Fundamental Limits of Covert Communications Covered by Randomly Activated Overt Users

This paper studies the fundamental limits of the amount of information that can be transmitted in covert communications covered by randomly activated overt users. First, an upper bound is developed with a novel recursive-iterative approximation on the intractable total variation distance (TVD) to measure the detection probability of the warden, which is tighter than the Kullback-Leibler divergence (KLD). On this basis, the collapse effect of the TVD is revealed, which shows that the TVD is strictly less than 1 if the covert user sets the transmit power to be an integer multiple of that of the overt users. Then, we find that up to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(N)$</tex> bits of information can be reliably and covertly transmitted over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> channel uses under the above power setting, which breaks the well-known square root law. If the above power setting is violated, still, the TVD will instantly approach 1 as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N\rightarrow \infty$</tex>, and only no more than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\sqrt{N})$</tex> bits can be covertly transmitted. To prove this, the detection method of the warden is modified to cope with the random activation of the users. The above conclusions also hold even if the transmit powers of the overt users are randomly distributed, which resembles realistic wireless transmissions.