# Covert Channel Based on Quasi-Orthogonal Coding

^{*}

## Abstract

**:**

## 1. Introduction

- (a)
- In [3], quadrature amplitude modulation (QAM) covert information constellation points are distributed around the QAM cover constellation points, forming the so-called dirty constellation. Thus, an additional constellation is formed based on the cover constellation.
- (b)
- (c)
- Hiding the data by moving the constellation points by a given angle (right or left) is shown in [1]. In this case, binary PSK (BPSK) modulation was used for practical implementation.
- (d)
- In [6], the use of 8 frequency-shift keying (8FSK) modulation to embed information in QAM constellation points is presented. In this case, artificial neural networks were used to extract hidden data.
- (e)
- The authors [7] noted that PSK modulations do not use the channel fully in terms of Shannon capacity. Therefore, the so-called residual capacity can be used to hide information. In order to hide this emission from potential observers, pseudo-noise asymmetric shift keying (PN-ASK) modulation is proposed.
- (f)
- (g)
- (h)
- The transmission of stealth information in the form of noise on a QAM basis is presented in [10]. In this case, the cover’s signal is not used to carry information.

## 2. Radio Physical Layer Steganography

#### 2.1. Creating a Covert Channel

- (a)
- the covert signal power should be significantly less than the cover signal (${P}_{1}\gg {P}_{2}$).
- (b)
- constellation points of the covert signal should have a pseudo-random (noise) characteristic.

#### 2.2. Channel Estimation Error

#### 2.3. Quasi-Orthogonality

## 3. Proposed Model

#### 3.1. Basic Assumptions

- At the transmitter, samples of the covert signal ${x}_{2}\left(t\right)$ are sorted based on a given sequence of the cover (QAM modulation) signal ${x}_{1}\left(t\right)$. In this way, the sorted signal ${x}_{{2}_{p}}$ has a pseudo-random (noise) form.
- At the receiver, the ${\widehat{x}}_{2}\left(t\right)$ signal re-sorting is performed after the SIC operation. Sorting in the receiver aims to:
- (a)
- restore the original sample order of the covert signal after the SIC operation ${\widehat{x}}_{{2}_{p}}$ to the original order (in an ideal case ${\widehat{x}}_{{2}_{i}}={x}_{2}$)
- (b)
- restoring the original order of ${\widehat{x}}_{{2}_{p}}$ is followed by a simultaneous change in the sample order of the residual signal associated with ${x}_{1}$ Thus, the residual signal becomes orthogonal (quasi-orthogonal) to the covert signal.
- (c)
- The covert signal ${\widehat{x}}_{{2}_{i}}$ is fed to the input of the covert channel demodulator

#### 3.2. Cover Signal Sorting

## 4. Simulation Tests

- (a)
- Averaged FFT spectrum of cover and covert signal (before and after sorting)
- (b)
- Averaged value and variance of cross-correlation of signals
- (c)
- Probability distribution of cross-correlation of signals estimated using histograms

#### 4.1. Simulation No. 1

#### 4.2. Simulation No. 2

#### 4.3. Simulation No. 3

#### 4.4. Simulation No. 4

#### 4.5. Simulation No. 5

#### 4.6. Simulation No. 6

#### 4.7. Simulation No. 7

## 5. Steganographic Analysis

## 6. Practical Implementation

## 7. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Transmission and reception of a covert signal as a superposition of cover and covert signals.

**Figure 3.**64QAM signal. (

**a**) Random and sorted (${x}_{1\_sorted},{f}_{0}$) 64QAM signal-real part of the signal, (

**b**) Random and sorted 64QAM signal-imaginary part of the signal (

**c**) Sorted 64QAM signal with frequencies of ${f}_{0}$ and 16∙${f}_{0}$ (d) Instantaneous FFT spectrum for 64 samples of random and sorted signals.

