-1{x}_{k-1}\alpha $ ${x}_{k}=\frac{2}{1-\alpha}\left({x}_{k-1}-1\right)+1\hspace{1em}\hspace{1em}\alpha <{x}_{k-1}<1$ | $-1<\alpha <1$ |

2D | Hénon | ${x}_{k}={y}_{k-1}+1-a{x}_{k-1}^{2}$ ${y}_{k}=b{x}_{k-1}$ | a = 1.4 b = 0.3 |

3D | Lorenz discrete (Euler approx. dt = h) | ${x}_{k}={x}_{k-1}-\sigma h({x}_{k-1}-{y}_{k-1})$ ${y}_{k}={y}_{k-1}+h(\rho {x}_{k-1}-{y}_{k-1}-{x}_{k-1}{z}_{k-1})$ ${z}_{k}={z}_{k-1}+h({x}_{k-1}{z}_{k-1}-\beta {z}_{k-1})$ | $\sigma =10$$\rho =28$ $\beta =8/3$h < 0.025 |

**Table 2.**Mean (m) and standard deviation (std) of the discrete Lorenz sequence for various initial conditions.

x_{0}; y_{0}; z_{0} | 0.1; 0.1; 0.1 | 0.5; 0.5; 0.5 | 1; 1; 1 | 2; 2; 2 |
---|---|---|---|---|

m | −2.19 | −0.64 | −1.00 | −1.44 |

std | 7.72 | 8.00 | 7,94 | 7.86 |

Case | Initial Magnitude | Convergence | Behavior | Examples |
---|---|---|---|---|

1 | ${r}_{0}<1/\phi $ | ${r}_{k}\stackrel{k\to \infty}{\to}0$ | Chaotic Transient | ${x}_{0}=0.13$${y}_{0}=0.6$${r}_{0}=0.614$ |

2 | $0.618<{r}_{0}<1.272$ | ${r}_{k}\stackrel{k\to \infty}{\to}1$ | Chaotic steady state | ${x}_{0}=0.6$${y}_{0}=0.6$${r}_{0}=0.848$ |

3 | $1.272<{r}_{0}<\phi $ | ${r}_{k}\stackrel{k\to \infty}{\to}0$ or ${r}_{k}\stackrel{k\to \infty}{\to}1$ | Chaotic Transient or Steady state | ${r}_{0}=1.44$ ➔ transient ${r}_{0}=1.58$ ➔ steady state |

4 | ${r}_{0}>\phi $ | ${r}_{k}\stackrel{k\to \infty}{\to}\infty $ | Divergence |

Logistic | Hénon | Proposed Chaotic Oscillator | |
---|---|---|---|

μ = 3.8 ${x}_{0}$ = 0.2 | a =1.4; b = 0.3 ${x}_{0}$ = 1; ${y}_{0}=0$ | ${\theta}_{0}=0.01$ | ${x}_{0}=0.5$ ${y}_{0}=0.8$ |

λ_{x} = 1.29 | λ_{x} = 0.89λ _{y} = 1.21 | λ_{x} = 2.14λ _{y} = 0.30 | λ_{x} = 1.63λ _{y} = 0.68 |

Data Format | LUT | FF | DSP | Dyn. Power | f_{s} max |
---|---|---|---|---|---|

Float 64 | 2999 | 128 | 40 | 35 mW | 33 MHz |

Float 32 | 1453 | 64 | 17 | 20 mW | 50 MHz |

Fix 23_21 | 33 | 32 | 6 | 6 mW | 100 MHz |

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**MDPI and ACS Style**

Berviller, Y.; Tisserand, E.; Poure, P.; Rabah, H.
Design and Implementation of a Digital Dual Orthogonal Outputs Chaotic Oscillator. *Electronics* **2020**, *9*, 264.
https://doi.org/10.3390/electronics9020264

**AMA Style**

Berviller Y, Tisserand E, Poure P, Rabah H.
Design and Implementation of a Digital Dual Orthogonal Outputs Chaotic Oscillator. *Electronics*. 2020; 9(2):264.
https://doi.org/10.3390/electronics9020264

**Chicago/Turabian Style**

Berviller, Yves, Etienne Tisserand, Philippe Poure, and Hassan Rabah.
2020. "Design and Implementation of a Digital Dual Orthogonal Outputs Chaotic Oscillator" *Electronics* 9, no. 2: 264.
https://doi.org/10.3390/electronics9020264