Abstract
Covert communication is critical to guarantee a strong security and secure user privacy. In this work, we consider adversary’s noise and channel uncertainties and analyze their impact on adversary’s optimum detection performance and the throughput of covert messages. We determine the throughput of covert messages and its gain and loss by having adversary’s channel uncertainty and fading channel, respectively. The results show that fading is essential to hide information, particularly for low noise uncertainty or at high SNR. The improvement of adversary’s optimum detection performance, hence the covert throughput, under channel uncertainty is more significant for larger noise uncertainty. The covert throughput is gained by having channel uncertainty roughly 12–19% when the noise uncertainty is about 1–2 dB.
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Notes
Hiding the covert information under existing transmissions in [9] will not be considered in this paper.
\(E\{\cdot \}\) notates the expectation operator and \(x \sim \mathcal {CN}(0,\nu )\) indicates a complex Gaussian with mean 0 and variance \(\nu \).
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Acknowledgements
Khuong Ho-Van would like to thank Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for the support of time and facilities for this study.
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A part of the current paper was shown in [1]
Appendices
Appendix A
Equation (12) is derived in this “Appendix”:
Appendix B
The pseudo-convexity of \(\xi (\hat{g})\) is proved in this “Appendix”. It follows from [28] that a function f(x) is strictly pseudo-convex if, for any value of \(x=x_0\) such that \(d f(x)/d x =0\), we have \(d^2 f(x)/d x^2 > 0\).
We have
and
respectively. Then, for any \(\lambda =\lambda _0\) such that
it is inferred from (50) that
Since \(I_0(z) = \sum ^\infty _{k=0} (z/2)^{2k}/(k!)^2\), we obtain from (10),
Due to the log-concavity of the non-central Chi-square [29], i.e. \((\ln (f_{X| \hat{g}}(x|\hat{g})))'' < 0\), (54) decreases with increasing x and hence, we obtain
which, since \(e^{-t} \ge 1-t\), yields \(f_{X|\hat{g}}(0|\hat{g}) \ge 0\). Hence, we obtain from (53) that \(d^2 \xi /d \lambda ^2|_{\lambda =\lambda _0} > 0\), yielding the pseudo-convexity of \(\xi (\hat{g})\).
Appendix C
In this “Appendix”, we derive (25). We have
which applying integration by part into (56) yields
Appendix D
We firstly prove \(\xi _{\min } < 0.9\) for \(1.0002 \le \alpha \le 3.16\) if \(\lambda ^\perp \ge \alpha \hat{\sigma }_w^2\). Then, we can conclude that for \(\xi _{\min } \ge 0.9\) and \(1.0002 \le \alpha \le 3.16\), we obtain \(\lambda ^\perp < \alpha \hat{\sigma }_w^2\).
Since the left hand side (LHS) of (26) increases with \(\lambda \) and is 0 when \(\lambda =\lambda ^\perp \), then
for \(\lambda ^\perp \ge \alpha \hat{\sigma }_w^2\). In (58), we applied \(\int ^b_a e^x dx /x^2 \ge (e^b-e^a)/b^2\). From (58), we obtain
which, from [26], yields
or equivalently,
Applying the inequality \(\int ^b_a e^x dx /x \ge (e^b-e^a)/b\), we have
which, from (28), yielding for \(\lambda ^\perp \ge \alpha \hat{\sigma }_w^2\),
which is derived from (61) and the fact that \(\sigma _g^2 \gamma \left( 1 - e^{-\frac{\alpha -\alpha ^{-1}}{\sigma _g^2 \gamma }} \right) \) is an increasing function of \(\sigma _g^2 \gamma \). Moreover, we obtain
for \(1.0002 \le \alpha \le 3.16\). Accordingly, \(\xi _{\min } < 0.9\) for \(1.0002 \le \alpha \le 3.16\).
Appendix E
Equation (40) is derived in this “Appendix”. Since the LHS of (26) increases with \(\lambda \), it follows from (26) and (38) that
for \(\lambda ^\perp \ge \sigma _g^2 P/[2 \epsilon \ln (\alpha )]\). Since \(\int e^z z^{-1} dt \simeq e^z ( z^{-1} + z^{-2}) \) for \(z \gg 1\) [25], we can approximate the LHS of (65) as
for \(\sigma _g^2 \gamma \ll 1\). Therefore, it follows from (65) and (66) that
yielding (40).
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Ta, H.Q., Pham, QV., Ho-Van, K. et al. Covert communication with noise and channel uncertainties. Wireless Netw 28, 161–172 (2022). https://doi.org/10.1007/s11276-021-02828-3
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DOI: https://doi.org/10.1007/s11276-021-02828-3