**Figure 5.**Testing the impact of cover orthogonalization on the non-ideal SIC process (

**a**) Averaged FFT spectrum of unsorted cover and 8FSK (

**b**) Averaged FFT spectrum of sorted cover and 8FSK (

**c**) Averaged cross-correlation value of unsorted cover and 8FSK (

**d**) Histogram of the cross-correlation value of the unsorted cover and 8FSK (

**e**) Histogram of the cross-correlation value of the sorted cover and 8FSK (

**f**) Reception of covert information (8FSK) for the unsorted and sorted cover (non-ideal SIC).

**Figure 6.**Testing the impact of cover orthogonalization on the non-ideal SIC process (

**a**) Averaged FFT spectrum of unsorted cover and 8FSK (

**b**) Averaged FFT spectrum of sorted cover and 8FSK (

**c**) Averaged cross-correlation value of unsorted cover and 8FSK (

**d**) Histogram of the cross-correlation value of the unsorted cover and 8FSK (

**e**) Histogram of the cross-correlation value of the sorted cover and 8FSK (

**f**) Reception of covert information (8FSK) for the unsorted and sorted cover (non-ideal SIC).

**Figure 8.**Testing the impact of cover orthogonalization on the non-ideal SIC process (

**a**) Averaged FFT spectrum of unsorted cover and 8FSK (

**b**) Averaged FFT spectrum of sorted cover and 8FSK (

**c**) Averaged cross-correlation value of unsorted cover and 8FSK (

**d**) Histogram of the cross-correlation value of the unsorted cover and 8FSK (

**e**) Histogram of the cross-correlation value of the sorted cover and 8FSK (

**f**) Reception of covert information (8FSK) for the unsorted and sorted cover (non-ideal SIC).

**Figure 9.**Use of 2-PSK modulation for covert transmission (

**a**) Averaged FFT spectrum of unsorted cover and 2PSK (

**b**) Averaged FFT spectrum of sorted cover and 2PSK (

**c**) Averaged cross-correlation value of unsorted cover and 2PSK (

**d**) Histogram of the cross-correlation value of the unsorted cover and 2PSK (

**e**) Histogram of the cross-correlation value of the sorted cover and 2PSK (

**f**) Reception of covert information (2PSK) for the unsorted and sorted cover (non-ideal SIC).

**Figure 10.**Use of 8-PSK modulation for covert transmission (

**a**) Averaged FFT spectrum of unsorted cover and 8PSK (

**b**) Averaged FFT spectrum of sorted cover and 8PSK (

**c**) Averaged cross-correlation value of unsorted cover and 8PSK (

**d**) Histogram of the cross-correlation value of the unsorted cover and 8PSK (

**e**) Histogram of the cross-correlation value of the sorted cover and 8PSK (

**f**) Reception of covert information (8PSK) for the unsorted and sorted cover (non-ideal SIC).

**Figure 11.**Use of 4-PSK modulation for covert transmission (

**a**) Averaged FFT spectrum of unsorted cover and 8PSK (

**b**) Averaged FFT spectrum of sorted cover and 4PSK (

**c**) Averaged cross-correlation value of unsorted cover and 4PSK (

**d**) Histogram of the cross-correlation value of the unsorted cover and 4PSK (

**e**) Histogram of the cross-correlation value of the sorted cover and 4PSK (

**f**) Reception of covert information (4PSK) for the unsorted and sorted cover (non-ideal SIC).

**Figure 12.**Use of 8-PSK modulation for covert transmission (

**a**) Averaged FFT spectrum of unsorted cover and 8PSK (

**b**) Averaged FFT spectrum of sorted cover and 8PSK (

**c**) Averaged cross-correlation value of unsorted cover and 8PSK (

**d**) Histogram of the cross-correlation value of the unsorted cover and 8PSK (

**e**) Histogram of the cross-correlation value of the sorted cover and 8PSK (

**f**) Reception of covert information (8PSK) for the unsorted and sorted cover (non-ideal SIC).

Covert Modulation | $\mathit{K}\mathit{S}\mathit{T}\mathit{E}\mathit{S}\mathit{T}$ | |
---|---|---|

SNR = 45 dB | SNR = 50 dB | |

2PSK | 0.153 | 0.321 |

4PSK | 0.087 | 0.214 |

8PSK | 0.082 | 0.188 |

2FSK | 0.081 | 0.186 |

4FSK | 0.081 | 0.186 |

8FSK | 0.080 | 0.185 |

Carrier Frequency | 850 MHz | |
---|---|---|

Cover (carrier) | Modulation | 16QAM |

Bandwidth | 8 MHz | |

Transmission rate | 32 Mb/s | |

Block length | 16 | |

Power | ${P}_{0}$ | |

Covert information | Modulation | 8FSK |

Number samples per symbol | $16$ | |

Number symbols in block | 1 | |

Transmission rate | 1.5 Mb/s | |

Power | $0.01\cdot {P}_{0}$ |

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**MDPI and ACS Style**

Grzesiak, K.; Piotrowski, Z.; Kelner, J.M.
Covert Channel Based on Quasi-Orthogonal Coding. *Electronics* **2023**, *12*, 2249.
https://doi.org/10.3390/electronics12102249

**AMA Style**

Grzesiak K, Piotrowski Z, Kelner JM.
Covert Channel Based on Quasi-Orthogonal Coding. *Electronics*. 2023; 12(10):2249.
https://doi.org/10.3390/electronics12102249

**Chicago/Turabian Style**

Grzesiak, Krystian, Zbigniew Piotrowski, and Jan M. Kelner.
2023. "Covert Channel Based on Quasi-Orthogonal Coding" *Electronics* 12, no. 10: 2249.
https://doi.org/10.3390/electronics12102249